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NCERT Solutions · Class 7 Maths · Ganita Prakash Part 1 · Chapter 3

Chapter 3: A Peek Beyond the Point (Decimals)

Step-by-step answers to every "Figure it Out" and in-text question of Chapter 3, A Peek Beyond the Point (NCERT Class 7 Maths, Ganita Prakash Part 1, 2026-27): tenths and hundredths, decimal place value, unit conversions, locating and comparing decimals on number lines, addition and subtraction of decimals, decimal sequences and estimation. All 55 questions are answered, with the key answer highlighted.

3.1 The Need for Smaller Units

Measure the second set of screws with the three scales and write their lengths. Which scale helped you measure the lengths accurately, and why?

Solution

Readings for the second screw on the three scales:

Scale marked only in cm: between 3 cm and 4 cm
Scale marked in half-cm: more than 3 cm but less than $3 \frac{1}{2}$ cm
Scale marked in tenths of a cm: $3 \frac{2}{10}$ cm

The third scale (each cm divided into 10 equal parts) helped us measure accurately, because its smaller divisions let us say exactly where the screw ends instead of only "between" two marks.

Between 3 and 4 cm; more than 3 but less than 3½ cm; 3 2/10 cm. The scale with each cm split into 10 parts gives the exact length.

What is the meaning of

2 \frac{7}{10}

cm? Why was the unit divided into smaller parts to measure the screws?

Solution

$2 \frac{7}{10}$ cm means 2 whole cm and 7 parts out of 10 equal parts of the next cm, i.e. 2 cm and seven one-tenths of a cm.

The two screws differ by less than 1 cm, so whole centimetres could not show the difference. Dividing a centimetre into smaller equal parts lets us measure lengths that are not whole numbers of cm.

2 cm and 7 tenths of a cm; the unit was split because the lengths (and their difference) were less than a whole cm apart.

Write the measurements of the objects shown in the picture (eraser, pencil, chalk).

Solution

Eraser: $2 \frac{4}{10}$ cm; pencil: $4 \frac{5}{10}$ cm; chalk: $1 \frac{4}{10}$ cm.

(Measure your own pen and sharpener the same way, e.g. pen $13 \frac{6}{10}$ cm, sharpener $2 \frac{5}{10}$ cm.)

Eraser 2 4/10 cm, pencil 4 5/10 cm, chalk 1 4/10 cm.

3.2 A Tenth Part

Write the lengths of the objects in two ways and read them aloud (USB cable: 4 and 8/10 units or 48/10 units).

Solution

Object	Units and tenths	Only tenths
USB cable	$4 \frac{8}{10}$ units (four and eight-tenths)	$\frac{48}{10}$ (forty-eight tenths)
Pencil box (height)	$2 \frac{8}{10}$ units	$\frac{28}{10}$
Thumb	$1 \frac{3}{10}$ units	$\frac{13}{10}$
Leaf	$10 \frac{9}{10}$ units	$\frac{109}{10}$

(The readings are taken from the pictures; small differences of one tenth are acceptable.)

USB 4 8/10 = 48/10; box about 2 8/10 = 28/10; thumb about 1 3/10 = 13/10; leaf about 10 9/10 = 109/10 units.

Arrange in increasing order: (a) 9/10 (b)

1 \frac{7}{10}

13 \frac{1}{10}

(e)

10 \frac{5}{10}

(f)

7 \frac{6}{10}

(g)

6 \frac{7}{10}

(h) 4/10

Solution

Note $\frac{130}{10} = 13$ .

$\frac{4}{10} < \frac{9}{10} < 1 \frac{7}{10} < 6 \frac{7}{10} < 7 \frac{6}{10} < 10 \frac{5}{10} < \frac{130}{10} < 13 \frac{1}{10}$

4/10, 9/10, 1 7/10, 6 7/10, 7 6/10, 10 5/10, 130/10, 13 1/10

Arrange in increasing order:

4 \frac{1}{10}

\frac{4}{10}

\frac{41}{10}

41 \frac{1}{10}

Solution

$\frac{41}{10} = 4 \frac{1}{10}$ . So $\frac{4}{10} < 4 \frac{1}{10} = \frac{41}{10} < 41 \frac{1}{10}$ .

4/10 < 4 1/10 = 41/10 < 41 1/10

The lengths of a honeybee's body parts are: head

2 \frac{3}{10}

, thorax

5 \frac{4}{10}

, abdomen

7 \frac{5}{10}

units. Find its total length.

Solution

$(2 + 5 + 7) + (\frac{3}{10} + \frac{4}{10} + \frac{5}{10}) = 14 + \frac{12}{10} = 14 + 1 + \frac{2}{10} = 15 \frac{2}{10}$ units

15 2/10 units

Shylaja's hand is

12 \frac{4}{10}

units and her palm

6 \frac{7}{10}

units. Find the length of the middle finger. Try computing by converting both to tenths.

Solution

$12 \frac{4}{10} - 6 \frac{7}{10}$ : split 1 unit of 12 into 10 tenths: $11 \frac{14}{10} - 6 \frac{7}{10} = 5 \frac{7}{10}$ .

In tenths: $\frac{124}{10} - \frac{67}{10} = \frac{57}{10} = 5 \frac{7}{10}$ units.

(In method (a) of the book, $6 - \frac{3}{10}$ is rewritten as $5 + 1 - \frac{3}{10} = 5 + \frac{10}{10} - \frac{3}{10} = 5 \frac{7}{10}$ : one unit is broken into ten tenths so that 3 tenths can be taken away.)

5 7/10 units

A Celestial Pearl Danio is

2 \frac{4}{10}

cm long and a Philippine Goby

\frac{9}{10}

cm. What is the difference in their lengths? How big are they compared to your finger?

Solution

$2 \frac{4}{10} - \frac{9}{10} = \frac{24}{10} - \frac{9}{10} = \frac{15}{10} = 1 \frac{5}{10}$ cm

A finger is about 6–7 cm long, so the Danio is about a third of a finger and the Goby is smaller than a fingernail.

1 5/10 cm; both fish are much smaller than a finger.

Identify the change after each term and extend the pattern: (a) 4,

4 \frac{3}{10}

4 \frac{6}{10}

, … (b)

8 \frac{2}{10}

8 \frac{7}{10}

9 \frac{2}{10}

, … (c)

7 \frac{6}{10}

8 \frac{7}{10}

, … (d)

5 \frac{7}{10}

5 \frac{3}{10}

, … (e)

13 \frac{5}{10}

, 13,

12 \frac{5}{10}

, … (f)

11 \frac{5}{10}

10 \frac{4}{10}

9 \frac{3}{10}

, …

Solution

(a) Add $\frac{3}{10}$ : $4 \frac{9}{10}$ , $5 \frac{2}{10}$ , $5 \frac{5}{10}$ , $5 \frac{8}{10}$

(b) Add $\frac{5}{10}$ : $9 \frac{7}{10}$ , $10 \frac{2}{10}$ , $10 \frac{7}{10}$ , $11 \frac{2}{10}$

(d) Subtract $\frac{4}{10}$ : $4 \frac{9}{10}$ , $4 \frac{5}{10}$ , $4 \frac{1}{10}$ , $3 \frac{7}{10}$

(e) Subtract $\frac{5}{10}$ : $12$ , $11 \frac{5}{10}$ , $11$ , $10 \frac{5}{10}$

(f) Subtract $1 \frac{1}{10}$ : $8 \frac{2}{10}$ , $7 \frac{1}{10}$ , $6$ , $4 \frac{9}{10}$

(a) +3/10 (b) +5/10 (c) +1 1/10 (d) −4/10 (e) −5/10 (f) −1 1/10; terms as above.

3.3 A Hundredth Part

A sheet

8 \frac{9}{10}

units long is folded in half. What is its length now? How many one-hundredths make one-tenth? Can we say the length is 4 units and 45 one-hundredths?

Solution

Half of $8 \frac{9}{10}$ = half of $\frac{890}{100}$ = $\frac{445}{100} = 4 \frac{4}{10} \frac{5}{100}$ units.

10 one-hundredths make one-tenth ( $\frac{1}{10} = \frac{10}{100}$ ). So 4 tenths and 5 hundredths $= \frac{40}{100} + \frac{5}{100} = \frac{45}{100}$ , and yes, the length is 4 units and 45 one-hundredths.

4 units, 4 tenths and 5 hundredths = 4 45/100 units; 10 hundredths make a tenth, so yes.

Fill the lengths in the empty boxes on the ruler (marked from 0 in tenths and hundredths).

Solution

Reading the pointers from left to right:

Box	1st	2nd	3rd	4th	5th
Length	$\frac{55}{100}$	$1 \frac{55}{100}$ $(\frac{155}{100})$	$1 \frac{74}{100}$ $(\frac{174}{100})$	$2 \frac{2}{100}$ $(\frac{202}{100})$	$2 \frac{4}{10}$ $(\frac{240}{100})$

About 55/100, 1 55/100, 1 74/100, 2 2/100 and 2 40/100 (= 2 4/10).

The wire's length is written as

1 \frac{1}{10} \frac{4}{100}

1 \frac{14}{100}

and

\frac{114}{100}

. Can you see how they denote the same length?

Solution

$1 \frac{1}{10} \frac{4}{100} = 1 + \frac{10}{100} + \frac{4}{100} = 1 + \frac{14}{100} = \frac{100}{100} + \frac{14}{100} = \frac{114}{100}$ .

One tenth is 10 hundredths, and 1 unit is 100 hundredths, so all three describe 114 hundredths.

1 + 1/10 + 4/100 = 1 + 14/100 = 114/100 (one tenth = 10 hundredths, one unit = 100 hundredths).

For the lengths shown below, write the measurements and read them in words.

Solution

Ruler	Measurement	In words
1	$5 \frac{37}{100}$	five and thirty-seven hundredths
2	$15 \frac{3}{100}$	fifteen and three hundredths
3	$7 \frac{52}{100}$	seven and fifty-two hundredths
4	$9 \frac{80}{100}$ $(= 9 \frac{8}{10})$	nine and eighty hundredths

5 37/100; 15 3/100; 7 52/100; 9 80/100

In each group, identify the longest and the shortest lengths: (a) 3/10, 3/100, 33/100 (b)

3 \frac{1}{10}

, 30/10,

1 \frac{3}{10}

3 \frac{6}{10}

3 \frac{6}{100}

3 \frac{6}{10} \frac{6}{100}

(e)

\frac{8}{10} \frac{2}{100}

, 9/100,

1 \frac{8}{100}

(f)

7 \frac{3}{10} \frac{5}{100}

7 \frac{5}{10}

7 \frac{41}{100}

(g)

\frac{65}{10} \frac{15}{100}

5 \frac{87}{100}

5 \frac{7}{100}

Solution

Write each in hundredths to compare.

	In hundredths	Longest	Shortest
(a)	30, 3, 33	33/100	3/100
(b)	310, 300, 130	$3 \frac{1}{10}$	$1 \frac{3}{10}$
(c)	45, 54, 50, 40	54/100	4/10
(d)	360, 306, 366	$3 \frac{6}{10} \frac{6}{100}$	$3 \frac{6}{100}$
(e)	82, 9, 108	$1 \frac{8}{100}$	9/100
(f)	735, 750, 741	$7 \frac{5}{10}$	$7 \frac{3}{10} \frac{5}{100}$
(g)	665, 587, 507	$\frac{65}{10} \frac{15}{100}$ (= 6.65)	$5 \frac{7}{100}$

Longest/shortest: (a) 33/100, 3/100 (b) 3 1/10, 1 3/10 (c) 54/100, 4/10 (d) 3 6/10 6/100, 3 6/100 (e) 1 8/100, 9/100 (f) 7 5/10, 7 3/10 5/100 (g) 65/10 15/100, 5 7/100

Are the two methods of adding

15 \frac{3}{10} \frac{4}{100} + 2 \frac{6}{10} \frac{8}{100}

different? Do you see similarities with 483 + 268 = (400 + 200) + (80 + 60) + (3 + 8)?

Solution

No, both methods do the same thing: add units to units, tenths to tenths and hundredths to hundredths, then regroup (10 hundredths make 1 tenth, 10 tenths make 1 unit). This is exactly what we do in 483 + 268, where 11 ones become 1 ten and 1 one, and 15 tens become 1 hundred and 5 tens. The sum is $18 \frac{2}{100}$ .

They are the same: add place by place and regroup tens, just like 483 + 268.

Find

25 \frac{9}{10} - 6 \frac{4}{10} \frac{7}{100}

by converting to hundredths. Also, what is the similarity between

15 \frac{3}{10} \frac{4}{100} - 2 \frac{6}{10} \frac{8}{100}

and 653 − 268?

Solution

$\frac{2590}{100} - \frac{647}{100} = \frac{1943}{100} = 19 \frac{4}{10} \frac{3}{100}$

In both subtractions, when a place has too few (4 hundredths < 8 hundredths, or 3 ones < 8 ones), we break one from the next higher place into 10 of this place: $15 \frac{3}{10} \frac{4}{100} - 2 \frac{6}{10} \frac{8}{100} = 12 \frac{6}{10} \frac{6}{100}$ , just as $653 - 268 = 385$ .

1943/100 = 19 4/10 3/100; both use borrowing (one of a higher place = 10 of the next lower place).

Figure it Out (page 58)

Find the sums and differences: (a)

\frac{3}{10} + 3 \frac{4}{100}

(b)

9 \frac{5}{10} \frac{7}{100} + 2 \frac{1}{10} \frac{3}{100}

(c)

15 \frac{6}{10} \frac{4}{100} + 14 \frac{3}{10} \frac{6}{100}

(d)

7 \frac{7}{100} - 4 \frac{4}{100}

(e)

8 \frac{6}{100} - 5 \frac{3}{100}

(f)

12 \frac{6}{10} \frac{2}{100} - \frac{9}{10} \frac{9}{100}

Solution

(a) $3 + \frac{3}{10} + \frac{4}{100} = 3 \frac{3}{10} \frac{4}{100} = 3 \frac{34}{100}$

(b) $(9 + 2) + \frac{6}{10} + \frac{10}{100} = 11 + \frac{6}{10} + \frac{1}{10} = 11 \frac{7}{10}$

(d) $(7 - 4) + \frac{7 - 4}{100} = 3 \frac{3}{100}$

(e) $(8 - 5) + \frac{6 - 3}{100} = 3 \frac{3}{100}$

(f) $\frac{1262}{100} - \frac{99}{100} = \frac{1163}{100} = 11 \frac{6}{10} \frac{3}{100}$

(a) 3 34/100 (b) 11 7/10 (c) 30 (d) 3 3/100 (e) 3 3/100 (f) 11 63/100

3.4 Decimal Place Value

How Big? (a) How many thousandths make one unit? (b) one tenth? (c) one hundredth? (d) How many tenths make one ten? (e) How many hundredths make one ten?

Solution

(a) 1000 (b) 100 (c) 10 (d) 100 (e) 1000

More questions: How many tenths make a hundred? (1000) How many thousandths make a ten? (10,000)

(a) 1000 (b) 100 (c) 10 (d) 100 (e) 1000

Can

4 \frac{2}{10}

be written as 42, skipping the

\frac{1}{10}

? If yes, how would we know whether 42 means 4 tens and 2 units or 4 units and 2 tenths?

Solution

No, not without a separator. 42 normally means 4 tens and 2 ones; if we also used it for 4 units and 2 tenths, we could not tell the two quantities apart. That is why we use a decimal point to show where the units end: 4 units and 2 tenths is written 4.2.

Not on its own; we need a mark (the decimal point) to tell 42 from 4.2.

Make a place value table and write each quantity in decimal form: (a) 2 ones, 3 tenths and 5 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 \frac{1}{100}

4 \frac{1}{10}

and

7 \frac{7}{1000}

Solution

	Hundreds	Tens	Units	Tenths	Hundredths	Thousandths	Decimal	Read as
(a)			2	3	5		2.35	two point three five
(b)		1	0	5			10.5	ten point five
(c)			4	0	6		4.06	four point zero six
(d)	1	0	1	0	1		101.01	one hundred one point zero one
(e)			0	9	8		0.98	zero point nine eight
(f)			0	0	5		0.05	zero point zero five
(g)			0	1			0.1	zero point one
(h)			2	0	1		2.01	two point zero one
			4	1			4.1	four point one
			7	0	0	7	7.007	seven point zero zero seven

(If the three numbers in (h) are added: $2.01 + 4.1 + 7.007 = 13.117$ .)

(a) 2.35 (b) 10.5 (c) 4.06 (d) 101.01 (e) 0.98 (f) 0.05 (g) 0.1 (h) 2.01, 4.1, 7.007

Write in decimal form: (a) 234 hundredths (b) 105 tenths

Solution

(a) $\frac{234}{100} = 2 + \frac{3}{10} + \frac{4}{100} =$ 2.34

(b) $\frac{105}{10} = 10 + \frac{5}{10} =$ 10.5

(a) 2.34 (b) 10.5

3.5 Units of Measurement

Fill in the blanks (mm ↔ cm): 70 mm = __; __ = 0.9 cm; 134 mm = __; __ = 203.6 cm

Solution

70 mm = 7 cm; 9 mm = 0.9 cm; 134 mm = 13.4 cm; 2036 mm = 203.6 cm

7 cm; 9 mm; 13.4 cm; 2036 mm

Fill in the blanks (cm ↔ m): 36 cm = __; 50 cm = __; __ = 0.89 m; 4 cm = __; 325 cm = __; __ = 2.07 m. How many mm does 1 metre have? Can we write 1 mm = 1/1000 m?

Solution

36 cm = 0.36 m; 50 cm = 0.5 m; 89 cm = 0.89 m; 4 cm = 0.04 m; 325 cm = 3.25 m; 207 cm = 2.07 m

1 m = 100 cm = 100 × 10 mm = 1000 mm. So yes, $1 mm = \frac{1}{1000}$ m $= 0.001$ m.

0.36 m, 0.5 m, 89 cm, 0.04 m, 3.25 m, 207 cm; 1 m = 1000 mm, so 1 mm = 1/1000 m = 0.001 m.

Fill in the blanks (g ↔ kg): 465 g = __; 68 g = __; 1560 g = __; 704 g = __; __ = 0.56 kg; __ = 2.5 kg

Solution

465 g = 0.465 kg; 68 g = 0.068 kg; 1560 g = 1.56 kg; 704 g = 0.704 kg; 560 g = 0.56 kg; 2500 g = 2.5 kg

0.465 kg, 0.068 kg, 1.56 kg, 0.704 kg, 560 g, 2500 g

Fill in the blanks (rupee ↔ paise): 10 p = __; __ p = ₹0.05; __ p = ₹0.36; __ = ₹0.50; 99 p = __; 250 p = __

Solution

10 p = ₹0.10; 5 p = ₹0.05; 36 p = ₹0.36; 50 p = ₹0.50; 99 p = ₹0.99; 250 p = ₹2.50

₹0.10, 5 p, 36 p, 50 p, ₹0.99, ₹2.50

3.6 Locating and Comparing Decimals

Name all the divisions between 1 and 1.1 on the number line. Identify the decimal numbers against the letters A, B, C, D (number line from 5 to 5.4).

Solution

Between 1 and 1.1: 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09.

On the second line each small division is 0.01:

Each small division is one-hundredth

A = 5.09, B = 5.13, C = 5.2 (5.20), D = 5.31

1.01 to 1.09; A = 5.09, B = 5.13, C = 5.2, D = 5.31

Sonu says 0.2 can also be written as 0.20 or 0.200; Zara thinks the zeros change the value. Which of 0.2, 0.20, 0.200, 0.02, 0.002 is the smallest and which the largest? Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50?

Solution

Sonu is right: zeros at the right end (after the decimal point) do not change the value. Largest: 0.2 = 0.20 = 0.200 (2 tenths). Smallest: 0.002 (2 thousandths).

Same values: 4.5 = 4.50 = 04.50, and 4.05 = 4.050. (0.405 and 4.005 are different from all others.)

Largest 0.2 (= 0.20 = 0.200), smallest 0.002; 4.5 = 4.50 = 04.50 and 4.05 = 4.050.

Identify the decimal number denoted by '?' in the last number line in Figure (b). Make such number lines for (a) 9.876 (b) 0.407.

Solution

In Figure (b) the magnified pieces are 3 to 4, then 3.0 to 3.1, then 3.05 to 3.06; the '?' is at the 9th mark of the last line: 3.059.

(a) Locating 9.876: 9 to 10, then 9.8 to 9.9, then 9.87 to 9.88

(b) Locating 0.407: 0 to 1, then 0.4 to 0.5, then 0.40 to 0.41

? = 3.059. For 9.876 zoom 9→10, 9.8→9.9, 9.87→9.88 and mark the 6th division; for 0.407 zoom 0.4→0.5, 0.40→0.41 and mark the 7th division.

What decimal numbers do the boxes a, b, c denote (number line from 5 to 10 with 10 divisions)? Find the numbers in boxes d, e (8 to 8.1) and f, g, h (4.3 to 4.8).

Solution

5 to 10 in 10 divisions: each division $= 5 \div 10 = 0.5$ . a = 6, b = 7.5, c = 9.5.
8 to 8.1 in 10 divisions: each $= 0.01$ . d = 8.01, e = 8.05.
4.3 to 4.8 in 10 divisions: each $= 0.5 \div 10 = 0.05$ . f = 4.35, g = 4.5, h = 4.85 (one division beyond 4.8).

a = 6, b = 7.5, c = 9.5; d = 8.01, e = 8.05; f = 4.35, g = 4.5, h = 4.85

Why can we stop comparing at the first place where the digits differ? Which is greater: (a) 1.23 or 1.32 (b) 3.81 or 13.800 (c) 1.009 or 1.090?

Solution

One unit of any place is 10 units of the next place, so it is more than everything all the later places can add up to (e.g. 0.01 is more than 0.009999…). So once one number has the bigger digit at a place, the remaining digits can never catch up.

(a) 1.32 (3 tenths > 2 tenths) (b) 13.800 (13 units > 3 units) (c) 1.090 (9 hundredths > 0 hundredths)

The later digits together are always less than one unit of the earlier place. (a) 1.32 (b) 13.800 (c) 1.090

Which of 0.9, 1.1, 1.01, 1.11 is closest to 1.09? Which is closest to 4: 3.56, 3.65, 3.099? Which is closest to 1: 0.8, 0.69, 1.08?

Solution

To 1.09: distances 0.19, 0.01, 0.08, 0.02 → 1.1
To 4: distances 0.44, 0.35, 0.901 → 3.65
To 1: distances 0.2, 0.31, 0.08 → 1.08

1.1; 3.65; 1.08

Use the digits 4, 1, 8, 2 and 5 exactly once to make a decimal number as close as possible to 25 in each layout: (i) □□.□□□ (ii) □.□□□ (iii) □□□.□□

Solution

(i) 25.148 (only 0.148 away; 24.851 is 0.149 away).

(ii) Only one digit before the point, so the number is less than 10; the closest is the largest: 8.542 (this layout has room for four of the digits).

(iii) Three digits before the point, so the number is at least 100; the closest is the smallest: 124.58.

(i) 25.148 (ii) 8.542 (iii) 124.58

3.7 Addition and Subtraction of Decimals

Write the detailed place value computation for 84.691 − 77.345, and its compact form.

Solution

	Tens	Units	Tenths	Hundredths	Thousandths
84.691	8	4	6	9	1
after regrouping	7	14	6	8	11
− 77.345	7	7	3	4	5
= 7.346	0	7	3	4	6

1 thousandth < 5 thousandths, so one hundredth is broken into 10 thousandths (11 − 5 = 6); 4 units < 7 units, so one ten is broken into 10 units (14 − 7 = 7).

$84.691 - 77.345 7.346$

84.691 − 77.345 = 7.346

Figure it Out (page 75)

Find the sums: (a) 5.3 + 2.6 (b) 18 + 8.8 (c) 2.15 + 5.26 (d) 9.01 + 9.10 (e) 29.19 + 9.91 (f) 0.934 + 0.6 (g) 0.75 + 0.03 (h) 6.236 + 0.487

Solution

(a) 7.9 (b) 26.8 (c) 7.41 (d) 18.11 (e) 39.10 (= 39.1) (f) 1.534 (g) 0.78 (h) 6.723

(a) 7.9 (b) 26.8 (c) 7.41 (d) 18.11 (e) 39.1 (f) 1.534 (g) 0.78 (h) 6.723

Find the differences: (a) 5.6 − 2.3 (b) 18 − 8.8 (c) 10.4 − 4.5 (d) 17 − 16.198 (e) 17 − 0.05 (f) 34.505 − 18.1 (g) 9.9 − 9.09 (h) 6.236 − 0.487

Solution

(a) 3.3 (b) 9.2 (c) 5.9 (d) 0.802 (e) 16.95 (f) 16.405 (g) 0.81 (h) 5.749

Decimal Sequences

Continue 4.4, 4.8, 5.2, 5.6, 6.0, … and find the change and next 3 terms for: (a) 4.4, 4.45, 4.5 (b) 25.75, 26.25, 26.75 (c) 10.56, 10.67, 10.78 (d) 13.5, 16, 18.5 (e) 8.5, 9.4, 10.3 (f) 5, 4.95, 4.90 (g) 12.45, 11.95, 11.45 (h) 36.5, 33, 29.5

Solution

Given sequence (+0.4): 6.4, 6.8, 7.2

	Change	Next three terms
(a)	+0.05	4.55, 4.6, 4.65
(b)	+0.5	27.25, 27.75, 28.25
(c)	+0.11	10.89, 11.00, 11.11
(d)	+2.5	21, 23.5, 26
(e)	+0.9	11.2, 12.1, 13
(f)	−0.05	4.85, 4.80, 4.75
(g)	−0.5	10.95, 10.45, 9.95
(h)	−3.5	26, 22.5, 19

6.4, 6.8, 7.2; (a) 4.55, 4.6, 4.65 (b) 27.25, 27.75, 28.25 (c) 10.89, 11, 11.11 (d) 21, 23.5, 26 (e) 11.2, 12.1, 13 (f) 4.85, 4.8, 4.75 (g) 10.95, 10.45, 9.95 (h) 26, 22.5, 19

Estimating Sums and Differences

Sonu claims that the sum of two decimals is always greater than the sum of their whole number parts and less than 2 more than it. Verify for 25.936 + 8.202 and for 25.93603259 + 8.202. Does it work for any two decimals? Find a similar range for differences.

Solution

$25.936 + 8.202 = 34.138$ , which lies between $25 + 8 = 33$ and $33 + 2 = 35$ . ✓ Also $25.93603259 + 8.202 = 34.13803259$ , again between 33 and 35. ✓

Why it always works: each number = its whole part + a fractional part that is at least 0 and less than 1. So the two fractional parts add up to at least 0 and less than 2. (The sum equals the whole-part sum only when both fractional parts are 0, as in 3.0 + 2.0.)

For a difference $A . \dots - B . \dots$ (with whole parts $A$ and $B$ ): the difference is more than $A - B - 1$ and less than $A - B + 1$ . Example: $25.936 - 8.202 = 17.734$ , which is between 16 and 18.

34.138 lies between 33 and 35; it is true for any two decimals because the two fractional parts add to less than 2. A difference lies between (A − B − 1) and (A − B + 1).

3.8 More on the Decimal System

"Overs left: 5.5" means 5 overs and 5 balls. Where else can we see such 'non-decimals' with a decimal-like notation?

Solution

Time: 4.30 on a clock or timetable means 4 hours 30 minutes, not 4.3 hours.
Height in feet and inches: "5.6" often means 5 feet 6 inches (not 5.6 feet, since 1 foot = 12 inches).
Dates: 15.08.2026 means 15th August 2026.
Book sections and versions: Section 3.12 comes after 3.9; app version 2.10 is after 2.9.
Cricket overs: 5.5 overs = 5 overs and 5 balls.

Clock times (4.30), heights in feet and inches (5.6), dates (15.08.2026), section numbers (3.12) and cricket overs.

Figure it Out (page 78)

Convert into decimals: (a) 5/100 (b) 16/1000 (c) 12/10 (d) 254/1000

Solution

(a) 0.05 (b) 0.016 (c) 1.2 (d) 0.254

Convert into a sum of tenths, hundredths and thousandths: (a) 0.34 (b) 1.02 (c) 0.8 (d) 0.362

Solution

(a) $\frac{3}{10} + \frac{4}{100}$ (b) $1 + \frac{0}{10} + \frac{2}{100}$ (c) $\frac{8}{10}$ (d) $\frac{3}{10} + \frac{6}{100} + \frac{2}{1000}$

(a) 3/10 + 4/100 (b) 1 + 2/100 (c) 8/10 (d) 3/10 + 6/100 + 2/1000

What decimal number does each letter represent on the number line (6.4 to 6.6, with 4 divisions in each tenth)?

Solution

Each tenth is split into 4 parts, so one division $= 0.1 \div 4 = 0.025$ .

Each division is 0.025

a = 6.45, c = 6.525, b = 6.55

a = 6.45, b = 6.55, c = 6.525

Arrange in descending order: (a) 11.01, 1.011, 1.101, 11.10, 1.01 (b) 2.567, 2.675, 2.768, 2.499, 2.698 (c) 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g (d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m

Solution

(a) 11.10 > 11.01 > 1.101 > 1.011 > 1.01

(b) 2.768 > 2.698 > 2.675 > 2.567 > 2.499

(d) 33.331 m > 33.313 m > 33.31 m > 33.133 m > 33.13 m

(a) 11.10, 11.01, 1.101, 1.011, 1.01 (b) 2.768, 2.698, 2.675, 2.567, 2.499 (c) 4.678, 4.666, 4.656, 4.600, 4.595 g (d) 33.331, 33.313, 33.31, 33.133, 33.13 m

Using the digits 1, 4, 0, 8 and 6, make (a) the decimal number closest to 30 (b) the smallest possible decimal number between 100 and 1000.

Solution

(a) There is no 2 or 3, so the tens digit is either 1 (number below 20) or 4 (number from 40). The best are 18.640 (11.36 away) and 40.168 (10.168 away). So 40.168 is closest to 30.

(b) Three digits before the point, smallest first digit 1, then 0, then 4: 104.68.

(a) 40.168 (b) 104.68

Will a decimal number with more digits be greater than a decimal number with fewer digits?

Solution

Not necessarily. For example, 2.05 has more digits than 2.5, but 2.5 > 2.05; and 0.999 < 5. What matters is the place value of the digits, not how many there are.

No: e.g. 2.5 > 2.05 and 5 > 0.999.

Mahi buys 0.25 kg beans, 0.3 kg carrots, 0.5 kg potatoes, 0.2 kg capsicums and 0.05 kg ginger. Find the total weight.

Solution

$0.25 + 0.3 + 0.5 + 0.2 + 0.05 = (0.25 + 0.05) + (0.3 + 0.5 + 0.2) = 0.3 + 1.0 =$ 1.3 kg

1.3 kg

Pinto supplies 3.79 L, 4.2 L and 4.25 L of milk in the first three days, and 25 L in 6 days. Find the milk supplied in the last three days.

Solution

First three days: $3.79 + 4.2 + 4.25 = 12.24$ L. Last three days: $25 - 12.24 =$ 12.76 L.

12.76 L

Tinku weighed 35.75 kg in January and 34.50 kg in February. Has he gained or lost weight? By how much?

Solution

$35.75 - 34.50 = 1.25$ . He has lost 1.25 kg.

Lost 1.25 kg.

Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, __, __

Solution

The rule is "add 0.9, then subtract 0.01" alternately: 5.5 → 6.4 → 6.39 → 7.29 → 7.28 → 8.18 → 8.17. (The printed "6.18, 6.17" should be 8.18, 8.17 to follow the rule.) Next terms: $8.17 + 0.9 =$ 9.07, then $9.07 - 0.01 =$ 9.06.

9.07, 9.06 (with 8.18, 8.17 before them)

How many millimetres make 1 kilometre?

Solution

1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm (one million).

10,00,000 mm

Railway travel insurance costs 45 paise per passenger. If 1 lakh people opt for it in a day, what is the total fee?

Solution

$45 \times 1, 00, 000 = 45, 00, 000$ paise $= 45, 00, 000 \div 100 =$ ₹45,000

₹45,000

Which is greater? (a) 10/1000 or 1/10 (b) one-hundredth or 90 thousandths (c) one-thousandth or 90 hundredths

Solution

(a) $\frac{10}{1000} = 0.01$ and $\frac{1}{10} = 0.1$ : 1/10 is greater.

(b) $0.01$ and $\frac{90}{1000} = 0.09$ : 90 thousandths is greater.

(a) 1/10 (b) 90 thousandths (c) 90 hundredths

Write the decimal forms: (a) 87 ones, 5 tenths and 60 hundredths = 88.10 (b) 12 tens and 12 tenths (c) 10 tens, 10 ones, 10 tenths and 10 hundredths (d) 25 tens, 25 ones, 25 tenths and 25 hundredths

Solution

(b) $120 + 1.2 =$ 121.2

(d) $250 + 25 + 2.5 + 0.25 =$ 277.75

(b) 121.2 (c) 111.1 (d) 277.75

Using each digit 0–9 not more than once, fill the boxes □.□□□ + □.□□□ so that the sum is closest to 10.5.

Solution

We can get exactly 10.5. For the thousandths: two digits adding to 10 (carry 1); hundredths: two digits adding to 9 (+1 = 10, carry 1); tenths: two digits adding to 14 (+1 = 15, write 5, carry 1); units: two digits adding to 9 (+1 = 10).

$0.643 + 9.857 10.500$

(Check: 3 + 7 = 10, 4 + 5 + 1 = 10, 6 + 8 + 1 = 15, 0 + 9 + 1 = 10; all eight digits are different.)

E.g. 0.643 + 9.857 = 10.500, exactly 10.5.

Write in decimal form: (a) 1/2 (b) 3/2 (c) 1/4 (d) 3/4 (e) 1/5 (f) 4/5

Solution

(a) $\frac{1}{2} = \frac{5}{10} = 0.5$ (b) $\frac{3}{2} = 1 \frac{5}{10} = 1.5$ (c) $\frac{1}{4} = \frac{25}{100} = 0.25$ (d) $\frac{3}{4} = \frac{75}{100} = 0.75$ (e) $\frac{1}{5} = \frac{2}{10} = 0.2$ (f) $\frac{4}{5} = \frac{8}{10} = 0.8$

(a) 0.5 (b) 1.5 (c) 0.25 (d) 0.75 (e) 0.2 (f) 0.8

← Chapter 2: Arithmetic Expressions Chapter 4: Expressions using Letter-Numbers →

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