NCERT Solutions Class 7 Maths Chapter 3 a Peek Beyond the Point

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 7 · M AT H S

NCERT Solutions

Chapter 3: A Peek Beyond the
Point

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

46 – 80 31 99 English

Solutions, notes, sample papers & more at 89 pages

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

CLASS 7 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 3: A Peek Beyond the Point
Chapter 3 of Ganita Prakash Grade 7 takes the Indian place value system one step to the right of the units
place. Splitting a unit into 10 one-tenths, and each tenth into 10 one-hundredths, gives us the decimal point
— and with it a way to measure a screw, a hair or a heap of rice as exactly as we like.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 7) 46 – 80

SECTIONS QUESTIONS

31 99

MEDIUM

English

In-text Questions — Pages 46–47

Page 1 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.1 The Need for Smaller Units

Q1 In the following figure, screws are placed above a scale. Measure them and write
their length in the space provided.

0 1 2 3 0 1 2 3

Between 2 cm and 3 cm

0 1 2 3 0 1 2 3

More than 2 ½ cm but less than 3 cm

0 1 2 3 0 1 2 3

2 ⁷⁄₁₀ cm

Page 47 — the same three scales, marked in whole centimetres, in half centimetres and
in tenths. The screw on the left is measured for you; the screw on the right is the one to
measure. Redrawn sketch.

The same screw is shown three times, above three different scales. The screw on the left is 2 7⁄10

cm long and the screw on the right is 3 2⁄10 cm long. What you can write depends on how finely
the scale is marked.

SCALE USED LEFT SCREW (GIVEN) RIGHT SCREW (TO FILL)

Only whole centimetres Between 2 cm and 3 cm Between 3 cm and 4 cm
marked

Half centimetres marked More than 2 1⁄2 cm but less than 3 More than 3 cm but less than 3 1⁄2
cm cm

Tenths of a centimetre marked 2 7⁄10 cm 3 2⁄10 cm

Page 2 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: the tip of the right-hand screw falls 2 small divisions past the 3 cm
mark. On the finest scale each small division is 1⁄10 cm, so the reading is 3 cm and 2
tenths — that is, 3 2⁄10 cm.

Tip: the difference between the two screws is 3 2⁄10 – 2 7⁄10 = 5⁄10 cm, i.e. only half a
centimetre. That tiny gap was enough to stop Sonu's mother from fixing the toy.

Q2 Which scale helped you measure the length of the screws accurately? Why?

The third scale — the one in which each centimetre is divided into 10 equal parts.

Why it happens: the tip of the screw does not land on a whole-centimetre mark. On
the first scale we can only say "between 2 cm and 3 cm". The second scale, with half-
centimetre marks, narrows it to "more than 2 1⁄2 cm but less than 3 cm". Only the
third scale has a mark exactly where the tip is, so it gives the exact length 2 7⁄10 cm.

Tip: the smaller the unit on the scale, the more accurate the measurement. That is
the whole idea behind this chapter.

Q3 What is the meaning of 2 7/10 cm (the length of the first screw)?

It means 2 whole centimetres and 7 one-tenths of a centimetre.

On the ruler, the length between two consecutive numbers is cut into 10 equal parts.

Each part = 1⁄10 cm

Go from 0 to 2 → 2 cm

Then take 7 more parts → 7 × 1⁄10 = 7⁄10 cm

Total = 2 cm + 7⁄10 cm

Page 3 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
Why it happens: a mixed number like 2 7⁄10 is just "2 units plus 7 of the small parts".

as e
. c om7 27
The same length can also be written as 27 one-tenths, because 2 cm itself is 20 one-
a g l
tenths:m ⁄10 + ⁄10 = ⁄10.
se
20

l a
ag
m
Read it as: "two and seven-tenth centimeters". Similarly 3 2⁄10 cm is read "three and
. co ag
two-tenth centimeters".
e m
g l as
a

co m
m.
Q4 Can you explain why the unit was divided into smaller parts to measure the screws?

m as e
ANSWER .co
a g l
a s emthe screws did not end exactly at a centimetre mark.
agl
Because

m
Why it happens: a measuring unit can only report lengths that are whole multiples
a s
m .co
of itself. With centimetres alone, both screws would be reported as "about 3 cm" and agl
l a se
would look identical — yet one was 2 7⁄10 cm and the other 3 2⁄10 cm. Cutting the
ag
centimetre into 10 equal parts gives a smaller unit (1⁄10 cm = 1 mm), and now the two
screws get different, exact readings.
co m
m .
o m l a se
.cyou know? The very same trouble made Sonu's mother's a g toy refuse to hold
m
Did

asetogether. In machines, a 5⁄10 cm error in a screw can be the difference between a
agl working part and a broken one.
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

com
m .
m ase
.co


a g l Page 4 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q5 Measure the following objects using a scale and write their measurements in
centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and
any other object of your choice.

0 1 2 3 0 1 2 3

Between 2 cm and 3 cm

0 1 2 3 0 1 2 3

More than 2 ½ cm but less than 3 cm

0 1 2 3 0 1 2 3

2 ⁷⁄₁₀ cm

Page 47 — the screws measured earlier above the three scales, showing how a length is
written.

This is a hands-on activity — do it with your own ruler. Line up the left end of the object with the
0 mark (not with the edge of the ruler), then read off the mark at the other end.

OBJECT SAMPLE READING IN TENTHS READ AS

Pen 13 5⁄10 cm 135⁄ thirteen and five-tenth centimeters
10 cm

Sharpener 2 4⁄10 cm 24⁄ two and four-tenth centimeters
10 cm

Eraser 3 8⁄10 cm 38⁄ three and eight-tenth centimeters
10 cm

Page 5 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Check it yourself: your own pen will not give exactly 13 5⁄10 cm — write down
whatever your ruler shows. Always give the answer in the form "whole centimetres +
tenths".

Q6 Write the measurements of the objects shown in the picture:

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

Page 48 — three objects, each with its left end at 0 of a scale marked in tenths of a
centimetre. Redrawn sketch.

Each object is drawn with its left end at 0, so read the mark at the right end.

OBJECT LENGTH SAME LENGTH IN TENTHS READ AS

Eraser 2 4⁄10 cm 24⁄ two and four-tenth centimeters
10 cm

Pencil 4 5⁄10 cm 45⁄ four and five-tenth centimeters
10 cm

Piece of chalk 1 4⁄10 cm 14⁄ one and four-tenth centimeters
10 cm

Page 6 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: the pencil ends exactly halfway between 4 cm and 5 cm. Five
tenths is half of ten tenths, so 4 5⁄10 cm is the same as 4 1⁄2 cm.

In-text Questions — Page 49

Page 7 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.2 A Tenth Part

Q1 For the objects shown below, write their lengths in two ways and read them aloud.
An example is given for the USB cable. (Note that the unit length used in each
diagram is not the same). The length of the USB cable is 4 and 8/10 units or 48/10
units.

(a)

0 1 2 3 4 5

(b) (c)

3

2

1

0
0 1

(d)

0 1 2 3 4 5 6 7 8 9 10 11

Page 49 — four objects, each against its own scale (the unit length is not the same in the
four diagrams); every unit is cut into 10 equal parts. Redrawn sketch.

Each object is drawn against its own scale, and each scale has every unit cut into 10 equal parts.
Read the mark at the far end.

Page 8 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
OBJECT AS UNITS AND AS READ ALOUD AS

as e
com
TENTHS TENTHS

. a g l
s em
USB cable
a
4 and 8⁄10 units 48⁄
10 units
four and eight-tenths / forty-eight

agl
(example) tenths

Pencil box (height) 2 and 8⁄10 units 28⁄ two and eight-tenths / twenty-eight
m
10 units

.co ag
tenths

a sem ⁄ units
agl
Fist (width) 1 and 3⁄10 units 13
10
one and three-tenths / thirteen tenths

Leaf 10 and 9⁄10 units 109⁄ ten and nine-tenths / one hundred nine
m
10 units

co
m.
tenths

o m l a se
Why it.chappens: one whole unit is 10 one-tenths. So 4 units a =g40 tenths, and 4 units
e m
l +s8 tenths = 48 tenths. Going the other way, 109⁄10 = 100⁄10 + 9⁄10 = 10 + 9⁄10.
a
a g
m a s
.co agl
Check it yourself: the unit length is different in each picture, so do not compare the

se m
a
four objects with one another — compare each one only with its own scale.

a g l

co m
Arrange these lengths in increasing order: (a) 9/10 (b) 1 7/10 (c) 130/10 (d) 13 1/10 (e)
m .
e
Q2

m l as
.co
10 5/10 (f) 7 6/10 (g) 6 7/10 (h) 4/10
a g
a s em
agl ANSWER

m
The quickest way is to turn every length into tenths and compare the numerators.

a se
. com a g l
m
ase
LENGTH (A) (B) (C) (D) (E) (F) (G) (H)

In tenths 9
ag17l 130 131 105 76 67 4

co m
m .
e
Increasing order of the numerators: 4 < 9 < 17 < 67 < 76 < 105 < 130 < 131
m l as
m4.co
So the answer is
a g
l a se
ag ⁄10 , 9⁄10 , 1 7⁄10 , 6 7⁄10 , 7 6⁄10 , 10 5⁄10 , 130⁄10 , 13 1⁄10

.c
i.e. (h), (a), (b), (g), (f), (e), (c), (d)
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 9 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: once every length is written with the same denominator 10,
comparing them is exactly like comparing whole numbers. Notice the trap in (c) and
(d): 130⁄10 = 13 while 13 1⁄10 = 131⁄10, so (d) is the larger of the two.

In-text Questions — Page 50
Section 3.2 A Tenth Part

Q1 Arrange the following lengths in increasing order: 4 1/10, 4/10, 41/10, 41 1/10.

Write all four in tenths.

4 1⁄10 = 41⁄10 → 41 tenths
4⁄
10 → 4 tenths
41⁄
10 → 41 tenths

41 1⁄10 = 411⁄10 → 411 tenths

Increasing order: 4⁄10 , 41⁄10 = 4 1⁄10 , 41 1⁄10

Why it happens: 4 1⁄10 and 41⁄10 are the same number written two ways, so they sit
at the same place on the number line. Only 4⁄10 and 41 1⁄10 are different.

Tip: the position of the ‘41’ matters a lot — 41 tenths is a little more than 4, but 41
units and a tenth is ten times bigger.

Q2 Sonu is measuring some of his body parts. The length of Sonu’s lower arm is 2 7/10
units, and that of his upper arm is 3 6/10 units. What is the total length of his arm?

Add the units to the units and the tenths to the tenths.

Page 10 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

(2 + 3) units and (7 + 6) tenths

= 5 units and 13 tenths

13 tenths = 1 unit and 3 tenths
= 5 + 1 units and 3 tenths

= 6 3⁄10 units

The same sum done entirely in tenths:

27⁄ 36⁄ 63⁄
10 + 10 = 10

= 60⁄10 + 3⁄10

= 6 + 3⁄10 = 6 3⁄10 units

Why it happens: whenever the tenths add up to 10 or more, we exchange 10 tenths
for 1 unit — exactly the same "carry" we use when 10 ones make a ten.

In-text Questions — Page 51

Page 11 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.2 A Tenth Part

MATH TALK

Q1 The lengths of the body parts of a honeybee are given. Find its total length. Head: 2
3/10 units, Thorax: 5 4/10 units, Abdomen: 7 5/10 units

Head Thorax Abdomen

Page 51 — a honeybee, with its head, thorax and abdomen marked. Schematic drawing.

Add the three lengths.

Units: 2 + 5 + 7 = 14

Tenths: 3 + 4 + 5 = 12

12 tenths = 1 unit and 2 tenths

Total = 14 + 1 units and 2 tenths

= 15 2⁄10 units

In tenths:

23⁄ 54⁄ 75⁄ 152⁄ 2
10 + 10 + 10 = 10 = 15 ⁄10 units

Page 12 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: 152 tenths = 150 tenths + 2 tenths = 15 units + 2 tenths, because
every 10 tenths make one whole unit.

Q2 The length of Shylaja’s hand is 12 4/10 units, and her palm is 6 7/10 units, as shown
in the picture. What is the length of the longest (middle) finger?

12 ⁴⁄₁₀ units

6 ⁷⁄₁₀ units

Page 51 — the outer rectangle spans the whole hand, from fingertip to wrist; the inner
one spans only the palm. Redrawn sketch.

Finger = hand – palm.

Page 13 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
(12 + 4⁄10) – (6 + 7⁄10)
e m.
om7 tenths from 4 tenths, so split one unit from 12:
We cannot.ctake g l as
m a
4⁄ se= 11 units and 14 tenths
12la
ag 10
14 tenths – 7 tenths = 7 tenths

co m
11 units – 6 units = 5 units
e m . ag
g l as
= 5 7⁄10 units
a

co m
m.
Doing it entirely in tenths:

as e
o⁄m = 57⁄ = 5 7⁄ units
124⁄ –.c67 a g l
s e m 10 10 10
gla
10

a
m
Why it happens: borrowing here is the same as borrowing in whole numbers. One
a s
m .co
unit is worth 10 tenths, so 12 units and 4 tenths can be re-read as 11 units and 14 agl
tenths without changing the value.
l a se
ag

co m
m .
Q3 Discuss what is being done here and why.
m as e
.co a g l
a s em
gl

a The two columns show the same subtraction 12 4⁄10 – 6 7⁄10, before and after regrouping.

se m
com
In the left column, the tenths line reads 4 – 7. Since 4 tenths is not enough, the subtraction
g l a
m . a
cannot start.
So one unit is taken from 12la
e
s changed into 10 tenths. Now 12 units 4 tenths is rewritten
g
a number has not changed, only the way it is split.
as 11 units 14 tenths — the
and

m
In the right column the tenths now read 14 – 7 = 7, and the units read 11 – 6 = 5, giving 5 7⁄10.
co
m .
m as e
.co a g l
Why it happens: we always subtract starting from the smallest place value. When

se m the top digit there is too small, we exchange one of the next bigger unit for ten of
g l a
a the smaller — exactly the "borrowing" used for 124 – 67 in whole numbers.
c
m .
m a s e
m . co
Check it yourself: add back — 5 7⁄10 + 6 7⁄10 = 11 14⁄10 = 12 4⁄10. The answer checks
e agl
out.
g l as
a

co m
m .
m ase
.co


a g l Page 14 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

In-text Questions — Page 52
Section 3.2 A Tenth Part

Q1 Try computing the difference by converting both lengths to tenths.

Change 12 4⁄10 and 6 7⁄10 into tenths first.

12 4⁄10 = 120⁄10 + 4⁄10 = 124⁄10

6 7⁄10 = 60⁄10 + 7⁄10 = 67⁄10
124⁄ 67⁄ 124 – 67⁄ 57⁄
10 – 10 = 10 = 10

= 50⁄10 + 7⁄10 = 5 7⁄10 units

Why it happens: once both numbers use the same unit (one-tenth), subtracting
them is an ordinary whole-number subtraction, 124 – 67 = 57. The denominator just
tags along.

Tip: this method needs no borrowing between "units" and "tenths" — all the
borrowing happens inside the plain subtraction 124 – 67.

Q2 A Celestial Pearl Danio’s length is 2 4/10 cm, and the length of a Philippine Goby is
9/10 cm. What is the difference in their lengths?

2 4⁄10 – 9⁄10

= 24⁄10 – 9⁄10

= 15⁄10

= 10⁄10 + 5⁄10

= 1 5⁄10 cm (that is, 1.5 cm)

Page 15 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: 9 tenths cannot be taken from 4 tenths, so one whole unit of the
Danio's length is opened into 10 tenths: 2 4⁄10 becomes 1 unit and 14 tenths. Then 14
– 9 = 5 tenths, leaving 1 5⁄10 cm.

Did you know? The Philippine Goby, at about 9⁄10 cm, is one of the smallest fish in
the world — shorter than the width of your little fingernail.

Q3 How big are these fish compared to your finger?

Measure your own index finger with a ruler and compare. A Class 7 student's index finger is
usually about 6 to 7 cm long.

FISH LENGTH COMPARED WITH A 6 CM FINGER

Celestial Pearl Danio 2 4⁄10 cm about 2⁄5 of the finger — a little less than half

Philippine Goby 9⁄
10 cm less than 1⁄6 of the finger — about the width of a fingernail

Check it yourself: mark 2 4⁄10 cm and 9⁄10 cm on a strip of paper and hold it against
your finger. Write down the comparison for your finger — the answer will be
different for everyone.

Q4 Observe the given sequences of numbers. Identify the change after each term and
extend the pattern: (a) 4, 4 3/10, 4 6/10, ___ (b) 8 2/10, 8 7/10, 9 2/10, ___ (c) 7 6/10, 8
7/10, ___ (d) 5 7/10, 5 3/10, ___ (e) 13 5/10, 13, 12 5/10, ___ (f) 11 5/10, 10 4/10, 9 3/10, ___

Find the change from one term to the next, then keep applying it four more times.

Page 16 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

CHANGE NEXT FOUR TERMS

(a) 4, 4 3⁄10, 4 6⁄10 add 3⁄10 4 9⁄10, 5 2⁄10, 5 5⁄10, 5 8⁄10

(b) 8 2⁄10, 8 7⁄10, 9 2⁄10 add 5⁄10 9 7⁄10, 10 2⁄10, 10 7⁄10, 11 2⁄10

(c) 7 6⁄10, 8 7⁄10 add 1 1⁄10 9 8⁄10, 10 9⁄10, 12, 13 1⁄10

(d) 5 7⁄10, 5 3⁄10 subtract 4⁄10 4 9⁄10, 4 5⁄10, 4 1⁄10, 3 7⁄10

(e) 13 5⁄10, 13, 12 5⁄10 subtract 5⁄10 12, 11 5⁄10, 11, 10 5⁄10

(f) 11 5⁄10, 10 4⁄10, 9 3⁄10 subtract 1 1⁄10 8 2⁄10, 7 1⁄10, 6, 4 9⁄10

Why it happens: the easiest way to run each pattern is to work in tenths. For (c), 76,
87, 98, 109, 120, 131 tenths — the numerator simply goes up by 11 each time, and
120⁄
10 is the whole number 12.

Tip: when the tenths reach 10, exchange them for one unit. In (f), 6 1⁄10 – 1 1⁄10 would
come after 6, but the term before 6 is 7 1⁄10, so the next term is exactly 6, and then 4
9⁄
10.

Q5 The length of a sheet of paper was 8 9/10 units, which can also be said as 8 units
and 9 one-tenths. It is folded in half along its length. What is its length now?

Folding in half means taking half of 8 9⁄10.

8 9⁄10 = 89⁄10 units

Half of 89⁄10 = 89⁄20 = 44.5⁄10

= 4 units + 4 tenths + half a tenth

= 4 4⁄10 5⁄100 units

Page 17 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: half of 8 units is 4 units; half of 9 tenths is 4 tenths and half a tenth
left over. A tenth-scale has no mark for half a tenth, so on that scale we can only say
the folded length lies between 4 4⁄10 and 4 5⁄10. Splitting each tenth into 10 equal
parts gives one-hundredths, and half a tenth is exactly 5⁄100.

Tip: this is exactly why the book moves on to hundredths in Section 3.3 — tenths
were not fine enough for this fold.

In-text Questions — Page 53
Section 3.3 A Hundredth Part

MATH TALK

Q1 What is the length of this smaller part? How many such smaller parts make a unit
length?

Each one-tenth has been cut into 10 equal parts.

1 unit = 10 one-tenths

1 one-tenth = 10 smaller parts

So 1 unit = 10 × 10 = 100 smaller parts

Length of one smaller part = 1⁄100 of a unit (one-hundredth)

Number of such parts in a unit = 100

Why it happens: cutting each of the 10 tenths into 10 pieces gives 10 groups of 10
pieces, i.e. 100 pieces in all. Each piece is therefore one-hundredth of the whole unit.

Did you know? On your ruler, 1 cm is divided into 10 mm. If each mm were divided
into 10 again, each tiny part would be 1⁄100 cm — about the thickness of a human
hair.

Page 18 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
How many one-hundredths make one-tenth? Can we also say that the length is 4
e
Q2

m l as
.co
units and 45 one-hundredths?

a g
se m
g l a
a

co m
1⁄ 10⁄

ag
10 =
.
100

em
So 10 one-hundredths make one-tenth.
g l as
a
Yes — the folded paper measures 4 4⁄10 5⁄100, and
co m
em.
m l as
.co a g
4⁄ 40
10 = ⁄100
m
40s⁄ e + 5⁄
a
gl 100 100 = ⁄100
45
a
s
Length = 4 units and 45 one-hundredths = 4 45⁄100
m a
em
.co agl
a s
a gl 5 hundredths" and "45 hundredths" describe the
Why it happens: "4 tenths and
same amount, just grouped differently — like saying "4 tens and 5 ones" or "45
ones".
co m
m .
o m l a se
g 4.45.
.cin decimal notation all three readings are the same number:
a
m
ase
Tip:

agl
se m
In-text Questions — Pages 54–55om
c g l a
m . a
a s e
agl

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

com
m .
m ase
.co


a g l Page 19 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.3 A Hundredth Part

Q1 Observe the figure below. Notice the markings and the corresponding lengths
written in the boxes when measured from 0. Fill the lengths in the empty boxes.

0 ¹⁄₁₀ ²⁄₁₀ 1 1 ³⁄₁₀ 2

¹⁄₁₀₀ ²⁰⁄₁₀₀ ⁹⁹⁄₁₀₀ ¹³⁰⁄₁₀₀

Page 54 — every long mark is one-tenth and every short mark is one-hundredth. Four
lengths are already written below the line; the five empty boxes are the ones to fill.

Every big mark on this scale is one-tenth and every small mark is one-hundredth. The boxes
below the line name the same marking in hundredths.

MARKING (IN TENTHS) SAME MARKING (IN HUNDREDTHS)

0 + 1 small mark 1⁄
100 (given)

2⁄ 20⁄
10 100 (given)

halfway between 5⁄10 and 6⁄10 55⁄
100

1 small mark before 1 99⁄
100 (given)

1 3⁄10 130⁄
100 (given)

1 5⁄10 + 5 small marks 155⁄
100

1 7⁄10 + 4 small marks 174⁄
100

2 + 2 small marks 202⁄
100

2 4⁄10 240⁄
100

Page 20 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

The five empty boxes, in order, are
55⁄ 155⁄ 174⁄ 202⁄ 240⁄
100 , 100 , 100 , 100 , 100

Why it happens: a whole unit is 100 small marks, so counting small marks from 0
gives the numerator straight away. For example 2 units and 2 small marks = 200 + 2
= 202 hundredths.

Q2 The length of the wire in the first picture is given in three different ways. Can you
see how they denote the same length? 1 1/10 4/100 — One and one-tenth and four-
hundredths; 1 14/100 — One and fourteen-hundredths; 114/100 — One Hundred and
Fourteen-hundredths

0 ¹⁄₁₀ ²⁄₁₀ ³⁄₁₀ ⁴⁄₁₀ ⁵⁄₁₀ ⁶⁄₁₀ ⁷⁄₁₀ ⁸⁄₁₀ ⁹⁄₁₀ 1 1 ¹⁄₁₀ 1 ²⁄₁₀

Page 54 — the wire laid along a scale whose unit is cut into tenths and each tenth into
hundredths. Redrawn sketch.

Yes. All three are 114 one-hundredths, only grouped differently.

1 1⁄10 4⁄100 = 1 + 10⁄100 + 4⁄100 = 1 + 14⁄100 = 1 14⁄100

1 14⁄100 = 100⁄100 + 14⁄100 = 114⁄100

All three = 1.14

Why it happens: 1 unit = 100 hundredths and 1 tenth = 10 hundredths. So we can
keep exchanging bigger pieces for hundredths without changing the length — just
as ₹1 and 14 paise, 114 paise, and ₹1.14 are the same money.

Page 21 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Tip: the third reading is the safest to check with: count the small marks from 0 and
you will count 114.

Q3 For the lengths shown below write the measurements and read out the measures in
words.

4 ⁹⁄₁₀ 5 5 ¹⁄₁₀ 5 ²⁄₁₀ 5 ³⁄₁₀ 5 ⁴⁄₁₀ 5 ⁵⁄₁₀ 5 ⁶⁄₁₀ 5 ⁷⁄₁₀ 5 ⁸⁄₁₀ 5 ⁹⁄₁₀ 6 6 ¹⁄₁₀

14 ⁴⁄₁₀ 14 ⁵⁄₁₀ 14 ⁶⁄₁₀ 14 ⁷⁄₁₀ 14 ⁸⁄₁₀ 14 ⁹⁄₁₀ 15 15 ¹⁄₁₀ 15 ²⁄₁₀ 15 ³⁄₁₀ 15 ⁴⁄₁₀ 15 ⁵⁄₁₀ 15 ⁶⁄₁₀

6 ⁷⁄₁₀ 6 ⁸⁄₁₀ 6 ⁹⁄₁₀ 7 7 ¹⁄₁₀ 7 ²⁄₁₀ 7 ³⁄₁₀ 7 ⁴⁄₁₀ 7 ⁵⁄₁₀ 7 ⁶⁄₁₀ 7 ⁷⁄₁₀ 7 ⁸⁄₁₀ 7 ⁹⁄₁₀

8 ⁹⁄₁₀ 9 9 ¹⁄₁₀ 9 ²⁄₁₀ 9 ³⁄₁₀ 9 ⁴⁄₁₀ 9 ⁵⁄₁₀ 9 ⁶⁄₁₀ 9 ⁷⁄₁₀ 9 ⁸⁄₁₀ 9 ⁹⁄₁₀ 10 10 ¹⁄₁₀

Pages 54–55 — four wires, each laid along a scale marked in tenths with hundredth
marks between. The red stroke on each ruler marks the far end of that wire. Redrawn
sketch. Note: in the printed book the first mark of the fourth ruler reads ‘4 ⁹⁄₁₀’; from the
scale it must be ‘8 ⁹⁄₁₀’.

On each ruler, the numbered marks are tenths and the tiny marks between them are
hundredths. Read where the red line falls.

Page 22 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

RULER MEASUREMENT DECIMAL IN WORDS

First (marks 4 9⁄10 to 6 1⁄10) 5 37⁄100 5.37 Five and thirty-seven hundredths

Second (marks 14 4⁄10 to 15 6⁄10) 15 3⁄100 15.03 Fifteen and three hundredths

Third (marks 6 7⁄10 to 7 9⁄10) 7 52⁄100 7.52 Seven and fifty-two hundredths

Fourth (marks 4 9⁄10 to 10 1⁄10) 9 80⁄100 9.80 Nine and eighty hundredths

Why it happens: in the first ruler the red line is 7 small marks past 5 3⁄10, so the
reading is 5 + 30 hundredths + 7 hundredths = 5 37⁄100. In the fourth ruler the line
lands exactly on the 9 8⁄10 mark, and 8 tenths = 80 hundredths.

Tip: 9 80⁄100 and 9 8⁄10 are the same length — the extra zero at the end changes
nothing.

In-text Questions — Pages 55–56
Section 3.3 A Hundredth Part

Q1 In each group, identify the longest and the shortest lengths. Mark each length on
the scale. (a) 3/10, 3/100, 33/100 (b) 3 1/10, 30/10, 1 3/10 (c) 45/100, 54/100, 5/10, 4/10
(d) 3 6/10, 3 6/100, 3 6/10 6/100 (e) 8/10 2/100, 9/100, 1 8/100 (f) 7 3/10 5/100, 7 5/10, 7
41/100 (g) 65/10 15/100, 5 87/100, 5 7/100

Write every length in hundredths (or as a decimal) and then compare like whole numbers.

Page 23 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
GROUP VALUES IN DECIMAL FORM LONGEST SHORTEST

m0.30, 0.03, 0.33 l a se⁄
.co ag
(a) 33⁄ 3

m
100 100

l a se
a g
(b) 3.1, 3.0, 1.3 3 1⁄10 1 3⁄10

com
(c) 0.45, 0.54, 0.50, 0.40 54⁄ 4⁄

. ag
100 10

s e m
a
agl
(d) 3.60, 3.06, 3.66 3 6⁄ 10
6⁄
100 3 6⁄ 100

(e) 0.82, 0.09, 1.08 1 8⁄100 9⁄

m
100

. c o
m7 ⁄ ⁄
ase
(f) 7.35, 7.50, 7.41 7 5⁄ 3 5

m
agl
10 10 100

. co
m
ase
(g) 6.65, 5.87, 5.07 65⁄ 15⁄ 5 7⁄100

l
10 100

agMarking them on the scale. Count small divisions from the labelled mark:

m a s
.co agl
(a) scale 0 to 1: 3⁄100 is the 3rd small mark, 3⁄10 the 30th, 33⁄100 the 33rd.

a s em
agl0.50 and 0.54.
(b) scale 0 to 10: mark 1.3, 3.0 and 3.1.
(c) scale 0 to 1: mark 0.40, 0.45,
(d) scale 30⁄10 to 40⁄10: 3.06 sits just after 3.0, 3.60 on the 36⁄10 mark, 3.66 six small marks later.

co m
(f) scale 68⁄10 to 78⁄10: mark 7.35, 7.41 and 7.50.
m .
m as e
.co a g l
a s em Why it happens: once all the lengths are converted to the same small unit, the

agl comparison is ordinary counting. In (d) the trap is 3 6⁄ 10 = 3.60 against 3
6⁄
100 = 3.06
— the 6 is worth ten times more in the first one.
se m
com g l a
m . a
ase
agl
Check it yourself: two of the printed scales are too short for their own group — in
(e) the scale runs only from 0 to 1, so 1 8⁄100 falls just beyond its right end, and in (g)
the scale starts at 57⁄10, so 5 7⁄100 falls before its left end. Extend the scale a little in
com
m .
e
your notebook and then mark them.
m l as
m .co a g
l a se
ag In-text Questions — Page 56
.c
s e m
m a
em . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 24 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.3 A Hundredth Part

Q1 What will be the sum of 15 3/10 4/100 and 2 6/10 8/100 ?

Units: 15 + 2 = 17

Tenths: 3 + 6 = 9

Hundredths: 4 + 8 = 12

= 17 + 9⁄10 + 12⁄100

12 hundredths = 1 tenth + 2 hundredths

= 17 + 9⁄10 + 1⁄10 + 2⁄100

= 17 + 10⁄10 + 2⁄100

= 18 2⁄100 (that is, 18.02)

Working entirely in hundredths gives the same answer:

1534⁄ 268⁄ 1802⁄ 1800⁄ 2 2
100 + 100 = 100 = 100 + ⁄100 = 18 ⁄100

Why it happens: two regroupings take place. First 12 hundredths become 1 tenth
and 2 hundredths; then the 10 tenths become 1 whole unit, pushing 17 up to 18.

In-text Questions — Page 57
Section 3.3 A Hundredth Part

MATH TALK

Q1 Are both these methods different?

No. Method 1 and Method 2 do exactly the same steps — Method 2 just writes them in a
column.

Page 25 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

STEP METHOD 1 (EXPANDED) METHOD 2 (COLUMN)

Add each place (15+2) + (3⁄10+6⁄10) + (4⁄100+8⁄100) 17 9⁄10 12⁄100

Regroup hundredths 17 + 9⁄10 + 1⁄10 + 2⁄100 17 10⁄10 2⁄100

Regroup tenths 18 + 2⁄100 18 2⁄100

Why it happens: both rest on the same two facts — 10 hundredths make 1 tenth,
and 10 tenths make 1 unit. Method 1 shows the reasoning; Method 2 is the short
form we normally write.

Page 26 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q2 Observe the addition done below for 483 + 268. Do you see any similarities between
the methods shown above?

(400 + 80 + 3) + (200 + 60 + 8)

= (400 + 200) + (80 + 60) + (3 + 8)

= 600 + 140 + 11

= 600 + 150 + 1

= 700 + 50 + 1

= 751

Page 57 — the addition 483 + 268 as it is worked out in the book.

(a) Method 1

(15 + 2) + (3⁄10 + 6⁄10) + (4⁄100 + 8⁄100)

= 17 + 9⁄10 + 12⁄100

= 17 + 9⁄10 + 1⁄10 + 2⁄100

= 17 + 10⁄10 + 2⁄100

= 18 2⁄100

10 hundredths is the same as 1 tenth.

(b) Method 2

15 3⁄ 4⁄
10 100

+ 2 6⁄ 8⁄
10 100

= 17 9⁄ 12⁄
10 100

= 17 10⁄ 2⁄
10 100

= 18 2⁄
100

Page 27 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Page 56 — the two methods shown just above, for the sum 15 3⁄10 4⁄100 + 2 6⁄10 8⁄100.

Yes — it is the very same procedure, one place value higher.

(400 + 80 + 3) + (200 + 60 + 8)

= 600 + 140 + 11

= 600 + 150 + 1 (10 ones → 1 ten)

= 700 + 50 + 1 (10 tens → 1 hundred)

= 751

WHOLE NUMBERS DECIMAL PARTS

10 ones = 1 ten 10 hundredths = 1 tenth

10 tens = 1 hundred 10 tenths = 1 unit

Add ones first, carry left Add hundredths first, carry left

Why it happens: in the Indian place value system every place is 10 times the place
on its right — and that stays true when we walk right of the units place into tenths
and hundredths. So the same carrying rule works everywhere.

Q3 What is the difference: 25 9/10 – 6 4/10 7/100 ?

The first number has no hundredths, so open one tenth into 10 hundredths first.

25 9⁄10 = 25 8⁄10 10⁄100

Hundredths: 10 – 7 = 3

Tenths: 8 – 4 = 4

Units: 25 – 6 = 19

= 19 4⁄10 3⁄100 (that is, 19.43)

Page 28 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
Why it happens: 25 9⁄10 means 25 units, 9 tenths and 0 hundredths. Since 7

as e
com g l
hundredths cannot be taken from 0, one of the 9 tenths is exchanged for 10

m . a
as e
hundredths, leaving 8 tenths and 10 hundredths.

a g l
Check it yourself: 19.43 + 6.47 = 25.90 ✓

co m
em . ag
g l as
In-text Questions — Page 58 a
Section 3.3 A Hundredth Part
co m
em.
m as
MATH TALK

.co a g l
a s em
gl
Solve this by converting to hundredths.
a
Q1

om a s
agl

. c
The difference asked for is 25 ⁄10 – 6 ⁄10m⁄100. Write both numbers in hundredths.
se
9 4 7

l a
ag
25 9⁄10 = 2500⁄100 + 90⁄100 = 2590⁄100
co m
m .
6 4⁄
e
7⁄ = 600⁄ + 40⁄ + 7⁄ = 647⁄
om– 647⁄100 = 1943⁄100 as
10 100 100 100 100 100

. c⁄100 a g l
m
ase
2590

agl = 1900⁄100 + 40⁄100 + 3⁄100

se m
= 19 4⁄10 3⁄100 (19.43)
com g l a
m . a
ase
agl
Why it happens: converting both numbers to the same unit turns the whole

m
problem into the plain subtraction 2590 – 647 = 1943. The answer is the same as by
the regrouping method, as it must be.
. co
em
m l as
.co a g
a s em
agl What is the difference 15 3/10 4/100 – 2 6/10 8/100 ?
c
Q2

m .
m a s e
m . co agl

as e
l
Regroup twice, then subtract place by place.
a g

co m
m .
m ase
.co


a g l Page 29 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

15 3⁄10 4⁄100

= 15 2⁄10 14⁄100 (1 tenth → 10 hundredths)

= 14 12⁄10 14⁄100 (1 unit → 10 tenths)

Hundredths: 14 – 8 = 6

Tenths: 12 – 6 = 6

Units: 14 – 2 = 12

= 12 6⁄10 6⁄100 (12.66)

In hundredths: 1534⁄100 – 268⁄100 = 1266⁄100 = 12 66⁄100.

Why it happens: 8 hundredths cannot come out of 4 hundredths, and after the first
exchange 6 tenths cannot come out of 2 tenths — so a second exchange is needed.
Both exchanges leave the value untouched.

Check it yourself: 12.66 + 2.68 = 15.34 ✓

Page 30 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q3 Observe the subtraction done below for 653 – 268. Do you see any similarities with
the methods shown above?

(600 + 50 + 3) – (200 + 60 + 8)

= (600 – 200) + (50 – 60) + (3 – 8)

= (600 – 200) + (40 – 60) + (13 – 8)

= (600 – 200) + (40 – 60) + 5

= (500 – 200) + (140 – 60) + 5

= 300 + 80 + 5

= 385

Page 58 — the subtraction 653 – 268 as it is worked out in the book.

3 4 2 14 12 14
15 15 14
10 100 10 100 10 100

6 8 6 8 6 8
−2 −2 −2
10 100 10 100 10 100

6 6
= 12
10 100

Page 58 — the subtraction 15 3⁄10 4⁄100 − 2 6⁄10 8⁄100, worked out just above: one
hundredth-place exchange (red) and then one tenth-place exchange (green).

Yes — it is the same double regrouping, one place value higher.

Page 31 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

(600 + 50 + 3) – (200 + 60 + 8)

= (600 – 200) + (50 – 60) + (3 – 8)

= (600 – 200) + (40 – 60) + (13 – 8) (1 ten → 10 ones)
= (500 – 200) + (140 – 60) + 5 (1 hundred → 10 tens)

= 300 + 80 + 5

= 385

653 – 268 15 3 ⁄10 4 ⁄100 – 2 6 ⁄10 8 ⁄100

Ones short → borrow a ten Hundredths short → borrow a tenth

Tens short → borrow a hundred Tenths short → borrow a unit

Answer 385 Answer 12.66

Why it happens: the borrowing rule only needs one fact — each place is worth 10 of
the place on its right. That fact holds on both sides of the decimal point, so the
procedure never changes.

Figure it Out — Page 58
Section 3.3 A Hundredth Part

Q1 Find the sums and differences: (a) 3/10 + 3 4/100 (b) 9 5/10 7/100 + 2 1/10 3/100 (c) 15
6/10 4/100 + 14 3/10 6/100 (d) 7 7/100 – 4 4/100 (e) 8 6/100 – 5 3/100 (f) 12 6/10 2/100 –
9/10 9/100

Line up units with units, tenths with tenths and hundredths with hundredths.

Page 32 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

IN DECIMAL FORM WORKING ANSWER

(a) 0.30 + 3.04 0 + 3 units, 3 + 0 tenths, 0 + 4 hundredths 3 34⁄100 = 3.34

(b) 9.57 + 2.13 11 units, 6 tenths, 10 hundredths → 11 7⁄10 11 7⁄10 = 11.70

(c) 15.64 + 14.36 29 units, 9 tenths, 10 hundredths → 30 30

(d) 7.07 – 4.04 3 units, 0 tenths, 3 hundredths 3 3⁄100 = 3.03

(e) 8.06 – 5.03 3 units, 0 tenths, 3 hundredths 3 3⁄100 = 3.03

(f) 12.62 – 0.99 borrow twice: 11 15⁄10 12⁄100 – 0 9⁄10 9⁄100 11 63⁄100 = 11.63

Full working for (b):

9 5⁄10 7⁄100 + 2 1⁄10 3⁄100 = 11 6⁄10 10⁄100

10 hundredths = 1 tenth

= 11 7⁄10 0⁄100 = 11 7⁄10

Full working for (c):

15 6⁄10 4⁄100 + 14 3⁄10 6⁄100 = 29 9⁄10 10⁄100

= 29 10⁄10 = 29 + 1 = 30

Why it happens: in (c) the hundredths make a whole tenth and the tenths then
make a whole unit, so the fractional part disappears completely and a clean whole
number 30 is left.

Tip: in (f) it helps to write 9⁄10 9⁄100 as 0.99 first. Then 12.62 – 0.99 = 12.62 – 1 + 0.01 =
11.63 in one mental step.

In-text Questions — Page 59

Page 33 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

Section 3.4 Decimal Place Value
co m
e m.
m l as
Q1
.co a g
Can we not split a unit into 4 equal parts, 5 equal parts, 8 equal parts, or any other
m
l a se
number of equal parts instead?

a g

co m
. ag
Yes, we can. Nothing in mathematics stops us.
e m
g l as
a
1
Split a unit into 4 → each part is ⁄4 (a quarter)

Split each quarter into 4 again → each part is 1⁄16
co m
em.
m l as
.co
16 such parts make 1 unit
a g
se m
g l a
a Why it happens: the same length can be described with any sized part. A

s
carpenter's tape in inches is split into halves, quarters, eighths and sixteenths, and it
m a
.co
measures perfectly well. What changes is only the name of the reading.
m agl
l a se
ag
Did you know? The old Indian rupee was once divided into 16 annas, and an anna

m
into 4 paise — a 16-part and 4-part system, not a 10-part one.
. co
e m
m l as
.co a g
a s em Then why split a unit into 10 parts every time?
gl
Q2

a
se m
com built on 10. l a

Because our number-writing system is .already a g
a s em
10 ones = 1 ten agl

co m
.
10 tens = 1 hundred
em
m l as
.co
10 hundreds = 1 thousand
a g
s e m Each place is 10 times the place to its right
agla
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 34 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: if we split the unit into 10, the new parts continue the same
pattern to the right of the units place — 10 tenths make a unit, 10 hundredths make
a tenth. So the same digits, the same carrying and borrowing, and the same
comparison rules keep working. If we split into 4 or 16 instead, we would need a
whole new set of rules and symbols.

Tip: this is why the system is called the decimal system — from Latin decem, ten, a
cousin of Sanskrit daśha.

In-text Questions — Page 60
Section 3.4 Decimal Place Value

Q1 Can we extend this further?

Yes — endlessly, in both directions.

… 1,00,000 10,000 1000 100 10 1 1⁄10 1⁄100 1⁄1000 1⁄10,000 …

Moving left → × 10 each step

Moving right → ÷ 10 each step

Why it happens: the rule "each place is ten times the next one on its right" does not
care where we start. Applying it again and again to the left gives bigger and bigger
place values; applying it to the right gives smaller and smaller ones, and it never has
to stop.

Page 35 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q2 What will the fraction be when 1/100 is split into 10 equal parts?

1⁄ 1
100 ÷ 10 = ⁄1000 — one-thousandth

1000 such parts make 1 unit

100 such parts make 1 tenth

10 such parts make 1 hundredth

Why it happens: a unit holds 100 hundredths. If each hundredth is cut into 10, the
unit now holds 100 × 10 = 1000 tiny parts, so each one is 1⁄1000 of a unit.

Did you know? One gram is exactly one-thousandth of a kilogram, and one
millimetre is one-thousandth of a metre — thousandths are all around us.

In-text Questions — Page 61

Page 36 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.4 Decimal Place Value — How Big? / Notation, Writing and Reading of Numbers

MATH TALK

Q1 We can ask similar questions about fractional parts: (a) How many thousandths
make one unit? (b) How many thousandths make one tenth? (c) How many
thousandths make one hundredth? (d) How many tenths make one ten? (e) How
many hundredths make one ten?

QUESTION WORKING ANSWER

(a) thousandths in one unit 1 ÷ 1⁄1000 1000

(b) thousandths in one tenth 1⁄ 100⁄ 100
10 = 1000

(c) thousandths in one hundredth 1⁄ 10⁄ 10
100 = 1000

(d) tenths in one ten 10 ÷ 1⁄10 = 10 × 10 100

(e) hundredths in one ten 10 ÷ 1⁄100 = 10 × 100 1000

Why it happens: every step you move to the right on the place value chart
multiplies the count by 10. From "ten" to "tenth" is two steps (ten → one → tenth), so
the count is 10 × 10 = 100. From "ten" to "hundredth" is three steps, giving 1000.

Tip: count the steps between the two place values on the chart. The answer is always
1 followed by that many zeros.

Q2 Make a few more questions of this kind and answer them.

Here are some you can use, with their answers. Then write five of your own for a friend.

Page 37 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

QUESTION STEPS ON THE CHART ANSWER

How many hundredths make one unit? 2 100

How many thousandths make one hundred? 5 1,00,000

How many tenths make one hundred? 3 1000

How many ten-thousandths make one tenth? 3 1000

How many hundredths make one thousand? 5 1,00,000

Try This: turn it around — "One hundredth is what part of one ten?" Answer: 1⁄1000.

Q3 Can the quantity 4 2/10 be written as 42 (skipping the 1/10 in 2 × 1/10)? If yes, how
would we know if 42 means 4 tens and 2 units or it means 4 units and 2 tenths?

No, we cannot. Writing it as 42 destroys the information.

4 tens and 2 units = 40 + 2 = 42

4 units and 2 tenths = 4 + 2⁄10 = 4.2

These are completely different quantities — one is ten times the other and more.

The same problem shows up with 705, which could mean:

7 hundreds, 0 tens and 5 ones = 700 + 0 + 5 = 705
7 tens, 0 units and 5 tenths = 70 + 0 + 5⁄10 = 70.5

7 units, 0 tenths and 5 hundredths = 7 + 0 + 5⁄100 = 7.05

Why it happens: in place value the meaning of a digit comes from where it sits. Once
we allow fractional places, the string of digits alone no longer tells us where the
units place ends. So we mark it with a separator — the decimal point — and write
4.2, 70.5, 7.05.

In-text Questions — Page 63

Page 38 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

Section 3.4 Decimal Place Value
co m
e m.
m l as
.co
Make a place value table similar to the one above. Write each quantity in decimal
a g
emand in terms of place value, and read the number: (a) 2 ones, 3 tenths and 5
Q1

a s
form

a gl hundredths (b) 1 ten and 5 tenths (c) 4 ones and 6 hundredths (d) 1 hundred, 1 one
and 1 hundredth (e) 8/100 and 9/10 (f) 5/100 (g) 1/10 (h) 2 1/100, 4 1/10 and 7 7/1000

co m
m . ag
se

l a
HUNDREDS TENS ONES ag
TENTHS HUNDREDTHS THOUSANDTHS DECIMAL READ
AS

co m
m.
(a) – – 2 3 5 – 2.35 two

e
m las
point

.co ag
three

sem
five

a
agl
(b) – 1 0 5 – – 10.5 ten
point

a s
five

m
m.co agl
se
(c) – – 4 0 6 – 4.06 four

l a
point

ag zero six

(d) 1 0 1 0 1 – 101.01 one

m
.co
hundred

em
one

m a s
point

.co ag l zero

sem
one

a
agl – – 0 9 8 – zero
(e) 0.98

m
point

se
nine

com g l a
.
eight

m a
ase
agl
(f) – – 0 0 5 – 0.05 zero
point
zero five

m
.co
(g) – – 0 1 – – 0.1 zero

em
point

m a s
one

. co ag l
se m (h) 2 – – 2 0 1 – 2.01 two

l a point
1

g
⁄100

a
zero

.c
m
one

m a s e
.co agl
(h) 4 – – 4 1 – – 4.1 four

se m
1
⁄10 point

l a
one

a g

co m
m .
m as e
.co


a g l Page 39 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

(h) 7 – – 7 0 0 7 7.007 seven
7
⁄1000 point
zero
zero
seven

In place value form:

(a) 2 × 1 + 3 × 1⁄10 + 5 × 1⁄100 = 2.35

(b) 1 × 10 + 5 × 1⁄10 = 10.5

(d) 1 × 100 + 1 × 1 + 1 × 1⁄100 = 101.01

(h) 7 × 1 + 7 × 1⁄1000 = 7.007

Why it happens: the zeros are doing real work. In (c) the 0 in the tenths place is
what keeps the 6 in the hundredths place — writing 4.6 instead of 4.06 would make
the number ten times bigger.

Tip: read the fractional part digit by digit — 7.007 is "seven point zero zero seven",
never "seven point seven".

In-text Questions — Page 64
Section 3.5 Units of Measurement — Length Conversion

Q1 Write these quantities in decimal form: (a) 234 hundredths, (b) 105 tenths.

(a) 234 hundredths = 234⁄100

= 200⁄100 + 30⁄100 + 4⁄100

= 2 + 3⁄10 + 4⁄100

= 2.34

Page 40 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

(b) 105 tenths = 105⁄10

= 100⁄10 + 5⁄10

= 10 + 5⁄10

= 10.5

Why it happens: this is the same trick as "23 hundreds = 2300". Break the
numerator into the largest possible groups of 100 (or 10), turn each group into a
whole, and let the leftover stay as the fractional part.

Tip: a quick shortcut — dividing by 100 moves every digit two places to the right, so
234 becomes 2.34.

Q2 How many cm is 1 mm?

1 cm = 10 mm

So 1 mm = 1⁄10 cm = 0.1 cm

Why it happens: if 10 equal millimetres fill one centimetre, then each millimetre
must be one-tenth of a centimetre. The millimetre marks on your ruler are the tenth-
marks of this chapter.

Page 41 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q3 How many cm is (a) 5 mm? (b) 12 mm?

(a) 5 mm = 5⁄10 cm = 0.5 cm

(b) 12 mm = 10 mm + 2 mm

= 1 cm + 2⁄10 cm

= 1.2 cm

Why it happens: once 10 mm are collected they make a whole centimetre, so 12
mm is 1 whole centimetre with 2 tenths left over.

Tip: to change mm into cm, just divide by 10 — that is, shift the decimal point one
place to the left.

Q4 How many mm is 5.6 cm?

5.6 cm = 5 cm + 0.6 cm

5 cm = 5 × 10 = 50 mm

0.6 cm = 6⁄10 cm = 6 mm

Total = 50 + 6 = 56 mm

Why it happens: going from cm to mm we multiply by 10, which shifts the decimal
point one place to the right: 5.6 → 56.

In-text Questions — Page 65

Page 42 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.5 Units of Measurement — Length Conversion

Q1 Fill in the blanks below (mm <–> cm): 12 mm = 1.2 cm; 56 mm = 5.6 cm; 70 mm =
_______; ________ = 0.9 cm; 134 mm = __________; ________ = 203.6 cm

GIVEN WORKING ANSWER

12 mm 12 ÷ 10 1.2 cm (given)

56 mm 56 ÷ 10 5.6 cm (given)

70 mm 70 ÷ 10 7.0 cm

? = 0.9 cm 0.9 × 10 9 mm

134 mm 134 ÷ 10 13.4 cm

? = 203.6 cm 203.6 × 10 2036 mm

Why it happens: 1 mm = 1⁄10 cm. So mm → cm means divide by 10 (decimal point
one step left) and cm → mm means multiply by 10 (decimal point one step right).

Tip: 7.0 cm and 7 cm are the same length — the zero after the point adds nothing,
but it does remind you that the measurement was made to the nearest millimetre.

Page 43 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
The illustration below shows how small some things are! Try taking an approximate
e
Q2

m l as
.co
measurement of each.

a g
se m
g l a
a

co m
e m . ag
l as
1.5 mm

a g

co m
m.
1 mm

m as e
.co a g l
se m
g l a
a
0.5 mm

m a s
m .co agl
ase
Actual size

a g l
Page 65 — three blue strokes stand for a bold, a medium and a fine pen stroke (1.5 mm,

m
1 mm and 0.5 mm thick), shown beside mustard seeds, the edge of a sheet of paper, a

. co
m
land snail, an ant and a human hair. Redrawn sketch.

m as e
.co a g l
a s em
agl ANSWER

se m
These are all smaller than one millimetre or just about one millimetre, so they must be
described using tenths and hundredths of a millimetre.
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
em . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 44 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

OBJECT SIZE AS PRINTED SAME SIZE IN CM

Fine pen stroke 0.5 mm 0.05 cm

Medium pen stroke 1 mm 0.1 cm

Bold pen stroke 1.5 mm 0.15 cm

Human hair (thickness) about 0.1 mm 0.01 cm

Newspaper (thickness) 0.05 to 0.08 mm 0.005 to 0.008 cm

Mustard seed (thickness) 1 – 2 mm 0.1 – 0.2 cm

Smallest ant, Carabera Bruni 0.8 – 1 mm 0.08 – 0.1 cm

Smallest land snail, Acmella Nana (shell) 0.7 mm 0.07 cm

Why it happens: a ruler's smallest mark is 1 mm, so none of these can be measured
directly. We say a hair is about 0.1 mm because roughly 10 hairs side by side span
one millimetre — the tenth again, one level smaller.

Try This: stack 20 sheets of a newspaper, measure the stack with a ruler, and divide
by 20. If the stack is 1.4 mm, one sheet is 1.4 ÷ 20 = 0.07 mm.

In-text Questions — Page 66

Page 45 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.5 Units of Measurement — Length Conversion

Q1 How many m is (a) 10 cm? (b) 15 cm?

1 m = 100 cm, so 1 cm = 1⁄100 m = 0.01 m

(a) 10 cm = 10⁄100 m = 1⁄10 m = 0.1 m

(b) 15 cm = 15⁄100 m

= 10⁄100 m + 5⁄100 m

= 1⁄10 m + 5⁄100 m

= 0.15 m

Why it happens: each centimetre is one-hundredth of a metre, so a number of
centimetres is that many hundredths of a metre. 15 hundredths regroups as 1 tenth
and 5 hundredths, which is written 0.15.

Page 46 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q2 Fill in the blanks below (cm <–> m): 36 cm = _______; 50 cm = _______; _______ = 0.89 m; 4
cm = _______; 325 cm = _______; ________ = 2.07 m

GIVEN WORKING ANSWER

36 cm 36⁄ 0.36 m
100 m

50 cm 50⁄ 0.5 m
100 m

? = 0.89 m 0.89 × 100 89 cm

4 cm 4⁄ 0.04 m
100 m

325 cm 325⁄ 25⁄ 3.25 m
100 m = 3 m + 100 m

? = 2.07 m 2.07 × 100 207 cm

Why it happens: cm → m divides by 100 (decimal point two steps left) and m → cm
multiplies by 100 (two steps right). Note 4 cm = 0.04 m, not 0.4 m — the zero in the
tenths place must be written.

Q3 How many mm does 1 meter have?

1 m = 100 cm

1 cm = 10 mm

1 m = 100 × 10 = 1000 mm

Why it happens: the metre is split into 100 centimetres and each of those into 10
millimetres, giving 100 groups of 10 — one thousand millimetres in all.

Page 47 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q4 Can we write 1 mm = 1/1000 m?

Yes.

1 m = 1000 mm

So 1 mm = 1⁄1000 m = 0.001 m

Why it happens: a thousand equal millimetres fill one metre, so each one is one-
thousandth of a metre. That is exactly the third decimal place, which is why 1 mm is
written 0.001 m.

Tip: the prefix "milli-" itself means one-thousandth: millimetre, milligram, millilitre.

In-text Questions — Pages 67–68
Section 3.5 Units of Measurement — Weight Conversion

Q1 How many kilograms is 5 g?

1 kg = 1000 g, so 1 g = 1⁄1000 kg = 0.001 kg

5 g = 5⁄1000 kg = 0.005 kg

Why it happens: 5 thousandths has no tenths and no hundredths, so both those
places must be filled with zeros: 0.005 kg.

Page 48 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
How many kilograms is 10 g?
e
Q2

m l as
m .co a g
a se

a g l
10 g = 10⁄1000 kg

co m
= 1⁄100 kg
e m . ag
g l as
= 0.010 kg = 0.01 kg
a

co m
Why it happens: 10 thousandths regroup into 1 hundredth, exactly as 10
em.
m l as
.co g
hundredths regroup into 1 tenth. The trailing zero in 0.010 may be kept or dropped
m a
se
— the value is the same.

g l a
a
m a s
.co
Fill in the blanks below (g <–> kg): 465 g = _______; 68 g = _________; 1560 g = ________; 704
m agl
se
Q3

l a
g = _______; ________ = 0.56 kg; _______ = 2.5 kg

ag

. om
cANSWER
e m
as
GIVEN WORKING

465 g.c
om a g l
m
ase
465⁄ 0.465 kg
1000 kg

agl 68 g 68⁄ 0.068 kg
m
1000 kg

a se
. co+m ⁄ a g l
em
1560 g 1000 g + 560 g = 1 kg 560 1.56 kg
1000 kg

a s
704 g 704⁄
agkgl
1000
0.704 kg

m
.co
? = 0.56 kg 0.56 × 1000 560 g

se m
com l a
? = 2.5 kg 2.5 × 1000 2500 g

. a g
m
ase
agl Sample working for 254 g:
c
m .
a s e
om
254 g = 254⁄1000 kg = (200⁄1000 + 50⁄1000 + 4⁄1000) kg

e
. c
m kg agl
= (2⁄ + 5⁄ + 4⁄
a s
1000) kg = 0.254

agl
10 100

co m
m .
m ase
.co


a g l Page 49 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Why it happens: g → kg divides by 1000, i.e. shifts the decimal point three places
left. That is why 68 g becomes 0.068 kg and not 0.68 kg — a zero must hold the
tenths place.

Did you know? In the rice picture the heaps are 1 g, 10 g, 100 g, 1000 g (1 kg) and
10000 g (10 kg). Together they weigh 11.111 kg — one digit for each place value!

In-text Questions — Page 69
Section 3.5 Units of Measurement — Rupee ─ Paise Conversion

TRY THIS

Q1 Fill in the blanks below (rupee <–> paise): 10 p = ___________; _______p = ₹ 0.05; ________p =
₹ 0.36; _________ = ₹ 0.50; 99 p = _________; 250 p = _________

GIVEN WORKING ANSWER

10 p 10⁄ ₹0.10
100 rupee

? = ₹0.05 0.05 × 100 5p

? = ₹0.36 0.36 × 100 36 p

? = ₹0.50 0.50 × 100 50 p

99 p 99⁄ ₹0.99
100 rupee

250 p 250⁄ 50⁄ ₹2.50
100 rupee = 2 rupees + 100

Why it happens: 1 rupee = 100 paise, so 1 paisa = 1⁄100 rupee = ₹0.01. The paise are
simply the two digits after the decimal point — which is why every price tag has
exactly two decimal places.

Tip: 75 paise = 75⁄100 = 7⁄10 + 5⁄100 = ₹0.75 — 7 ten-paise coins and 5 single paise.

Page 50 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q2 Discuss with adults at home/school the prices of different products and services
during their childhood. Try to find old coins and stamps.

This is a project to do at home. Note down each price in rupees and in paise, then compare with
today's price.

ITEM PRICE IN THE 1970S IN RUPEES ROUGHLY TODAY

Masala dosa 50 paise ₹0.50 ₹50 – ₹80

One banana 20 – 25 paise ₹0.20 – ₹0.25 ₹5 – ₹10

A handful of peppermints 2 – 3 paise ₹0.02 – ₹0.03 ₹5

1 kg rice ₹2.45 ₹2.45 ₹50 – ₹70

Why it happens: all coins of 25 paise and below were withdrawn in 2011 because
they could no longer buy anything. Paise survive today only inside written amounts
— bills, bank statements and petrol prices.

Try This: ask an elder what their first salary was and write it in rupees and paise.
Then work out how many masala dosas it could have bought in the 1970s, and how
many it would buy today.

In-text Questions — Page 70

Page 51 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Section 3.6 Locating and Comparing Decimals / There is Zero Dilemma!

Q1 Name all the divisions between 1 and 1.1 on the number line.

1.04

1 1.1 1.2 1.3

Page 70 — the stretch of the number line from 1 to 1.3, with the unit from 1 to 1.1 cut
into 10 equal parts.

The gap from 1 to 1.1 is one tenth, and it has been cut into 10 equal parts. Each part is therefore
one hundredth.

The nine marks in between are

1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09

Why it happens: 1 = 1.00 and 1.1 = 1.10. Counting in hundredths from 1.00 to 1.10
gives 1.01, 1.02, … , 1.09 in between — nine marks, because 10 parts have 9 dividing
points.

Tip: the arrow in the figure points at the 4th mark, which is 1.04 — one unit, zero
tenths and four hundredths.

Q2 Identify and write the decimal numbers against the letters.

The big marks are tenths (5, 5.1, 5.2, 5.3, 5.4) and the small marks between them are
hundredths.

Page 52 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

LETTER POSITION DECIMAL NUMBER

A 9 small marks after 5 5.09

B 3 small marks after 5.1 5.13

C 10 small marks after 5.1 5.2

D 1 small mark after 5.3 5.31

Why it happens: each small division is 1⁄100 = 0.01. So counting small marks from
the nearest labelled tenth gives the hundredths digit directly. A is just short of 5.1,
which is why it reads 5.09 and not 5.9.

Q3 Sonu says that 0.2 can also be written as 0.20, 0.200; Zara thinks that putting zeros
on the right side may alter the value of the decimal number. What do you think?

Sonu is right. Zeros added at the right end of a decimal do not change its value.

DECIMAL UNITS TENTHS HUNDREDTHS THOUSANDTHS VALUE
NUMBER

0.2 0 2 – – 2⁄
10

0.20 0 2 0 – 20⁄ 2
100 = ⁄10

0.200 0 2 0 0 200⁄
1000 =
2⁄
10

0.02 0 0 2 – 2⁄
100

0.002 0 0 0 2 2⁄
1000

Why it happens: a zero written after the last digit only says "and no hundredths",
"and no thousandths" — it adds nothing. But a zero written before the 2 pushes the 2
into a smaller place, dividing it by 10 each time. That is why 0.2, 0.02 and 0.002 are
all different.

Page 53 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
Tip: ₹0.2 and ₹0.20 are both 20 paise, but ₹0.02 is only 2 paise.

m as e
.co a g l
a s em
agl Questions — Page 71
In-text
Section 3.6 Locating and Comparing Decimals

co m
e m . ag
Q1
l as
Can you tell which of these is the smallest and which is the largest?
g
a
m

co
Comparing 0.2, 0.20, 0.200, 0.02 and 0.002:
em.
m l as
m .co a g
l a se
0.2 = 0.20 = 0.200 = 200⁄1000
ag
0.02 = 20⁄1000

m a s
0.002 = 2⁄1000
m .co agl
l a se
a
Order: 0.002 < 0.02 < 0.2 = 0.20 = 0.200
g
Largest: 0.2 (= 0.20 = 0.200) Smallest: 0.002

co m
m .
m as e
.co
Why it happens: writing all five with three decimal places makes them easy to
a g l
a s em compare — 200, 20 and 2 thousandths. The largest is ten times the next one and a

agl hundred times the smallest.
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 54 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q2 Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50?

NUMBER UNITS TENTHS HUNDREDTHS THOUSANDTHS

4.5 4 5 0 0

4.05 4 0 5 0

0.405 0 4 0 5

4.050 4 0 5 0

4.50 4 5 0 0

4.005 4 0 0 5

04.50 4 5 0 0

4.5 = 4.50 = 04.50

4.05 = 4.050

0.405 and 4.005 are different from all the others.

Why it happens: zeros at the two ends — before the first digit or after the last — are
only decoration. Zeros between digits are not: the 0 in 4.05 keeps the 5 in the
hundredths place, and the two 0s in 4.005 push the 5 all the way to the thousandths
place.

Page 55 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q3 Identify the decimal number in the last number line in Figure (b) denoted by ‘?’.

0 10 0 10

4 5

4.1 4.2

4.18 4.185 4.19 ?

(a) (b)

Page 71 — Figure (a) on the left and Figure (b) on the right. At each level the boxed part
of the number line is magnified to make the next line.

Follow the magnifications one level at a time.

Level 1: the box lies between 3 and 4 → the number starts with 3
Level 2 (3 to 4): the box lies between 3.0 and 3.1 → 3.0…

Level 3 (3.0 to 3.1): the box lies between 3.05 and 3.06 → 3.05…

Level 4 (3.05 to 3.06): the arrow is on the 9th of 10 marks

? = 3.05 + 9 × 0.001 = 3.059

Why it happens: each magnification cuts the previous gap into 10 equal parts, so
each level fixes one more decimal digit — first the units, then the tenths, then the
hundredths, then the thousandths.

Tip: compare with Figure (a), where the same four levels zoom in on 4.185: 4 → 4.1
→ 4.18 → 4.185.

Page 56 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q4 Make such number lines for the decimal numbers: (a) 9.876 (b) 0.407.

Draw four number lines, each one magnifying a part of the one above it.

LEVEL (A) FOR 9.876 (B) FOR 0.407

1 0 to 10, mark the part between 9 and 10 0 to 10, mark the part between 0 and 1

2 9 to 10 in tenths, mark 9.8 to 9.9 0 to 1 in tenths, mark 0.4 to 0.5

3 9.8 to 9.9 in hundredths, mark 9.87 to 9.88 0.4 to 0.5 in hundredths, mark 0.40 to 0.41

4 9.87 to 9.88 in thousandths, arrow on the 0.40 to 0.41 in thousandths, arrow on the
6th mark → 9.876 7th mark → 0.407

zoom into 9 – 10

0 10
zoom into 9.8 – 9.9

9 10
9.876

9.87 9.88
Each line magnifies the marked part of the line above it.

Zooming in on 9.876 — the same four-step magnification used in the book's Figure (a).

Why it happens: a decimal number is located by trapping it between two marks,
then splitting that gap into 10 and trapping it again. Each split reveals the next digit.

Page 57 of 89

Class 7 Maths Chapter 3 A Peek Beyond the Point AglaSem · NCERT Solutions

Q5 In the number line shown below, what decimal numbers do the boxes labelled ‘a’,
‘b’, and ‘c’ denote? The box with ‘b’ corresponds to the decimal number 7.5; are you
able to see how? … What numbers do ‘a’ and ‘c’ denote?

5 10

a b c

Page 71 — the stretch of the number line between 5 and 10 divided into 10 equal parts,
with three marks named a, b and c.

There are 5 units between 5 and 10, and the gap is cut into 10 equal parts.

Size of each division = 5 ÷ 10 = 1⁄2 unit = 0.5

a is at the 2nd division → 5 + 2 × 0.5 = 6

b is at the 5th division → 5 + 5 × 0.5 = 7.5 ✓

c is at the 9th division → 5 + 9 × 0.5 = 9.5

Why it happens: the number of divisions does not have to match the number of
units. Here every two divisions make one unit, so each single division is half a unit —
0.5. Always work out the size of one division before reading any mark.

Check it yourself: from 5, counting 0.5 at a time gives 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5,
10 — ten steps, landing exactly on 10.

Page 58 of 89

as e
Class 7 Maths Chapter 3 A Peek Beyond the Point
a g l AglaSem · NCERT Solutions

co m
m.
In-text Questions — Page 72
m as e
.co
Section 3.6 Locating and Comparing Decimals
a g l
a s em
aQ1gl Using similar reasoning find out the decimal numbers in the boxes below.

com
. ag

e m
l as
First find the size of one division on each number line, then count divisions.
g
First line (8 to 8.1, 10 divisions): a

co m
em.
Size of one division = 0.1 ÷ 10 = 0.01
m l as
.co
d is at the 1st division → 8 + 0.01 = 8.01
m a g
esiseat the 5th division → 8 + 5 × 0.01 = 8.05
a
agl
m a s
agl
Second line (4.3 to 4.8, 10 divisions):

m.co
l a se
Size of one division = 0.5 ÷ 10 = 0.05
ag
f is at the 1st division → 4.3 + 0.05 = 4.35

co m
g is at the 4th division → 4.3 + 4 × 0.05 = 4.5
m .
m as e
.co
h is at the 11th division → 4.3 + 11 × 0.05 = 4.85
a g l
se m
g l a
a Why it happens: the marks look identical on both lines, but they stand for different
se m
omthat must be worked out every time.
amounts — 0.01 on the first and 0.05 on the second. The value of a division is
(labelled gap) ÷ (number of parts),.cand g l a
em a
a s
agl
Tip: h lies one division beyond the 4.8 mark, so it is 4.8 + 0.05 = 4.85.

co m
m .
as e
. comWhich is larger: 6.456 or 6.465? a g l
sem
Q2

a
agl c
m .
e

m a s
co agl
6.465 is larger. Compare digit by digit, starting from the highest place value.

m .
as e
a g l

co m
m .
m ase
.co


a g l Page 59 of 89