NCERT Solutions Class 8 Maths Chapter 8 Fractions in Disguise

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 8 · M AT H S

NCERT Solutions

Chapter 8: Fractions in Disguise

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

Part II, 1 – 31 20 84 English

Solutions, notes, sample papers & more at 72 pages

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

CLASS 8 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 8: Fractions in Disguise
The opening chapter of Ganita Prakash Part II shows that a percentage is nothing new — it is a fraction
wearing a denominator of 100. From that one idea the chapter builds comparison of proportions, percentage
increase and decrease, profit, loss, discount, GST, compound growth and depreciation.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 8) Part II, 1 – 31

SECTIONS QUESTIONS

20 84

MEDIUM

English

In-text Questions — Page 2
Section 1.1 Fractions as Percentages

Q1 Can you tell what percentage of the colour was made using yellow?

25%.
The mixture is made of red paint and yellow paint only, so the two shares must together make
up the whole mixture.

Red = 3/4 of the mixture = 75%

Whole mixture = 1 = 100%

Yellow = 1 – 3/4 = 1/4

1/4 = 1 × 25 / 4 × 25 = 25/100 = 25%

Why it happens: A percentage is a share of one fixed whole. Here the whole is the
deep orange paint, and it has only two parts. So the two percentages must add to
100 — you can get the second one either by subtracting the fraction (1 – 3/4) or by
subtracting the percentage (100% – 75%). Both give the same answer because 100%
is the whole.

Page 1 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

In-text Questions — Page 3
Section 1.1 Fractions as Percentages

Q1 Try completing Method 3 by filling the boxes.

1 2 3 4 5
0 =1
5 5 5 5 5

Total prize
Savings for canvas
money

0% 100%

Method 3, page 3 — the bar is marked 0, 1⁄5, …, 5⁄5 = 1 along the top and 0% … 100% along
the bottom. The four empty boxes are the ones to fill.

The boxes are 20%, 40%, 60% and 80%, and the savings for the canvas is 40% of the prize
money.

FRACTION OF PRIZE MONEY 0 1/5 2/5 3/5 4/5 5/5 = 1

PERCENTAGE 0% 20% 40% 60% 80% 100%

Surya saves 2/5 of the money, and the shaded band “Savings for canvas” reaches the mark 2/5.
So the savings are 40%.

Why it happens: The whole bar is 100%. Cutting it into 5 equal parts cuts 100% into
5 equal parts too, so each part is 100 ÷ 5 = 20%. That is exactly what the owl is
pointing out: finding what percentage 2/5 is, is the same as finding 2/5 th of 100.

Tip: This gives you a fast mental rule. Fifths jump in 20s (20, 40, 60, 80, 100), quarters
jump in 25s, tenths jump in 10s, eighths jump in 12.5s.

Figure it Out — Pages 3–4

Page 2 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Section 1.1 Fractions as Percentages

Q1 Express the following fractions as percentages. (i) 3/5 (ii) 7/14 (iii) 9/20 (iv) 72/150 (v)
1/3 (vi) 5/11

A fraction is of one unit and a percentage is per 100 units, so multiply each fraction by 100.

(i) 3/5 × 100 = 300/5 = 60%

(ii) 7/14 = 1/2, and 1/2 × 100 = 50%

(iii) 9/20 × 100 = 900/20 = 45%

(iv) 72/150 = 12/25, and 12/25 × 100 = 1200/25 = 48%

(v) 1/3 × 100 = 100/3 = 33.33% (exactly 33⅓%)

(vi) 5/11 × 100 = 500/11 = 45.45% (45.4545… %)

Why it happens: Writing 3/5 as a percentage means finding the equivalent fraction
with 100 in the denominator: 3/5 = 60/100. Multiplying by 100 does exactly that in
one step, because (3/5) × 100 is the numerator that sits over 100.

Check it yourself: (v) and (vi) do not stop. 1/3 and 5/11 have denominators with
prime factors other than 2 and 5, so no equivalent fraction of theirs can have
denominator 100 — their percentages are non-terminating decimals, and we round.

Q2 Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are
white? (i) 10% (ii) 15% (iii) 25% (iv) 60% (v) 40% (vi) None of these

(iv) 60%.

White marbles as a fraction of all marbles = 15/25

15/25 = 15 × 4 / 25 × 4 = 60/100

= 60%

Page 3 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
m.
Why it happens: Option (ii) 15% is the trap. 15 is the count of white marbles, not the

m l a se
percentage — it would only be 15% if the total were 100. The whole here is 25, so
o g 15 × 4 = 60%.
.c is worth 4% of the collection, and 15 marbles are worth
a
m
each marble

l a se
ag

co m
ag
In a school, 15 of the 80 students come to school by walking. What percentage of
.
Q3

e m
as
the students come by walking?

a g l

co m
em.
as
Fraction who walk = 15/80 = 3/16

. com= 3/16 × 100 = 300/16 a g l
em
Percentage
a s
a gl = 18.75%

m a s
m .co
Tip: A quick estimate first: 15 out of 80 is a little less than 20 out of 80, and 20 out of
agl
l a se
80 is exactly a quarter, i.e. 25%. So the answer must be a bit under 25% — 18.75%
fits. a g

co m
m .
m ase
Q4
.co a g l
A group of friends is participating in a long-distance run. The positions of each of

a s em them after 15 minutes are shown in the following picture. Match (among the given

ag l options) what percentage of the race each of them has approximately completed.

se m
com g l a
m . a
as e
agl
A B C D

m
.co
Start Finish

a s em84%
com gl
55% 20% 38% 72% 93%
m . a
ase
agl Page 4 — where A, B, C and D had reached after 15 minutes, with the six options printed
c
m .
e
below the track.

m a s
e m . co agl
g l as
a
Reading each runner's position as a fraction of the whole Start-to-Finish line:

co m
m .
m as e
.co


a g l Page 4 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

RUNNER POSITION ON THE LINE MATCHING OPTION

A a little over a third of the way 38%

B just past the halfway mark 55%

C about three-quarters of the way 72%

D almost at the Finish 93%

A B C D

Start Finish
38% 55% 72% 93%

The four runners placed on the Start-to-Finish line, with the option each one matches.

The options 20% and 84% are not used — no runner is that close to the Start, and none sits
between C and D.

Why it happens: The whole race is the whole line, so “percentage completed” is just
the runner's distance from Start divided by the total length, times 100. Because the
options are far apart, you can match them by eye: halfway is 50%, quarter-way is
25%, and so on.

Q5 Pairs of quantities are shown below. Identify and write appropriate symbols ‘>’, ‘<’,
‘=’ in the blanks. Try to do it without calculations. (i) 50% ____ 5% (ii) 5/10 ____ 50% (iii)
3/11 _____ 61% (iv) 30% ____ 1/3

(i) 50% > 5% — 50 hundredths against 5 hundredths

(ii) 5/10 = 50% — 5/10 = 50/100
(iii) 3/11 < 61% — 3/11 is under 3/10, i.e. under 30%

(iv) 30% < 1/3 — 1/3 = 33⅓%

Page 5 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Why it happens: None of these needs long division. (iii): 3/11 is smaller than 3/10
because a bigger denominator cuts the whole into smaller pieces, and 3/10 is only
30% — far below 61%. (iv): a third of 100 is 33⅓, and 30 is less than that. Comparing
sizes is often faster than computing values.

In-text Questions — Page 4
Section 1.1 Fractions as Percentages

Q1 Well, if percentages are just a particular type of fraction, why do we need them?
Why can’t we just continue using fractions?

Because fractions with different denominators cannot be compared at a glance, while
percentages always share the denominator 100.
Take the book's own example — sugar makes up 9/34 of Variety 1 and 13/45 of Variety 2.

9/34 × 100 = 26.47%

13/45 × 100 = 28.88%

So Variety 2 is the more sugary one.

Why it happens: To compare 9/34 with 13/45 as fractions you must first bring them
to a common denominator — 34 × 45 = 1530 — and compare 405/1530 with
442/1530. A percentage does that same job once and for all by fixing the common
denominator at 100 in advance. That is the whole point: percentages are fractions
already prepared for comparison.

Q2 If we want to have the same denominator, why choose 100 in particular? Why not
10, 50, 1000, or 43? Think.

Any denominator would work in principle. 100 is chosen because it balances two things a
comparison needs — fine enough detail and easy mental handling.

Page 6 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

It fits our number system. Our numerals are base ten, so 10, 100 and 1000 turn straight
into decimals: 31% = 31/100 = 0.31. Moving between fraction, decimal and percentage costs
nothing.
Per 10 is too coarse. 9/34 would be 2.647 “per decem” — you need decimals before you
have even started.
Per 1000 is too heavy. 9/34 becomes 264.7 per mille — accurate, but hard to picture.
43 is worse than useless. Nothing in our notation makes 43 easy to divide by or to imagine.

Why it happens: A whole number out of 100 already carries two-digit accuracy,
which is enough for marks, discounts, taxes and shares of a population — and 100 is
small enough that “37 out of 100” is a picture you can hold in your head. Larger
bases are used where finer detail matters: rates of disease are quoted per lakh
precisely because the numbers per 100 would all round to 0.

In-text Questions — Page 7

Page 7 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Section 1.2 Percentage of Some Quantity — Free-hand Computations

MATH TALK

Q1 Try to calculate (without using pen and paper) the indicated percentages of the
values shown in the table below. Write your answers in the table.

100 200 50 80 10 35 287

25% 25

10%

20%

5%

The table printed on page 7. The book has filled in only one entry — 25% of 100.

100 200 50 80 10 35 287

25% 25 50 12.5 20 2.5 8.75 71.75

10% 10 20 5 8 1 3.5 28.7

20% 20 40 10 16 2 7 57.4

5% 5 10 2.5 4 0.5 1.75 14.35

None of these needs written work if you use the friendly fractions behind the percentages:

10% of a number = the number ÷ 10 (shift the decimal point one place left)

20% = double the 10% value

5% = half the 10% value

25% = one quarter, i.e. halve, then halve again

So for 287: 10% is 28.7 → 20% is 57.4 → 5% is 14.35 → 25% is 57.4 + 14.35 = 71.75.

Page 8 of 72

ase
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
m.
Why it happens: 10% is 10/100 = 1/10, and dividing by 10 in a base-ten system is

m l a se
only a shift of the decimal point. Every other percentage in this table is built from
o
m .c step by doubling, halving or adding — which is whyagthe table can be
that one easy
a e
smentally.
ag l
filled

o m
e
. c
m the methods with the class. Do you find ag
s
How did you find these values? Discuss
a
Q2

agl
anything interesting in the table?

co m 287
m.
100 200 50 80 10 35

m as e
.co
25% 25
a g l
se m
g l a 10%
a
s
20%

m a
5%
em
.co agl
a s
agl page 7 — the same one you have just filled in.
The table from

co m
m .
m run through the table: as e
l

. c o a g
e m
Several
s
patterns

agla The 20% row is exactly double the 10% row, in every column.
m
The 5% row is exactly half the 10% row.

a se
com g l
20% row + 5% row = 25% row, column by column (40 + 10 = 50, 16 + 4 = 20, 7 + 1.75 = 8.75,

m . a
e
…).

g l asvalues fall in the same order as the percentages — because
Reading down any column, the
a
the base is the same in that column.

co m
m .
e
(20/100 × y) + (5/100 × y) = (25/100 × y)
m l as
m .co a g
l a se
ag Why it happens: All four rows are the same number y multiplied by different
hundredths. So the row values are proportional to the percentages themselves:
.c
s e m
m a
doubling the percentage doubles the value, and adding two percentages adds their

m . co
values. In algebra this is just the distributive law — (20/100)y + (5/100)y = (20 + 5)/100
e agl
× y.
g l as
a

co m
m .
m ase
.co


a g l Page 9 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Careful: This adding rule works only while the base y stays the same. 20% of one
quantity plus 5% of a different quantity is not 25% of anything.

Q3 Using this understanding, mentally calculate how much 40% of the values in the
table above would be.

100 200 50 80 10 35 287

25% 25

10%

20%

5%

The table on page 7. The values along the top row are the ones to take 40% of.

40% is double 20%, which is itself double 10%. So take a tenth and double it twice.

100 200 50 80 10 35 287

10% 10 20 5 8 1 3.5 28.7

40% 40 80 20 32 4 14 114.8

40% of 287 → 10% is 28.7 → 20% is 57.4 → 40% is 114.8

Check it yourself: 40% is also 2/5. And 2/5 of 287 = 574/5 = 114.8. Same answer,
different route.

In-text Questions — Page 8

Page 10 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Section 1.2 — The FDP Trio: Fractions, Decimals, and Percentages

MATH TALK

Q1 Using this observation, mentally calculate how much 15% of the values in the table
would be.

100 200 50 80 10 35 287

25% 25

10%

20%

5%

The table from page 7. The values along the top row are the ones to take 15% of.

15% = 10% + 5%. Take a tenth, then add half of it.

100 200 50 80 10 35 287

10% 10 20 5 8 1 3.5 28.7

5% 5 10 2.5 4 0.5 1.75 14.35

15% 15 30 7.5 12 1.5 5.25 43.05

15% of 287 = 28.7 + 14.35 = 43.05

Q2 Suppose you have to mentally calculate the following percentages of some value:
75%, 90%, 70%, 55%. How would you do it? Discuss.

Break each one into the two easy pieces, 10% and 25%, and use adding or subtracting from the
whole.

75% = 100% – 25%. Or three quarters: halve, halve again, take three of those parts.
90% = 100% – 10%. Take a tenth away from the whole.

Page 11 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

70% = 7 × 10%. Or 100% – 30%.
55% = 50% + 5%. Half, plus a twentieth.

Try it on 240:

75% → 240 – 60 = 180

90% → 240 – 24 = 216

70% → 7 × 24 = 168

55% → 120 + 12 = 132

Why it happens: Subtracting from 100% is legitimate because the part and the rest
always add to the whole: (100 – x)% of y = y – (x% of y). For percentages near 100,
taking the small piece away is far less work than building the large piece up.

Q3 Similarly, to find 10% of a quantity, what decimal value should be multiplied?

0.1

10% = 10/100 = 1/10 = 0.1

So 10% of 350 = 0.1 × 350 = 35

Why it happens: Every percentage has one decimal twin, found by dividing by 100
— that is, shifting the decimal point two places left. 50% → 0.5, 10% → 0.1, 7% →
0.07, 125% → 1.25. Multiplying by that decimal and taking that percentage are the
same operation written two ways.

Page 12 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q4 Complete the following table:

Per cent 50% 100% 25% 75% 10% 1% 5% 43%

Fraction 50⁄
100

Decimal 0.5

The table printed on page 8. The first column is filled in as an example.

PER CENT 50% 100% 25% 75% 10% 1% 5% 43%

FRACTION 50/100 = 100/100 25/100 = 75/100 = 10/100 = 1/100 5/100 = 43/100
1/2 =1 1/4 3/4 1/10 1/20

DECIMAL 0.5 1.0 0.25 0.75 0.1 0.01 0.05 0.43

Why it happens: The middle row is written straight from the definition x% = x/100;
the bottom row is the same fraction carried out as a division. Nothing is converted
twice — the three rows are three notations for one number. That is why 100% is 1:
the whole thing.

Tip: 43/100 cannot be simplified, because 43 is prime and does not divide 100. That
is fine — a percentage does not have to reduce.

Q5 Activity: How Close Can You Get? Make a pair. Each of you choose a number.
Suppose, the numbers chosen are a and b. Share your numbers with each other.
Both of you should estimate the percentage equivalent to the fraction a/b (where a
< b) and announce your answers by a fixed time, say, 5 seconds. The one whose
estimate is the closest wins this round. Play this for 10 rounds.

The skill this game trains is bracketing — trapping the answer between two percentages you
already know, instead of dividing.
Suppose the pair chooses a = 7, b = 24.

Page 13 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
e m.
Half of 24 is 12, and 7 < 12, so 7/24 < 50%
m l as
.co
A quarter of 24 is 6, and 7 > 6, so 7/24 > 25%
m a g
l a se
g
A third of 24 is 8, and 7 < 8, so 7/24 < 33.3%
aSo the answer lies between 25% and 33%, nearer the top → estimate about 29%

co m
. ag
(the exact value is 700/24 = 29.17%)
e m
g l as
a
Why it happens: Landmarks like 1/2, 1/3, 1/4 and 1/10 of the denominator are quick
to compute, and each one you check cuts the range of possible answers. Two or
co m
three checks narrow the estimate to within a couple of per cent — enough to win a
em.
m l as
.co g
five-second round.
m a
l a se
a g Tip: Keep score over the 10 rounds and, afterwards, work out the exact percentages.

s
Compare how far off each estimate was; you will find your estimates tighten as you
m a
.co agl
play.

se m
g l a
a
In-text Questions — Page 11
co m
m .
as e
Section 1.2 — Percentages Greater than 100

m l
.co a g
a s em On Days 5 and 6 his sales were ₹7800 and ₹9550 respectively. Calculate the
gl
Q1

a percentage of the target achieved on these days.

se m
com g l a
. a

m
ase
Kishanlal's daily target is ₹5000, so the target is the base — the 100%.

agl
m
Day 5: 7800/5000 × 100 = 78/50 × 100 = 156%

. co
Day 6: 9550/5000 × 100 = 955/500 × 100 = 191%
em
m l as
.co
mAnother way to see the same thing: a g
l a se
ag
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 14 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Day 5: 7800 = 5000 + 2800, and 2800/5000 = 56%

so 100% + 56% = 156%

Day 6: 9550 = 5000 + 4550, and 4550/5000 = 91%

so 100% + 91% = 191%

Why it happens: A percentage above 100 simply means the amount is larger than
the base it is compared with. There is nothing unusual about it — 156% of the target
is the same as 1.56 times the target. He exceeded the target by 56% on Day 5 and by
91% on Day 6.

In-text Questions — Page 12
Section 1.2 — Percentages Greater than 100

Q1 On Day 7, he achieved 150% of his target. On Day 8, he achieved 210% of his target.
Find the sales made on these days.

This reverses the earlier question: the percentage is known, the sales amount is not.

Day 7: 150% of 5000 = 150/100 × 5000 = 1.5 × 5000 = ₹7500

Day 8: 210% of 5000 = 210/100 × 5000 = 2.1 × 5000 = ₹10,500

Mentally: 100% is ₹5000 and 10% is ₹500, so 150% = 5000 + 2500 = 7500, and 210% = 10,000 +
500 = 10,500.

Why it happens: Reading “150% of the target” as “1.5 times the target” turns the
percentage question into a plain multiplication. That is the most useful habit in this
whole chapter: r% of a quantity is (r ÷ 100) times that quantity, whether r is 15, 150
or 1500.

Page 15 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q2 Complete the table below. Mark the approximate locations in the following
diagram.

Percent 90% 110% 200% 250% 15% 173% 358% 28.9% 305%

Fraction

Decimal

The table printed on page 12.

90%

0% 100% 200% 300% 400%
(0) (1) (2) (3) (4)

Page 12 — the diagram to mark on. The solid block is 0% to 100%; the dashed blocks
carry on to 400%. The arrow shows where 90% has been marked as an example.

PER CENT 90% 110% 200% 250% 15% 173% 358% 28.9% 305%

FRACTION 90/100 110/100 200/100 250/100 15/100 173/100 358/100 289/1000 305/100
= 9/10 = 11/10 =2 = 5/2 = 3/20 = = 61/20
179/50

DECIMAL 0.9 1.1 2.0 2.5 0.15 1.73 3.58 0.289 3.05

On the diagram the marks sit at the decimal values, since 100% is the point 1:

Page 16 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

15% just after the start | 28.9% about a third of the way to (1)

90% just before (1) | 110% just after (1)

173% between (1) and (2), nearer (2) | 200% exactly at (2)

250% halfway between (2) and (3) | 305% just after (3)

358% between (3) and (4), nearer (4)

Why it happens: The diagram's two scales — 0%…400% and (0)…(4) — are the same
scale written twice, because a percentage divided by 100 is its position on the
number line. 28.9% needs a denominator of 1000 as a fraction (28.9/100 = 289/1000)
only because its numerator is not a whole number; as a decimal it stays simple,
0.289.

Figure it Out — Pages 12–14
Section 1.2 Percentage of Some Quantity

MATH TALK

Q1 Estimate first before making any computations to solve the following questions. Try
different methods including mental computations. Find the missing numbers. The
first problem has been worked out. [(i) a bar of 5 equal parts, one part marked 20%;
the second bar of 5 parts totals 75 with 4 parts marked 60. (ii) bars of 10 equal parts,
one part marked ?; the second bar totals 90 with 6 parts marked ?. (iii) bars of 4
equal parts, one part marked ?; the second bar totals 140 with 3 parts marked ?.]

In each pair, the left bar tells you what one part is worth as a percentage, and the right bar uses
that to find a value.

PARTS IN THE BAR ONE PART SHADED PORTION TOTAL MISSING VALUE

(i) 5 20% 4 parts = 80% 75 60 (worked out)

(ii) 10 10% 6 parts = 60% 90 54

(iii) 4 25% 3 parts = 75% 140 105

Page 17 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

(ii) One part = 100% ÷ 10 = 10%

Shaded 6 parts = 60% of 90 = 0.6 × 90 = 54

(iii) One part = 100% ÷ 4 = 25%

Shaded 3 parts = 75% of 140 = 3/4 × 140 = 105

Why it happens: The bar is one whole, so cutting it into n equal parts makes each
part 100/n per cent — no matter what number the bar stands for. That is why the
same picture works for 75 in (i) and for 140 in (iii): the percentages depend only on
how the bar is cut, and the values depend on what the whole bar is worth.

Check it yourself: In (iii), 105 out of 140 should come back to 75%. 105/140 = 3/4 =
75%. It does.

Q2 Find the value of the following and also draw their bar models. (i) 25% of 160 (ii) 16%
of 250 (iii) 62% of 360 (iv) 140% of 40 (v) 1% of 1 hour (vi) 7% of 10 kg

(i) 25% of 160 = 1/4 × 160 = 40

(ii) 16% of 250 = 0.16 × 250 = 40

(iii) 62% of 360 = 0.62 × 360 = 223.2

(iv) 140% of 40 = 1.4 × 40 = 56

(v) 1% of 1 hour = 0.01 × 60 min = 0.6 min = 36 seconds

(vi) 7% of 10 kg = 0.07 × 10 kg = 0.7 kg = 700 g

Page 18 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
e m.
as
(i) 25% of 160

. com a g l
sem
40
a
agl0% 25% 100% = 160

m
.co
(iv) 140% of 40

ag
140% = 56

sem
gla
100% = 40 +40% = 16

a
Two bar models: a percentage below 100 shades part of the bar; a percentage above 100 runs past
the end of it.
co m
em.
m l as
.co a g
For the rest, draw a bar for the whole and shade the part: 16% of 250 shades roughly a sixth of

a s em
the bar; 62% of 360 shades a little under two-thirds; 1% of an hour is a sliver; 7% of 10 kg is a

a l strip near the start.
gthin
m a s
co250. agl
Why it happens: (i) and (ii) both come to 40 even though the wholes differ — 40 is a
quarter of 160 but only about a sixth.of
e m This is the chapter's central warning: a

g l asyou say of what. And (iv) shows why bar models
must be allowed to extendapast 100% — the answer, 56, is larger than the whole, 40.
percentage is meaningless until

co m
m .
m as e
Q3 .co
a g l
em made up 3/4 of the deep orange paint?
Surya made 60 ml of deep orange paint, how much red paint did he use if red paint

l a s
ag
se m
com g l a
m. a
ase
agl
Red = 3/4 of 60 ml

= 3 × 60 / 4 = 180/4

co m
.
= 45 ml

se m
o m l a
gearlier.
m .c remaining 15 ml is yellow — which matches the 25% found
The
a
a se
agl c
.
Check it yourself: 45 ml out of 60 ml is 45/60 = 3/4 = 75%, and 15/60 = 25%. The two
shares add to 60 ml and to 100%.
s e m
m a
em . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 19 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q4 Pairs of quantities are shown below. Identify and write appropriate symbols ‘>’, ‘<’,
‘=’ in the boxes. Visualising or estimating can help. Compute only if necessary or for
verification. (i) 50% of 510 __ 50% of 515 (ii) 37% of 148 __ 73% of 148 (iii) 29% of 43 __
92% of 110 (iv) 30% of 40 __ 40% of 50 (v) 45% of 200 __ 10% of 490 (vi) 30% of 80 __
24% of 64

(i) 50% of 510 < 50% of 515 — same percentage, bigger base

(ii) 37% of 148 < 73% of 148 — same base, bigger percentage

(iii) 29% of 43 < 92% of 110 — smaller share of a smaller base

(iv) 30% of 40 = 12 < 40% of 50 = 20

(v) 45% of 200 = 90 > 10% of 490 = 49

(vi) 30% of 80 = 24 > 24% of 64 = 15.36

Why it happens: (i) and (ii) need no arithmetic at all, because only one of the two
factors changes. (iii) needs none either — 29% of 43 is under half of 43, so under 22,
while 92% of 110 is nearly all of 110. (v) and (vi) are worth a rough check: in (v) 45%
of 200 is nearly half of 200, while 10% of 490 is only 49; in (vi) both the percentage
and the base are larger on the left, so the left must win.

Tip: A comparison only becomes a calculation when the percentage rises while the
base falls (or the reverse). Otherwise the answer can be seen.

Q5 Fill in the blanks appropriately: (i) 30% of k is 70, 60% of k is _____, 90% of k is _____,
120% of k is ______. (ii) 100% of m is 215, 10% of m is _____, 1% of m is ______, 6% of m is
______. (iii) 90% of n is 270, 9% of n is ______, 18% of n is _____, 100% of n is ______. (iv)
Make 2 more such questions and challenge your peers.

None of these needs you to find k, m or n first — the percentages themselves are in simple
ratios.

Page 20 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

(i) 60% is twice 30% → 140

90% is three times 30% → 210

120% is four times 30% → 280

(ii) 10% is one tenth of 100% → 21.5

1% is one hundredth of 100% → 2.15

6% = 6 × 1% → 12.9

(iii) 9% is one tenth of 90% → 27

18% is twice 9% → 54

100% = 90% + 10%, and 10% = 30 → 300

(iv) Two questions of the same kind:

40% of t is 96. Find 10% of t, 25% of t and 150% of t. (24, 60, 360)
75% of w is 45. Find 25% of w, 100% of w and 5% of w. (15, 60, 3)

Why it happens: Every one of these values is the same unknown multiplied by a
hundredth, so the values are proportional to the percentages. Double the percentage
and the value doubles; take a tenth of the percentage and the value becomes a
tenth. Solving for k (which is 700/3, not a whole number) would only make the
arithmetic worse.

Q6 Fill in the blanks: (i) 3 is ____ % of 300. (ii) _____ is 40% of 4. (iii) 40 is 80% of _____.

(i) 3/300 × 100 = 1 → 3 is 1% of 300

(ii) 40% of 4 = 0.4 × 4 = 1.6

(iii) 80% of x = 40 → x = 40/0.8 = 50

Page 21 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Why it happens: Each blank sits in a different place of the same relationship, part =
(per cent ÷ 100) × whole. In (i) the part and the whole are known; in (ii) the per cent
and the whole; in (iii) the part and the per cent. Knowing any two always gives the
third.

Careful: In (iii) do not take 80% of 40. That would give 32 — a number smaller than
40, when the whole must be larger than 40, since 40 is only 80% of it.

Q7 Is 10% of a day longer than 1% of a week? Create such questions and challenge your
peers.

Yes — 10% of a day is longer.

1 day = 24 hours = 1440 minutes

10% of a day = 0.1 × 1440 = 144 minutes = 2 hours 24 min

1 week = 7 × 1440 = 10,080 minutes

1% of a week = 0.01 × 10,080 = 100.8 minutes = 1 hour 40.8 min

144 min > 100.8 min

Why it happens: A week is only 7 times a day, but 10% is 10 times 1%. Ten beats
seven, so the smaller base with the larger percentage wins. Had the comparison
been 10% of a day against 1% of a fortnight (14 days), the fortnight would have won
— 1% of 14 days is 201.6 minutes.

Try This: Is 5% of a kilometre longer than 50% of a metre? Is 2% of a kilogram
heavier than 25% of 100 g? Is 1% of a year longer than 20% of a fortnight?

Page 22 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q8 Mariam’s farm has a peculiar bull. One day she gave the bull 2 units of fodder and
the bull ate 1 unit. The next day, she gave the bull 3 units of fodder and the bull ate
2 units. The day after, she gave the bull 4 units and the bull ate 3 units. This
continued, and on the 99th day she gave the bull 100 units and the bull ate 99 units.
Represent these quantities as percentages. This task can be distributed among the
class. What do you observe?

On day n she gives (n + 1) units and the bull eats n units, so the percentage eaten is n/(n + 1) ×
100.

DAY GIVEN EATEN FRACTION PERCENTAGE PERCENTAGE
EATEN EATEN LEFT

1 2 1 1/2 50% 50%

2 3 2 2/3 66.67% 33.33%

3 4 3 3/4 75% 25%

4 5 4 4/5 80% 20%

9 10 9 9/10 90% 10%

19 20 19 19/20 95% 5%

49 50 49 49/50 98% 2%

99 100 99 99/100 99% 1%

What we observe: the percentage eaten climbs steadily — 50%, 66.67%, 75%, 80%, … , 99% —
but it never reaches 100%.

Why it happens: The bull always leaves exactly 1 unit uneaten, every single day.
What changes is not the waste but the size of the meal it is compared with. The
percentage left over is 1/(n + 1) × 100, and as the meal grows, that one wasted unit
becomes a smaller and smaller share of it. Since 1/(n + 1) is never 0, the percentage
eaten is never 100% — it only creeps closer. This is exactly why a fixed amount looks
serious in a small budget and trivial in a large one.

Page 23 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
m.
Workers in a coffee plantation take 18 days to pick coffee berries in 20% of the
se
Q9

o m l a
plantation. How many days will they take to complete the picking work for the
g is this
entire.cplantation, assuming the rate of work stays the same?aWhy
se m
g l aassumption necessary?
a
m

. co ag
em
20% of the plantation takes 18 days
g l as
100% is 5 times 20%
a

co m
m.
Time for the whole plantation = 5 × 18 = 90 days

as e
. com a g l
em
Or as a proportion:

a s
a gl
20/100 = 18/d → d = 18 × 100 / 20 = 90 days

m a s
e m . c o agl
s
Why the assumption is necessary: multiplying the days by 5 because the area is 5 times bigger

a g la of the plantation costs the same amount of time as the
only works if every additional stretch
first. That needs the number of workers, their hours, the density of berries and the terrain to

m
stay the same throughout.

. co
se m
o m l a
Why it happens: Proportional reasoning is a claim about the world, not just about
g part of the
the.cnumbers. If the first 20% happened to be the easiest,aflattest
m
aseplantation, the remaining 80% would take more than 72 days and the answer would
agl be an underestimate. Stating the assumption is part of the answer.
se m
com g l a
m . a
gl ase
Q10
a
The badminton coach has planned the training sessions such that the ratio of
warm up : play : cool down is 10% : 80% : 10%. If he wants to conduct a training of
90 minutes. How long should each activity be done?
co m
m .
m as e
.co
a g l
se m
g l a
a Warm up = 10% of 90 min = 0.1 × 90 = 9 minutes
c
m .
Play = 80% of 90 min = 0.8 × 90 = 72 minutes
m a s e
e m . co agl
as
Cool down = 10% of 90 min = 9 minutes

a g l
Check: 9 + 72 + 9 = 90 minutes, and 10 + 80 + 10 = 100%.

com
m .
m ase
.co


a g l Page 24 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Tip: Once you have 10% of 90, which is 9, everything else follows — 80% is eight
times that, 8 × 9 = 72. Finding 10% first is almost always the cheapest first move.

Q11 An estimated 90% of the world’s population lives in the Northern Hemisphere. Find
the (approximate) number of people living in the Northern Hemisphere based on
this year’s worldwide population.

The chapter itself gives the world population in 2025 as about 8.2 billion.

90% of 8.2 billion = 0.9 × 8.2

= 7.38 billion

≈ 7.4 billion people (about 738 crore)

That leaves roughly 0.8 billion — about 82 crore people — in the Southern Hemisphere.

Tip: Use whatever current figure your class agrees on; the method does not change.
With 8 billion the answer is 7.2 billion, with 8.2 billion it is 7.38 billion. Since the 90%
itself is only an estimate, quoting the answer as “about 7.4 billion” is honest —
writing 7,380,000,000 would suggest a precision the data do not have.

Q12 A recipe for the dish, halwa, for 4 people has the following ingredients in the given
proportions — Rava: 40%, Sugar: 40%, and Ghee: 20%. (i) If you want to make halwa
for 8 people, what is the proportion of each of the above ingredients? (ii) If the
total weight of the ingredients is 2 kg, how much rava, sugar and ghee are
present?

(i) The proportions do not change: Rava 40%, Sugar 40%, Ghee 20%.
You need twice as much of everything, but doubling all three amounts leaves each one the
same share of the total.

Page 25 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Suppose 4 people need 100 g in all: rava 40 g, sugar 40 g, ghee 20 g

For 8 people: rava 80 g, sugar 80 g, ghee 40 g, total 200 g

Rava's share = 80/200 = 40% — unchanged

(ii) Total = 2 kg = 2000 g.

Rava = 40% of 2000 g = 800 g

Sugar = 40% of 2000 g = 800 g

Ghee = 20% of 2000 g = 400 g

Check: 800 + 800 + 400 = 2000 g ✓

Why it happens: A recipe is a ratio, 2 : 2 : 1. Scaling every part of a ratio by the same
factor leaves the ratio — and therefore the percentages — untouched. This is exactly
why recipes are written in proportions in the first place: one recipe then serves any
number of people.

In-text Questions — Page 15
Section 1.3 Using Percentages — Know Your Contents (KYC)

Q1 They are at a shop to buy badam drink mix. They are looking at two products and
wondering which has a larger share of badam. Can you figure it out? Which product
uses a smaller proportion of food chemicals? [DEF: sugar 99 g, milk solids 30 g,
badam powder 12 g, food chemicals 9 g, total weight 150 g. Zacni: sugar 272 g, milk
solids 64 g, badam powder 40 g, food chemicals 24 g, total weight 400 g.]

Zacni has the larger share of badam. Neither uses a smaller proportion of food chemicals —
the two are equal.

Page 26 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Badam in DEF = 12/150 × 100 = 8%

Badam in Zacni = 40/400 × 100 = 10%

Food chemicals in DEF = 9/150 × 100 = 6%

Food chemicals in Zacni = 24/400 × 100 = 6%

Why it happens: Reading the raw grams misleads you twice over. Zacni lists 40 g of
badam against DEF's 12 g — more than three times as much — but the packet is also
much larger, so the share rises only from 8% to 10%. And Zacni's 24 g of food
chemicals looks far worse than DEF's 9 g, yet as a proportion of what you actually
drink the two are identical at 6%. Comparing packets of different sizes is exactly the
situation percentages were invented for.

Q2 Complete this table by calculating the percentages to answer the questions:

Sugar Milk Solids Badam Powder Food Chemicals

DEF 66%

Zacni

The table printed on page 15. DEF’s sugar entry is filled in as the worked example.

SUGAR MILK BADAM FOOD
SOLIDS POWDER CHEMICALS

DEF (TOTAL 150 G) 66% 20% 8% 6%

ZACNI (TOTAL 400 68% 16% 10% 6%
G)

DEF: 99/150 = 66% | 30/150 = 20% | 12/150 = 8% | 9/150 = 6%

Zacni: 272/400 = 68% | 64/400 = 16% | 40/400 = 10% | 24/400 = 6%

Page 27 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Did you know? In both packets sugar is the largest ingredient by a long way — two-
thirds of the contents. Madhu's joke, that it should be called a “sugar drink mix”, is
arithmetically fair.

Q3 Check if the percentages of each product add up to 100.

DEF: 66 + 20 + 8 + 6 = 100 ✓

Zacni: 68 + 16 + 10 + 6 = 100 ✓

The weights agree as well: 99 + 30 + 12 + 9 = 150 g, and 272 + 64 + 40 + 24 = 400 g.

Why it happens: Each percentage is one ingredient divided by the total weight.
Adding them adds the numerators over a common denominator, and the
ingredients add back to the total weight — so the sum must be total/total = 1 =
100%. If a label's percentages did not add to 100, either an ingredient has been left
off the list or one of the figures is wrong. That makes this a genuinely useful check
on any packet you pick up.

In-text Questions — Page 17
Section 1.3 — Profit and Loss

Q1 Find the profit percentage of the wholesaler and the manufacturer. [From the
sweater's journey — Manufacturing unit: CP ₹230, MP ₹255, SP ₹253. Wholesale
store: CP ₹253, MP ₹310, SP ₹300. Retail store: CP ₹300, MP ₹480, SP ₹430.]

Profit percentage is always taken on the cost price of that seller.

Page 28 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
e m.
Manufacturer: CP ₹230, SP ₹253
m l as
.co
Profit = 253 – 230 = ₹23
m a g
l a se
g
Profit % = 23/230 × 100 = 10%
a

co m
. ag
Wholesaler: CP ₹253, SP ₹300
e m
Profit = 300 – 253 = ₹47
g l as
a
Profit % = 47/253 × 100 = 4700/253 = 18.58% (18.577…%)

co m
em.
m l as
.co
Retailer (Kishanlal, for comparison): CP ₹300, SP ₹430
a g
se m
Profit = ₹130, Profit % = 130/300 × 100 = 43.33%
g l a
a
m a s
agl
Why it happens: Notice that one sweater carries three different “cost prices” and

m .co
three different “selling prices” — the wholesaler's cost price, ₹253, is the

l a se
g
manufacturer's selling price. The labels CP, MP and SP are not properties of the
a
sweater; they describe a particular transaction. So each seller's profit percentage
must be worked out against their own buying price. The percentages differ widely
co m
(10%, 18.58%, 43.33%) even though every step adds a modest amount of rupees.
m .
m as e
.co a g l
s e m it yourself: Nobody sells at the marked price. The manufacturer marks ₹255
Check

agla and sells at ₹253; the wholesaler marks ₹310 and sells at ₹300; Kishanlal marks ₹480
and sells at ₹430. The gap between MP and SP is the discount given after bargaining.
se m
com g l a
m . a
ase
Q2 agl
Shambhavi owns a stationery shop. She procures 200 page notebooks at ₹36 per
book. She sells them with a profit margin of 20%. Find the selling price.

co m
m .
m as e
l

.co
m Cost price = ₹36, profit margin = 20% of the cost price a g
l a se
ag
.c
e m
Profit = 20% of 36 = 0.20 × 36 = ₹7.20
m a s
. co agl
Selling price = 36 + 7.20 = ₹43.20

se m
l a
agthat the selling price is 120% of the cost price:
In one step, using the fact

co m
m .
m ase
.co


a g l Page 29 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

SP = 1.2 × 36 = ₹43.20

Why it happens: “A profit margin of 20%” means the profit is 20% of what she paid.
So her selling price is the whole cost (100%) plus the profit (20%) — that is, 120% of
₹36. Adding the percentage to 100 and multiplying once is quicker and less error-
prone than finding the profit and adding it separately.

Q3 She sells crayon boxes at ₹50 per box with a profit margin of 25%. How much did
Shambhavi buy them from the wholesaler?

Here the selling price is known and the cost price is not — so we must undo the mark-up.

SP = 125% of CP

1.25 × CP = 50

CP = 50/1.25 = ₹40

Check: profit = 50 – 40 = ₹10, and 10/40 × 100 = 25%. ✓

Why it happens: A common mistake is to take 25% of ₹50, get ₹12.50, and answer
₹37.50. That is wrong because the 25% is measured on the cost price, not the selling
price — the base is the smaller number. Checking confirms it: 12.50 on a cost of
37.50 would be a margin of 33.33%, not 25%.

Q4 Could we have just calculated the loss percentage per kg instead? Would it be the
same? [Raghu bought rice at ₹35 per kg and sold 10 kg for ₹300, a loss of ₹50 on
₹350, i.e. 14.28%.]

Yes — the loss percentage per kilogram is exactly the same, 14.28%.

Page 30 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Per kg: CP = ₹35, SP = 300/10 = ₹30

Loss per kg = 35 – 30 = ₹5

Loss % = 5/35 × 100 = 500/35 = 14.28% (14.2857…%)

For 10 kg: loss = ₹50 on a cost of ₹350

Loss % = 50/350 × 100 = 14.28%

Why it happens: Going from 1 kg to 10 kg multiplies the loss by 10 and the cost
price by 10. In the fraction 50/350 both the numerator and the denominator carry
that same factor of 10, so it cancels: 50/350 = 5/35. A percentage is a ratio, and a
ratio does not notice how much you scale both of its parts. That is why a shopkeeper
can quote one profit percentage for a whole sack and for a single kilogram.

In-text Questions — Page 18
Section 1.3 — Profit, Loss and Discount

Q1 Due to heavy rains, Snehal could not transport strawberries to Hyderabad from his
farm in Panchgani. He sells some of his stock at ₹80 per kg with a 12% loss. What is
the cost price?

A 12% loss means the selling price is 100% – 12% = 88% of the cost price

0.88 × CP = 80

CP = 80/0.88 = 8000/88 = 1000/11

= ₹90.91 per kg (₹90.90 to the nearest paisa is 90.909…)

Check: 12% of 90.91 = ₹10.91, and 90.91 – 10.91 = ₹80. ✓

Page 31 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Why it happens: The 12% is a loss on what Snehal paid, so the cost price is the base
— the unknown. Taking 12% of ₹80 instead would give ₹9.60 and a cost price of
₹89.60, which is wrong: a loss of ₹9.60 on ₹89.60 is 10.7%, not 12%. Whenever the
percentage refers to a quantity you do not yet know, write the relationship first (SP =
0.88 × CP) and solve.

Q2 A utensil store is offering a 35% discount on the cooker with an MRP ₹1800. What is
the selling price? If the cost price was ₹900, what is the percentage profit made
after the sale?

Selling price: a 35% discount leaves 65% of the MRP

SP = 0.65 × 1800 = ₹1170

Profit percentage: CP = ₹900

Profit = 1170 – 900 = ₹270

Profit % = 270/900 × 100 = 30%

Why it happens: Two percentages appear in this one sale and they are measured on
two different bases. The 35% discount is taken off the marked price of ₹1800; the
30% profit is measured against the cost price of ₹900. Mixing the bases is the classic
error — 35% of ₹900 or 30% of ₹1800 would both be meaningless here. Note also
that the store still earns a healthy profit after the discount, because the MRP was set
at twice the cost price.

In-text Questions — Page 19

Page 32 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Section 1.3 — Taxes

TRY THIS

Q1 Check if the calculations are correct in the bill shown. [XY Electricals sales receipt,
06/07/2025 — CFL Bulb, Qty 3, Price ₹150.00, Amount ₹450.00; Sub Total ₹450.00;
CGST 9% ₹40.50; SGST 9% ₹40.50; TOTAL ₹531.00.]

Every figure on the bill is correct.

LINE ON THE BILL CHECK VERDICT

Amount ₹450.00 3 × 150 = 450 correct

Sub Total ₹450.00 only one item correct

CGST 9% ₹40.50 0.09 × 450 = 40.50 correct

SGST 9% ₹40.50 0.09 × 450 = 40.50 correct

TOTAL ₹531.00 450 + 40.50 + 40.50 = 531 correct

Total tax = 40.50 + 40.50 = ₹81

Tax as a percentage of the sub total = 81/450 × 100 = 18%

Why it happens: GST on this item is 18%, but the bill splits it into two halves of 9%
each — CGST, which goes to the Central government, and SGST, which goes to the
State government. Both are 9% of the same base, the sub total of ₹450, so both
come to ₹40.50. Adding the two 9% shares to the sub total is the same as multiplying
it by 1.18: 450 × 1.18 = ₹531.

Tip: A fast check on any GST bill — the total should be the sub total plus a fixed
percentage of it. Here 10% of 450 is 45, so 18% is a little under 81… and it is exactly
81.

Page 33 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
m.
You may share any bills you have at home with the class. Observe the different
e
Q2

m l as
.co
elements present in the bills. Are there any similarities or differences in these bills?

a g
se m
g l a
a

Collect a few bills — a grocery bill, a restaurant bill, a mobile recharge receipt, an electricity bill

m
— and compare them.
. co ag
What is usually the same:
e m
g l as
a
The seller's name, the date and a bill number.
Item, quantity, rate per unit, and amount (= quantity × rate).
A sub total, then tax lines, then a grand total.
co m
se
GST split as CGST + SGST when buyer and seller are in the same state.
m.
o m g l a
m .c
What differs:
a
l a se GST rate. It is not one number for everything — most fresh food carries 0%, many
ag household goods 5% or 12%, most appliances and services 18%.
The

m a s
agl
IGST instead of CGST + SGST on bills from another state (online orders, for instance).

m .co
Some bills show a discount line before the tax, some after.

l a se
g
Some prices are marked “inclusive of all taxes”, so no tax line appears at all.
a
m
Why it happens: When a price is inclusive of tax, the tax is not 18% of the printed
price — it is 18% of the price before tax. If a bill says ₹236 inclusive of 18% GST, then
. co
se m
o m l a
1.18 × base = 236, so the base is ₹200 and the tax ₹36. Working backwards through
g is the same piece of
the.cmark-up, exactly as with the cost price of the strawberries,
m a
asereasoning again.
agl
se m
com g l a
Figure it Out — Pages 19–20em. a
a s
agl
Section 1.3 — Profit, Loss, Discount and Percentage Change

co m
Q1
m .
If a shopkeeper buys a geometry box for ₹75 and sells it for ₹110, what is his profit

as e
com
margin with respect to the cost?

. a g l
e m
as
agl

.c
s e m
m a
agl
Profit = 110 – 75 = ₹35
. co
e m
as
Profit % = 35/75 × 100 = 3500/75

a g
= 46.67% (exactly 46⅔%) l

com
m .
m ase
.co


a g l Page 34 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Tip: Estimate first. ₹35 is a little less than half of ₹75, so the margin must be a little
under 50%. 46.67% fits.

Q2 I am a carpenter and I make chairs. The cost of materials for a chair is ₹475 and I
want to have a profit margin of 50%. At what price should I sell a chair?

SP = CP + 50% of CP = 150% of CP
= 1.5 × 475

= ₹712.50

Equivalently: profit = half of 475 = ₹237.50, and 475 + 237.50 = ₹712.50.

Careful: A 50% margin does not mean “half the selling price is profit”. Here the profit
₹237.50 is exactly a third of the selling price ₹712.50 — because the 50% is
measured against the cost, not the sale.

Q3 The total sales of a company (also called revenue) was ₹2.5 crore last year. They had
a healthy profit margin of 25%. What was the total expenditure (costs) of the
company last year?

Reading “profit margin” the way this chapter has used it so far — a percentage of the cost —
the revenue is 125% of the expenditure.

1.25 × expenditure = ₹2.5 crore
Expenditure = 2.5/1.25 = ₹2 crore

Profit = 2.5 – 2 = ₹0.5 crore, and 0.5/2 × 100 = 25% ✓

Page 35 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Why it happens: Questions 1 and 2 of this set both measure the margin against the
cost, so we keep that meaning here. But the chapter also introduces a second
convention — profit as a percentage of the revenue — used when a business asks
“how much profit did I make on my overall revenue?”. On that reading the profit
would be 25% of ₹2.5 crore = ₹0.625 crore and the expenditure ₹1.875 crore. Both
are defensible arithmetic; what is not optional is saying which base you used. In
business writing, “margin” usually means the second one, so always check.

Q4 A clothing shop offers a 25% discount on all shirts. If the original price of a shirt is
₹300, how much will Anwar have to pay to buy this shirt?

He pays 100% – 25% = 75% of the marked price

= 3/4 × 300

= ₹225

Or: discount = 25% of 300 = ₹75, so he pays 300 – 75 = ₹225.

Q5 The petrol price in 2015 was ₹60 and ₹100 in 2025. What is the percentage increase
in the price of petrol? (i) 50% (ii) 40% (iii) 60% (iv) 66.66% (v) 140% (vi) 160.66%

(iv) 66.66%.

Increase = 100 – 60 = ₹40

Percentage increase = increase / original × 100

= 40/60 × 100 = 200/3

= 66.66% (exactly 66⅔%)

Page 36 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Why it happens: Option (ii) 40% is the trap: ₹40 is the amount of the rise, not its
percentage. Option (v) 140% is the second trap: the new price is 140% of the old
price, but the increase is 140% – 100% = 40 percentage points on a base of 60, i.e.
66.66%. Percentage change is always measured against the original amount — the
2015 price of ₹60, not the 2025 price of ₹100.

Check it yourself: If the price had fallen back from ₹100 to ₹60, the percentage
decrease would be 40/100 = 40% — a different number for the same ₹40, because
the base has changed.

Q6 Samson bought a car for ₹4,40,000 after getting a 15% discount from the car dealer.
What was the original price of the car? [Printed as question 3 on page 20; the book
restarts its numbering there.]

After a 15% discount he pays 85% of the original price

0.85 × original = 4,40,000

original = 4,40,000 / 0.85 = 44,00,000/8.5

= ₹5,17,647 (5,17,647.06 to the nearest paisa)

Check: 15% of 5,17,647 = ₹77,647, and 5,17,647 – 77,647 = ₹4,40,000. ✓

Why it happens: Adding 15% of ₹4,40,000 (which is ₹66,000) to get ₹5,06,000 is
wrong, and it is worth seeing exactly why: the discount was 15% of the original price,
a bigger number than ₹4,40,000, so 15% of it is more than ₹66,000. Whenever a
percentage has already been applied and you are working backwards, divide — do
not add the same percentage back.

Note: From this question onwards the book's printed numbering restarts at 3 on
page 20, so its “3, 4, 5, …, 9” repeat numbers already used on page 19. The questions
are numbered here continuously, Q6 to Q12.

Page 37 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q7 1600 people voted in an election and the winner got 500 votes. What percent of the
total votes did the winner get? Can you guess the minimum number of candidates
who stood for the election? [Printed as question 4 on page 20.]

Winner's share = 500/1600 × 100 = 5/16 × 100

= 31.25%

Minimum number of candidates: 4.

Votes left for everyone else = 1600 – 500 = 1100

No other candidate can have 500 or more (or they would not have lost)

With 3 candidates: 2 others share 1100 votes

→ at least one of them gets 1100 ÷ 2 = 550 > 500 — impossible

With 4 candidates: 3 others share 1100 votes
→ e.g. 400, 350, 350 — all below 500. Possible.

Why it happens: The winner did not get a majority — 31.25% is under half — so the
other 68.75% of the votes must have been split among enough people that no one
of them beat 500. Two rivals cannot split 1100 votes without one of them crossing
550. Three rivals can. This is a real feature of first-past-the-post elections: the more
candidates there are, the smaller the share a winner needs.

Page 38 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
m.
The price of 1 kg of rice was ₹38 in 2024. It is ₹42 in 2025. What is the rate of
e
Q8

com l
inflation? (Inflation is the percentage increase in prices.) [Printed as question 5 on
page .20.] g as
em a
a s
a gl

co m
Increase = 42 – 38 = ₹4
e m . ag
g l as
a
Rate of inflation = 4/38 × 100 = 400/38 = 200/19

= 10.53% (10.526…%)

co m
em.
m l as
.co a g
Did you know? Inflation is always quoted against the earlier price, which is why the

a s em ₹4 rise means different things at different price levels. A ₹4 rise on ₹38 rice is
same

a gl 10.53% inflation; the same ₹4 on ₹200 pulses would be only 2%.
m a s
m .co agl
l a se
g
Q9 A number increased by 20% becomes 90. What is the number? [Printed as question 6
on page 20.] a

co m
m .
m as e
c o
. the number be x. Increasing it by 20% makes it 120% of x a g l
e m
as
Let

agl 1.2x = 90

se m
com a
x = 90/1.2 = 900/12

. a g l
m
ase
= 75

a gl
Check: 20% of 75 = 15, and 75 + 15 = 90. ✓

co m
m .
e
Why it happens: Taking 20% of 90 and subtracting gives 72 — and that is wrong, as
m l as
.co g
the check shows: 20% of 72 is 14.4, giving 86.4, not 90. The 20% was added to the

em a
s
smaller starting number, so undoing it means dividing by 1.2, not subtracting 20% of
l a
ag the result.
c
m .
m a s e
e m . co agl
g l as
a

com
m .
m ase
.co


a g l Page 39 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q10 A milkman sold two buffaloes for ₹80,000 each. On one of them, he made a profit
of 5% and on the other a loss of 10%. Find his overall profit or loss. [Printed as
question 7 on page 20.]

He made an overall loss of about ₹5,079, which is about 3.08%.
The two selling prices are equal, but the two cost prices are not — so work each one out first.

Buffalo 1 (5% profit): SP = 105% of CP

1.05 × CP₁ = 80,000 → CP₁ = 80,000/1.05 = ₹76,190.48

Buffalo 2 (10% loss): SP = 90% of CP

0.90 × CP₂ = 80,000 → CP₂ = 80,000/0.90 = ₹88,888.89

Total cost price = 76,190.48 + 88,888.89 = ₹1,65,079.37

Total selling price = 80,000 + 80,000 = ₹1,60,000

Loss = 1,65,079.37 – 1,60,000 = ₹5,079.37

Loss % = 5,079.37 / 1,65,079.37 × 100 = 3.08%

Why it happens: It is tempting to say “+5% and –10% average out to –2.5%”. They do
not, because the two percentages sit on different bases. The buffalo sold at a loss
had cost him ₹88,889 — far more than the ₹76,190 buffalo — so the 10% loss is
taken on a bigger amount than the 5% profit. Percentages can only be added when
they refer to the same whole; here they refer to two different cost prices.

Check it yourself: Profit on the first = 80,000 – 76,190.48 = ₹3,809.52. Loss on the
second = 88,888.89 – 80,000 = ₹8,888.89. The difference, 8,888.89 – 3,809.52 =
₹5,079.37, is the overall loss.

Page 40 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q11 The population of elephants in a national park increased by 5% in the last decade.
If the population of the elephants last decade is p, the population now is (i) p × 0.5
(ii) p × 0.05 (iii) p × 1.5 (iv) p × 1.05 (v) p + 1.50 [Printed as question 8 on page 20.]

(iv) p × 1.05.

Population now = p + 5% of p

= p + 0.05p

= p(1 + 0.05)

= 1.05p

Why it happens: Look at what the wrong options would mean. (ii) p × 0.05 is only
the increase, 5% of p — not the new total. (i) p × 0.5 would be a 50% fall. (iii) p × 1.5
would be a 50% rise, ten times too much. (v) p + 1.50 adds a fixed 1.5 elephants,
which is not a percentage at all. Growing by r means multiplying by (1 + r) — the
whole plus the extra.

Q12 Which of the following statement(s) mean the same as — “The demand for
cameras has fallen by 85% in the last decade”? (i) The demand now is 85% of the
demand a decade ago. (ii) The demand a decade ago was 85% of the demand now.
(iii) The demand now is 15% of the demand a decade ago. (iv) The demand a decade
ago was 15% of the demand now. (v) The demand a decade ago was 185% of the
demand now. (vi) The demand now is 185% of the demand a decade ago. [Printed
as question 9 on page 20.]

Only (iii) means the same thing.

Let the demand a decade ago be d.

Fallen by 85% → the fall is 0.85d

Demand now = d – 0.85d = 0.15d = 15% of d

Page 41 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

STATEMENT WHAT IT SAYS SAME?

(i) now = 0.85d — a fall of only 15% No

(ii) d = 0.85 × now, so demand has risen No

(iii) now = 0.15d Yes

(iv) d = 0.15 × now, so demand has risen sharply No

(v) d = 1.85 × now, a fall of about 46% No

(vi) now = 1.85d, a rise of 85% No

Why it happens: Two separate traps are at work. The first is confusing “fallen by
85%” with “is 85% of” — the first leaves 15%, the second leaves 85%. The second is
swapping which quantity is the base. The correct reverse statement is not (v): if now
= 0.15d then d = now/0.15 = 6.67 × now, that is, the demand a decade ago was about
667% of the demand now — not 185%.

Figure it Out — Pages 22–24

Page 42 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Section 1.3 — Growth and Compounding

MATH TALK

Q1 Bank of Yahapur offers an interest of 10% p.a. Compare how much one gets if they
deposit ₹20,000 for a period of 2 years with compounding and without
compounding annually.

Without compounding — the principal stays ₹20,000 every year

Interest each year = 10% of 20,000 = ₹2000
Interest for 2 years = 2 × 2000 = ₹4000

Amount = 20,000 + 4000 = ₹24,000

With compounding — each year's interest joins the principal

After year 1: 20,000 × 1.1 = ₹22,000

After year 2: 22,000 × 1.1 = ₹24,200

(or 20,000 × 1.1² = 20,000 × 1.21 = 24,200)

Compounding gives ₹200 more.

Why it happens: The extra ₹200 is exactly 10% of the first year's interest of ₹2000. In
the compounding account that ₹2000 stays in the deposit and earns interest of its
own during the second year; in the other account it is paid out and earns nothing.
That is the whole difference between the two options — interest on interest.

Page 43 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
m.
Bank of Wahapur offers an interest of 5% p.a. Compare how much one gets if one
e
Q2

com annually.
deposits ₹20,000 for a period of 4 years with compounding and without
g l as
. a
em
compounding

a s
a gl

co m
Without compounding
e m . ag
g l as
Interest each year = 5% of 20,000 = ₹1000
a
Interest for 4 years = 4 × 1000 = ₹4000

co m
m.
Amount = ₹24,000
m as e
.co a g l
a s emcompounding
gl
With
a Amount = 20,000 × (1.05)⁴

m a s
.co agl
1.05² = 1.1025, so 1.05⁴ = 1.1025 × 1.1025 = 1.21550625

se m
l a
= 20,000 × 1.21550625 = ₹24,310.125 = ₹24,310.13 to the nearest paisa
g
a
YEAR OPENING AMOUNT INTEREST AT 5%
co m
CLOSING AMOUNT

m .
m as e
.co g l
1 ₹20,000 ₹1000 ₹21,000

m a
l a se 2 ₹21,000 ₹1050 ₹22,050

a g
m
₹22,050 ₹1102.50 ₹23,152.50

se
3

com₹1157.63 g l a
m. a
4 ₹23,152.50 ₹24,310.13

ase
Compounding gives about ₹310 more.
agl

. com
m a s emtwo questions above?
gl
Do you observe anything interesting in the solutions of the
co Share and discuss.
Q3

. a
a s em
agl c
.

s e m
m a
agl
Without compounding both deposits give exactly the same amount, ₹24,000 — but with

. co
m
compounding they do not.

as e
a g l

co m
m .
m ase
.co


a g l Page 44 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

10% FOR 2 YEARS 5% FOR 4 YEARS

Total simple interest ₹4000 ₹4000

Amount, no compounding ₹24,000 ₹24,000

Amount, compounded ₹24,200 ₹24,310.13

Without compounding the amount is p(1 + rt), and rt is the same for both:

0.10 × 2 = 0.20 and 0.05 × 4 = 0.20

With compounding the amount is p(1 + r)ᵗ, and these differ:

(1.10)² = 1.2100 but (1.05)⁴ = 1.2155

Why it happens: Without compounding, only the product rt matters — interest is
added on the same principal every time, so a high rate for a short time and a low
rate for a long time balance out. With compounding, the number of times the
interest is folded back matters too. The 5% account folds interest back four times
instead of two, and each fold earns interest on all the earlier interest. More frequent
compounding beats a higher rate applied fewer times, even when the plain totals
agree.

Q4 Jasmine invests amount ‘p’ for 4 years at an interest of 6% p.a. Which of the
following expression(s) describe the total amount she will get after 4 years when
compounding is not done? (i) p × 6 × 4 (ii) p × 0.6 × 4 (iii) p × (0.6/100) × 4 (iv) p ×
(0.06/100) × 4 (v) p × 1.6 × 4 (vi) p × 1.06 × 4 (vii) p + (p × 0.06 × 4)

Only (vii).

Without compounding, amount = p + interest

Interest = p × r × t = p × 0.06 × 4 = 0.24p

Amount = p + 0.24p = 1.24p — which is what (vii) says

Page 45 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

OPTION WHAT IT ACTUALLY COMPUTES

(i) p × 6 × 4 24p — treats 6% as the number 6

(ii) p × 0.6 × 4 2.4p — uses 60%, not 6%

(iii) p × (0.6/100) × 4 0.024p — a hundred times too small

(iv) p × (0.06/100) × 4 0.0024p — divides by 100 twice over

(v) p × 1.6 × 4 6.4p

(vi) p × 1.06 × 4 4.24p — multiplies the whole amount by 4 instead of the interest

(vii) p + (p × 0.06 × 4) 1.24p ✓

Why it happens: Options (iii) and (iv) are the same slip made twice — 6% is already
0.06, so writing 0.06/100 divides by 100 a second time. Option (vi) is the subtler
error: p × 1.06 is the amount after one year, and multiplying that by 4 quadruples the
whole deposit rather than repeating the interest. Without compounding the interest
is added four times, but the principal is counted only once.

Page 46 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q5 The post office offers an interest of 7% p.a. How much interest would one get if one
invests ₹50,000 for 3 years without compounding? How much more would one get if
it was compounded?

Without compounding

Interest = p × r × t = 50,000 × 0.07 × 3
= ₹10,500

With compounding

Amount = 50,000 × (1.07)³

1.07² = 1.1449, so 1.07³ = 1.1449 × 1.07 = 1.225043

= 50,000 × 1.225043 = ₹61,252.15

Interest = 61,252.15 – 50,000 = ₹11,252.15

Extra from compounding = 11,252.15 – 10,500 = ₹752.15

Check it yourself: Year by year — 50,000 → 53,500 → 57,245 → 61,252.15. The
yearly interest rises from ₹3500 to ₹3745 to ₹4007.15, because the principal it is
charged on keeps growing.

Q6 Giridhar borrows a loan of ₹12,500 at 12% per annum for 3 years without
compounding and Raghava borrows the same amount for the same time period at
10% per annum, compounded annually. Who pays more interest and by how much?

Giridhar pays more — by ₹362.50.

Page 47 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Giridhar — 12%, no compounding

Interest = 12,500 × 0.12 × 3 = ₹4500

Raghava — 10%, compounded annually

Amount = 12,500 × (1.1)³ = 12,500 × 1.331 = ₹16,637.50

Interest = 16,637.50 – 12,500 = ₹4137.50

Difference = 4500 – 4137.50 = ₹362.50

Why it happens: Compounding is not automatically the costlier deal. Over 3 years,
10% compounded multiplies the debt by 1.331 — an effective 33.1% — while 12%
simple multiplies it by 1.36, an effective 36%. The 2 percentage points of extra rate
outweigh what compounding adds over such a short term. Over a longer period the
balance tips the other way: at 10% compounded for 10 years the multiplier is 2.594,
well past 12% simple, which reaches only 2.2.

Q7 Consider an amount ₹1000. If this grows at 10% p.a., how long will it take to double
when compounding is done vs. when compounding is not done? Is compounding an
example of exponential growth and not-compounding an example of linear growth?

Without compounding: 10 years. With compounding: 8 years. And yes — compounding is
exponential growth, and not-compounding is linear growth.

No compounding: interest is ₹100 every year

Amount = 1000 + 100t. Doubling needs 100t = 1000

t = 10 years

With compounding: Amount = 1000 × (1.1)ᵗ

(1.1)⁷ = 1.9487 → ₹1948.72 — not yet doubled

(1.1)⁸ = 2.1436 → ₹2143.59 — past ₹2000

So it doubles during the 8th year → 8 years on annual compounding

Page 48 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m10
m.
END OF YEAR 1 2 4 6 7 8

m a e
s1800
.co
NO COMPOUNDING (₹) 1100 1200 1400 1600 1700
ag l 2000

se m
g l a
a
COMPOUNDED (₹) 1100 1210 1464.10 1771.56 1948.72 2143.59 2593.74

. c om
Why it happens: Without compounding the amount is p(1 + rt) — t appears once,
ag
s
multiplied by a constant, so the growth
e mis a fixed ₹100 every year and the graph is a
aWith compounding the amount is p(1 + r)ᵗ — t is
agl
straight line. That is linear growth.
now an exponent, so each year multiplies the previous amount by 1.1 rather than
adding to it. The yearly increase itself grows: ₹100 in year 1, ₹110 in year 2, ₹121 in
co m
year 3. That is exponential growth, and it is why the compounded column pulls
e m.
m l as
.co g
further and further ahead the longer you leave it.
m a
l a se
a g
The population of a city is rising by about 3% every year. If the current population is
m a s
agl
Q8

.co
1.5 crore, what is the expected population after 3 years?
m
l a se
ANSWER a g
A yearly percentage rise compounds, because each year's growth is measured on the

co m
.
population at the start of that year.
em
com after 3 years = 1.5 crore × (1.03)³ g l as
m . a
ase
Population

agl 1.03² = 1.0609, so 1.03³ = 1.0609 × 1.03 = 1.092727

= 1.5 × 1.092727 crore = 1.6390905 crore
se m
com g l a
. a
em
≈ 1.64 crore (about 1,63,90,905 people)
a s
agl
Why it happens: Simply adding 3% three times — 9% of 1.5 crore = 1.635 crore — is

. com
close but slightly low. The extra 0.0041 crore (about 41,000 people) comes from the

m a s
growth of the people added in years 1 and 2, who go on having emchildren themselves.
.cOver
m the difference is enormous. aglagainst a simple 1.90, and
o 3 years the gap is small; over 30 years, 1.03³⁰ = 2.43
l a se
ag
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 49 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Q9 In a laboratory, the number of bacteria in a certain experiment increases at the
rate of 2.5% per hour. Find the number of bacteria at the end of 2 hours if the initial
count is 5,06,000.

Count after 2 hours = 5,06,000 × (1.025)²

(1.025)² = 1.050625

= 5,06,000 × 1.050625

= 5,31,616.25 ≈ 5,31,616 bacteria

Step by step:

After 1 hour: 5,06,000 × 1.025 = 5,18,650

After 2 hours: 5,18,650 × 1.025 = 5,31,616.25

Careful: The first hour adds 12,650 bacteria and the second adds 12,966.25 — more,
because the second 2.5% is taken on the larger count of 5,18,650. Since bacteria
come in whole numbers, reporting about 5,31,616 is right; the decimal is an artefact
of using a smooth percentage on a discrete count.

In-text Questions — Page 24
Section 1.3 — Growth and Compounding

MATH TALK

Q1 Suppose we want to know the expression/formula to find the total interest amount
gained at the end of the maturity period. What would be the formula for each of
the two options?

Interest is whatever the deposit has gained above the principal, so in both cases subtract p from
the maturity amount.

Page 50 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

AMOUNT AT MATURITY TOTAL INTEREST

WITHOUT COMPOUNDING p(1 + rt) p(1 + rt) – p = prt

WITH COMPOUNDING p(1 + r)t p(1 + r)t – p = p[(1 + r)t – 1]

Check with the chapter's numbers — p = 6000, r = 0.1, t = 3

Without compounding: prt = 6000 × 0.1 × 3 = ₹1800 ✓ (7800 – 6000)
With compounding: 6000[(1.1)³ – 1] = 6000 × 0.331 = ₹1986 ✓ (7986 – 6000)

Why it happens: In the first formula the interest prt is a plain product — triple the
years and you triple the interest. In the second, the years sit in the exponent, so the
interest is p times ((1 + r)ᵗ – 1), and that bracket grows faster than t does. The “– 1” is
doing the same job as the “– p” before it was factored out: it removes the original
principal, which was never interest.

In-text Questions — Page 25
Section 1.3 — Tricky Percentages: Would You Rather?

MATH TALK

Q1 You have won a contest. The organisers offer you two options to choose from:
Option A: You deposit ₹100 and you get back ₹300. Option B: You deposit ₹1000 and
you get back ₹1500. What is the percentage gain each option gives? You can choose
any option only once. Which option would you choose? Why?

Option A: gain = 300 – 100 = ₹200 on a deposit of ₹100
Percentage gain = 200/100 × 100 = 200%

Option B: gain = 1500 – 1000 = ₹500 on a deposit of ₹1000

Percentage gain = 500/1000 × 100 = 50%

Page 51 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Which to choose? Option A wins on percentage, but Option B puts more money in your pocket
— ₹500 against ₹200. Since you may use an option only once, Option B is the better choice,
provided you have ₹1000 to deposit. If you only have ₹100, Option B is not available at all and
A is the answer.

Why it happens: A percentage gain tells you how hard each rupee worked, not how
many rupees you end up with. Option A quadruples a small stake; Option B adds
half again to a large one. If the offer could be repeated, the 200% rate would quickly
overtake — ₹100 → ₹300 → ₹900 → ₹2700 in three rounds, while Option B needs
₹1000 up front each time. The “only once” condition is what makes the absolute gain
decide it. This is the chapter's warning in a single question: when comparing
percentages, always check what whole they refer to.

In-text Questions — Page 26
Section 1.3 — Tricky Percentages

Q1 A provision store is offering a stock clearance sale. Customers can choose one of the
two options — 20% discount or ₹50 discount — for any purchase above ₹150. Which
option would you choose if you want to: (i) buy items worth ₹180 (ii) buy items
worth ₹225 (iii) buy items worth ₹300

PURCHASE 20% DISCOUNT ₹50 DISCOUNT BETTER CHOICE YOU PAY

(i) ₹180 ₹36 ₹50 ₹50 off ₹130

(ii) ₹225 ₹45 ₹50 ₹50 off ₹175

(iii) ₹300 ₹60 ₹50 20% off ₹240

The two offers are worth the same at one particular bill:

20% of x = 50

0.2x = 50 → x = ₹250

Page 52 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Why it happens: The ₹50 discount is a fixed amount — it does not grow with the bill.
The 20% discount grows in step with the bill, so it is worth ₹30 on a ₹150 purchase
and ₹60 on a ₹300 one. Below ₹250 the fixed discount is larger; above ₹250 the
percentage discount overtakes it. Knowing the ₹250 break-even point lets you decide
any bill at a glance, without working out both offers each time.

Q2 A bakery called Cakely is offering a 30% + 20% discount on all cakes. Another bakery
called Cakify is offering a 50% discount on all cakes. Would you rather choose
Cakely or Cakify if you want the cheaper cost?

Cakify — its single 50% discount is genuinely bigger than Cakely's “30% + 20%”.

For a cake worth ₹200:

Cakely: 30% off ₹200 → 200 – 60 = ₹140

then 20% off ₹140 → 140 – 28 = ₹112

Cakify: 50% off ₹200 → ₹100

Cakely's offer is really a single discount of 44%, not 50%:

Price paid = 0.70 × 0.80 × MP = 0.56 × MP

So the customer pays 56% and the discount is 44%

Why it happens: In shopping, “30% + 20%” means the discounts are applied one
after the other, and the second one is taken on the already reduced price — 20% of
₹140, not 20% of ₹200. Percentages taken in succession multiply their remaining
fractions (0.7 × 0.8 = 0.56); they do not add their discounts. The missing 6% is
precisely 20% of the first ₹60 discount — a discount you never get, because that ₹60
has already been taken off.

Tip: Order does not matter. 20% first and then 30% also gives 0.8 × 0.7 = 0.56, i.e.
₹112. What matters is that a reduced price becomes the base for the next reduction.

Page 53 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
m.
In-text Questions — Page 27
m as e
Section 1.3 — A Mishap
.co a g l
se m
l a
TRY THIS

a g
Ariba and Arun have some marbles. Ariba says, “The number of marbles with me is

com
Q1

. ag
120% of the marbles Arun has”. What would be an appropriate statement Arun

a s
could make comparing the numberemof marbles he has with Ariba’s?
agl

co m
m.
Arun could say: “The number of marbles with me is 83⅓% of the marbles Ariba has” — or,

m
equivalently, “I have 16⅔% fewer marbles than Ariba.”
as e
.co a g l
s m have n marbles. Then Ariba has 1.2n.
eArun
gl a
Let
a Arun as a fraction of Ariba = n / 1.2n = 1/1.2 = 10/12 = 5/6
m a s
.co agl
5/6 × 100 = 500/6 = 83.33%

se m
g l a
a
Shortfall = 100% – 83.33% = 16.67% (exactly 16⅔%)

c o m
.
m = 83.33%, and
s e
A quick check with numbers: if Arun has 50 marbles, Ariba has 60. Then 50/60

.
Arun has
com
10 fewer, which is 10/60 = 16.67% of Ariba's.
a gla
a s em
agl Why it happens: The tempting answer, “Arun has 20% fewer marbles than Ariba”, is
wrong — and the reason is the base. Ariba's extra 10 marbles are 20% of Arun's 50,
se m
com
but only 16.67% of Ariba's 60. The same gap, measured against two different wholes,
g l a
m . a
ase
gives two different percentages. Reversing a percentage comparison is never as

agl
simple as changing the sign: if A is (1 + r) times B, then B is 1/(1 + r) times A, and
1/1.2 is not 0.8.

co m
m .
as e
. comit Out — Pages 28–30 a g l
em
Figure
l a s
ag c
m .
m a s e
e m . co agl
g l as
a

com
m .
m ase
.co


a g l Page 54 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Section 1.3 Using Percentages — chapter-end set

TRY THIS

Q1 The population of Bengaluru in 2025 is about 250% of its population in 2000. If the
population in 2000 was 50 lakhs, what is the population in 2025?

Population in 2025 = 250% of 50 lakhs

= 2.5 × 50 lakhs

= 125 lakhs = 1.25 crore

Tip: 250% means “two and a half times”. The increase alone is 150%, i.e. 75 lakhs —
the city grew by 75 lakh people in 25 years.

Q2 The population of the world in 2025 is about 8.2 billion. The populations of some
countries in 2025 are given. Match them with their approximate percentage share
of the worldwide population. [Hint: Writing these numbers in the standard form
and estimating can help]. [Germany 83 million, India 1.46 billion, Bangladesh 175
million, USA 347 million; options 13%, 8%, 18%, 10%, 1%, 35%, 2%, 2%, 0.1%.]

Write every population in billions first, then divide by 8.2.

COUNTRY POPULATION IN SHARE OF 8.2 OPTION
BILLIONS BILLION

Germany 83 million 0.083 0.083/8.2 = 1.01% 1%

India 1.46 billion 1.46 1.46/8.2 = 17.8% 18%

Bangladesh 175 million 0.175 0.175/8.2 = 2.13% 2%

USA 347 million 0.347 0.347/8.2 = 4.23% no exact match — nearest
is 2%

Page 55 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

Estimating instead of dividing:

1% of 8.2 billion = 82 million → Germany, at 83 million, is almost exactly 1%

10% of 8.2 billion = 820 million; India's 1460 million is nearly twice that → about 18%
Bangladesh's 175 million is a little over 2 × 82 million → about 2%

USA's 347 million is a little over 4 × 82 million → about 4%

Note: The option list has no 4% card, so the USA's share of about 4.2% cannot be
matched properly — the closest of the nine options is the second 2%. Take the USA's
share as approximately 4%; the remaining options (13%, 8%, 10%, 35%, 0.1%) are
distractors.

Why it happens: The hint about standard form matters because the figures are
given in two different units. Comparing 83 million with 8.2 billion directly invites a slip
of a factor of 1000. Once everything is in billions the arithmetic is easy, and the
benchmark “1% of the world is 82 million people” lets you read off every answer by
eye.

Q3 The price of a mobile phone is ₹8,250. A GST of 18% is added to the price. Which of
the following gives the final price of the phone including the GST? (i) 8250 + 18 (ii)
8250 + 1800 (iii) 8250 + 18/100 (iv) 8250 × 18 (v) 8250 × 1.18 (vi) 8250 + 8250 × 0.18 (vii)
1.8 × 8250

(v) and (vi) — both give the correct final price.

GST = 18% of 8250 = 0.18 × 8250 = ₹1485

Final price = 8250 + 1485 = ₹9735

(v) 8250 × 1.18 = 9735 ✓

(vi) 8250 + 8250 × 0.18 = 8250 + 1485 = 9735 ✓

Page 56 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

OPTION VALUE WHAT IS WRONG

(i) 8250 + 18 8268 adds ₹18, not 18%

(ii) 8250 + 1800 10,050 18% of 8250 is 1485, not 1800

(iii) 8250 + 18/100 8250.18 adds 0.18 rupees

(iv) 8250 × 18 1,48,500 18 times the price

(vii) 1.8 × 8250 14,850 an 80% tax, not 18%

Why it happens: (v) and (vi) are the same calculation written two ways — 8250 ×
1.18 is 8250 × (1 + 0.18), which the distributive law expands into 8250 + 8250 × 0.18.
Option (vii) is the sharpest trap: 1.8 and 1.18 look alike but mean 180% and 118% of
the price. The decimal for a percentage always needs two places shifted — 18% is
0.18, so the multiplier is 1.18.

Q4 The monthly percentage change in population (compared to the previous month) of
mice in a lab is given: Month 1 change was +5%, Month 2 change was –2%, and
Month 3 change was –3%. Which of the following statement(s) are true? The initial
population is p. (i) The population after three months was p × 0.05 × 0.02 × 0.03. (ii)
The population after three months was p × 1.05 × 0.98 × 0.97. (iii) The population
after three months was p + 0.05 – 0.02 – 0.03. (iv) The population after three months
was p. (v) The population after three months was more than p. (vi) The population
after three months was less than p.

(ii) and (vi) are true.

Page 57 of 72

Class 8 Maths Chapter 8 Fractions in Disguise AglaSem · NCERT Solutions

+5% → multiply by 1.05

–2% → multiply by 0.98

–3% → multiply by 0.97

Population after 3 months = p × 1.05 × 0.98 × 0.97

1.05 × 0.98 = 1.029

1.029 × 0.97 = 0.99813

= 0.99813p, which is less than p

STATEMENT VERDICT REASON

(i) False 0.05 × 0.02 × 0.03 uses the changes, not the multipliers

(ii) True each change multiplies the running population

(iii) False adds bare decimals to a population count

(iv) False 0.99813p ≠ p

(v) False 0.99813p < p

(vi) True the population ends 0.187% below where it started

Why it happens: +5, –2 and –3 add up to zero, which is why option (iv) looks so
plausible. But percentage changes are multiplications, not additions, and each one is
measured on a different population. The +5% was applied to p, while the –2% and –
3% were applied to larger populations — so the falls remove slightly more mice than
the rise put in. The net effect, ×0.99813, leaves the colony just under where it began.

Q5 A shopkeeper initially set the price of a product with a 35% profit margin. Due to
poor sales, he decided to offer a 30% discount on the selling price. Will he make a
profit or a loss? Give reasons for your answer.

He makes a loss of 5.5%.

Page 58 of 72

as e
Class 8 Maths Chapter 8 Fractions in Disguise
a g l AglaSem · NCERT Solutions

co m
e m.
Let the cost price be x.
m l as
.co
Marked price with a 35% margin = 1.35x
m a g
l a se
g
After a 30% discount, he receives 70% of that:
a0.70 × 1.35x = 0.945x

co m
em . ag
0.945x < x, so it is a loss
g l as
Loss = x – 0.945x = 0.055x a

co m
m.
Loss % = 0.055x / x × 100 = 5.5%

o m l a se
g ₹405, sale price ₹945
.c if the cost price were ₹1000: marked price ₹1350, discount
a
m
With numbers,
e of ₹55, which is 5.5% of ₹1000.
— asloss
a
ag l
s
Why it happens: The 35% and the 30% are measured on different bases. The 35% is
m a
.co
added to the cost price; the 30% is taken off the marked price, which is larger. So the
em agl
a s
discount removes more rupees than the margin added — 30% of ₹1350 is ₹405,
gl For the shopkeeper to break even, the discount
while the margin was onlya₹350.
would have to be 35/135 = 25.9% at most.

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 59 of 72