Class 7 Mathematics · Part 1 · Chapter 3

3. A Peek Beyond the Point

Answers to all Figure it Out and in-text questions on tenths, hundredths, decimal place value, conversions and decimal sums. CBSE / NCERT Class 7 Ganita Prakash.

3.1 The Need for Smaller Units

?Measure the screws and write their lengths.

Scale 1 (only whole cm marked): the screw is between 3 cm and 4 cm.

Scale 2 (half cm marked): it is more than 3 cm but less than 312 cm.

Scale 3 (tenths marked): it is 3210 cm.

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

The third scale. Each centimetre on it is divided into 10 equal parts, so we can read the exact length.

?Can you explain why the unit was divided into smaller parts to measure the screws?

The screws were longer than 2 cm but shorter than 3 cm. Whole centimetres could not show the exact length, so smaller parts were needed.

?Write the measurements of the objects shown in the picture.

Eraser: 2410 cm

Pencil: 4510 cm

Chalk: 1410 cm

3.2 A Tenth Part

?Write the lengths of the objects in two ways (USB cable: 4 and 810 units or 4810 units).

Pencil box: about 2910 units or 2910 units.

Thumb: about 1310 units or 1310 units.

Leaf: about 10910 units or 10910 units.

These are read from the pictures, so small differences are fine.

?Arrange these lengths in increasing order.

410 < 910 < 1710 < 6710 < 7610 < 10510 < 13010 < 13110

?Arrange in increasing order: 4110, 410, 4110, 41110.

410 < 4110 = 4110 < 41110

(4110 is 4 units and 1 tenth, the same as 4110.)

?Find the total length of the honeybee: head 2310, thorax 5410, abdomen 7510 units.

Units: 2 + 5 + 7 = 14. Tenths: 3 + 4 + 5 = 12 tenths = 1 unit and 2 tenths.

Total length = 15210 units.

?Try computing the finger length (12410 − 6710) by converting both lengths to tenths.

12410 − 6710 = 5710 = 5710 units.

?A Celestial Pearl Danio is 2410 cm and a Philippine Goby is 910 cm long. What is the difference?

2410 − 910 = 1510 = 1510 cm.

Both fish are smaller than a finger. The goby is shorter than 1 cm.

?Identify the change after each term and extend the pattern.

(a) Add 310: 4, 4310, 4610, 4910, 5210, 5510, 5810

(b) Add 510: 8210, 8710, 9210, 9710, 10210, 10710, 11210

(d) Subtract 410: 5710, 5310, 4910, 4510, 4110, 3710

(e) Subtract 510: 13510, 13, 12510, 12, 11510, 11, 10510

(f) Subtract 1110: 11510, 10410, 9310, 8210, 7110, 6, 4910

3.3 A Hundredth Part

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

Each smaller part is 1100 of a unit. 100 such parts make one unit.

?How many one-hundredths make one-tenth? Can we say the length is 4 units and 45 one-hundredths?

10 one-hundredths make one-tenth.

Yes. 4 tenths = 40 hundredths. So, 44105100 = 4 units and 45 one-hundredths.

?Fill the lengths in the empty boxes on the number line.

Reading the markings: the boxes show about 55100, 155100, 174100 and 202100.

Read the exact small mark under each pointer in your book. Count tenths first, then hundredths.

?The length of the wire is given in three ways. How do they denote the same length?

11104100 = 1 + 10 hundredths + 4 hundredths = 114100.

Also 1 unit = 100 hundredths, so 114100 = 114100 (one hundred fourteen hundredths).

?Write the measurements of the lengths shown and read them in words.

53107100 — five and thirty-seven hundredths

153100 — fifteen and three hundredths

75102100 — seven and fifty-two hundredths

9810 — nine and eight tenths

?In each group, identify the longest and the shortest lengths.

Group	Longest	Shortest
(a)	33100	3100
(b)	3110	1310
(c)	54100	410
(d)	36106100	36100
(e)	18100	9100
(f)	7510	73105100
(g)	651015100	57100

?Are both methods of finding 153104100 + 26108100 different? Do you see similarities with 483 + 268?

No. Both add units, tenths and hundredths separately, then carry 10 hundredths as 1 tenth.

It is just like 483 + 268, where 10 ones are carried as 1 ten. Answer: 182100.

?Solve 25910 − 64107100 by converting to hundredths.

2590100 − 647100 = 1943100 = 194103100

?Observe 653 − 268. Do you see any similarities with the methods above?

Yes. When a place has a smaller digit, we break 1 from the next higher place, just as we break 1 tenth into 10 hundredths.

FIOFigure it Out: Find the sums and differences.

(a) 310 + 34100 = 33104100

(b) 95107100 + 21103100 = 1161010100 = 11710

(d) 77100 − 44100 = 33100

(e) 86100 − 53100 = 33100

(f) 1261002100 − 9109100 = 128100 − 99100 = 119100

In (f), if the first number is meant to be 126102100, the answer is 116103100.

3.4 Decimal Place Value

?What will the fraction be when 1100 is split into 10 equal parts?

11000. A thousand such parts make one unit.

?How Big?

(a) 1000 thousandths make one unit.

(b) 100 thousandths make one tenth.

(d) 100 tenths make one ten.

(e) 1000 hundredths make one ten.

?Can 4210 be written as 42? How would we know if 42 means 4 tens and 2 units or 4 units and 2 tenths?

No. 42 already means 4 tens and 2 units. We need a mark to show where the units end. So, we use a decimal point: 4.2.

?Write each quantity in decimal form and read it.

Quantity	Decimal	Read as
(a) 2 ones, 3 tenths, 5 hundredths	2.35	two point three five
(b) 1 ten and 5 tenths	10.5	ten point five
(c) 4 ones and 6 hundredths	4.06	four point zero six
(d) 1 hundred, 1 one, 1 hundredth	101.01	one hundred one point zero one
(e) 8100 and 910	0.98	zero point nine eight
(f) 5100	0.05	zero point zero five
(g) 110	0.1	zero point one
(h) 21100, 4110, 771000	2.01, 4.1, 7.007	two point zero one; four point one; seven point zero zero seven

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

(a) 234 hundredths = 234100 = 2.34

(b) 105 tenths = 10510 = 10.5

3.5 Units of Measurement

?Fill in the blanks (mm ↔ cm).

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

?Fill in the blanks (cm ↔ m).

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

?How many mm does 1 metre have? Can we write 1 mm = 11000 m?

1 m = 100 cm = 100 × 10 mm = 1000 mm. So, yes, 1 mm = 11000 m = 0.001 m.

?Fill in the blanks (g ↔ kg).

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

?Fill in the blanks (rupee ↔ paise).

10 p = ₹0.10; 5 p = ₹0.05; 36 p = ₹0.36;

50 p = ₹0.50; 99 p = ₹0.99; 250 p = ₹2.50

3.6 Locating and Comparing Decimals

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

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

?Identify the decimal numbers against the letters (between 5 and 5.4).

A = 5.09, B = 5.13, C = 5.2, D = 5.31

?Sonu says 0.2 = 0.20 = 0.200. Which of 0.2, 0.20, 0.200, 0.02, 0.002 is the smallest and which is the largest?

Sonu is right. Zeros on the right do not change the value.

0.2 = 0.20 = 0.200 are the largest (2 tenths). 0.002 is the smallest (2 thousandths).

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

4.5 = 4.50 = 04.50

4.05 = 4.050

0.405 and 4.005 are different from all others.

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

The lines zoom into 3 to 4, then 3.0 to 3.1, then 3.05 to 3.06. The ‘?’ is at the 9th mark, so it is 3.059.

?Make such number lines for (a) 9.876, (b) 0.407.

(a) 0 to 10 → 9 to 10 → 9.8 to 9.9 → 9.87 to 9.88; mark the 6th small division: 9.876.

(b) 0 to 1 → 0.4 to 0.5 → 0.40 to 0.41; mark the 7th small division: 0.407.

?What decimal numbers do ‘a’, ‘b’, ‘c’ denote (5 to 10 in 10 divisions)?

Each division is 12 unit = 0.5.

a = 6, b = 7.5, c = 9.5

?Find the decimal numbers in the boxes d, e, f, g, h.

8 to 8.1 has 10 divisions, so each is 0.01: d = 8.01, e = 8.05.

4.3 to 4.8 has 10 divisions, so each is 0.05: f = 4.35, g = 4.5, h = 4.85.

?Why can we stop comparing at the first place where the digits differ?

A bigger digit in a higher place is worth more than all the digits after it put together.

Example: 0.01 is more than 0.009999…. So, the later digits cannot change the result.

?Which decimal number is greater?

(a) 1.32

(b) 13.800

?Closest decimals

Closest to 1.09 (from 0.9, 1.1, 1.01, 1.11): 1.1 (only 0.01 away).

Closest to 4 (from 3.56, 3.65, 3.099): 3.65.

Closest to 1 (from 0.8, 0.69, 1.08): 1.08.

?Use the digits 4, 1, 8, 2 and 5 and make a decimal number as close as possible to 25.

Two digits before the point and three after: 25.148 (only 0.148 away).

One digit before the point and three after: 8.542 (with one digit, the number is less than 10).

Three digits before the point and two after: 124.58.

3.7 Addition and Subtraction of Decimals

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

(8 × 10 + 4 × 1 + 6 × 110 + 9 × 1100 + 1 × 11000) − (7 × 10 + 7 × 1 + 3 × 110 + 4 × 1100 + 5 × 11000)

Thousandths: 1 − 5 is not possible. Break 1 hundredth into 10 thousandths: 11 − 5 = 6.

Hundredths: 8 − 4 = 4. Tenths: 6 − 3 = 3. Units: 4 − 7 is not possible. Break 1 ten: 14 − 7 = 7. Tens: 7 − 7 = 0.

84.691 − 77.345 7.346

FIOFigure it Out

1. Find the sums.

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

2. Find the differences.

(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: identify the change and write the next 3 terms.

4.4, 4.8, 5.2, 5.6, 6.0, 6.4, 6.8, 7.2 (add 0.4)

(a) add 0.05: 4.55, 4.6, 4.65

(b) add 0.5: 27.25, 27.75, 28.25

(d) add 2.5: 21, 23.5, 26

(e) add 0.9: 11.2, 12.1, 13

(f) subtract 0.05: 4.85, 4.80, 4.75

(g) subtract 0.5: 10.95, 10.45, 9.95

(h) subtract 3.5: 26, 22.5, 19

?Sonu’s claim: the sum of two decimals is more than the sum of their whole number parts and less than 2 more than it. Is it true?

25.936 + 8.202 = 34.138. It is more than 25 + 8 = 33 and less than 35. So, it is true here.

It works for any 2 decimals that have a fractional part. Each fractional part is less than 1, so together they add less than 2.

For 25.93603259 + 8.202 = 34.13803259, it is also between 33 and 35.

?Narrow down the range for the difference of two decimal numbers.

Find the difference of the whole number parts. The answer lies between 1 less and 1 more than this.

Example: 14.8 − 6.3 = 8.5. The whole parts give 14 − 6 = 8, and 8.5 lies between 7 and 9.

3.8 More on the Decimal System

?Where else can we see ‘non-decimals’ written in a decimal-like notation?

Time: 2.30 on a clock means 2 hours 30 minutes, not 2.3 hours.

Height: 5.6 (5 ft 6 inches). Book chapters: 3.2 means Section 2 of Chapter 3.

FIOFigure it Out

1. Convert into decimals.

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

2. Convert into a sum of tenths, hundredths and thousandths.

(a) 0.34 = 310 + 4100

(b) 1.02 = 1 + 2100

(d) 0.362 = 310 + 6100 + 21000

3. What decimal number does each letter represent?

From 6.4 to 6.5 there are 4 divisions, so each is 0.025.

a = 6.45, c = 6.525, b = 6.55

4. Arrange in descending order.

(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

5. Using the digits 1, 4, 0, 8 and 6, make:

(a) the decimal number closest to 30: 40.168

(b) the smallest decimal number between 100 and 1000: 104.68

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

Not always. 2.5 has fewer digits than 2.05, but 2.5 is greater.

7. Total weight of Mahi’s items

0.25 + 0.3 + 0.5 + 0.2 + 0.05 = 1.3 kg

8. Milk supplied in the last three days

First three days: 3.79 + 4.2 + 4.25 = 12.24 L.

Last three days: 25 − 12.24 = 12.76 L.

9. Has Tinku gained or lost weight?

He has lost weight. Change = 35.75 − 34.50 = 1.25 kg.

10. Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, …

The rule is: add 0.9, then subtract 0.01. So, after 7.28 come 8.18, 8.17, 9.07, 9.06.

The book prints 6.18, 6.17. By the pattern, these should be 8.18, 8.17.

11. How many millimetres make 1 kilometre?

1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm.

12. Insurance fee for 1 lakh people at 45 paise each

45 × 1,00,000 = 45,00,000 paise = ₹45,000.

13. Which is greater?

(a) 101000 = 1100, so 110 is greater.

(b) 90 thousandths = 0.09 and one-hundredth = 0.01. So, 90 thousandths is greater.

14. Write the decimal forms.

(a) 87 + 0.5 + 0.60 = 88.10 (given)

(b) 12 tens and 12 tenths = 120 + 1.2 = 121.2

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

15. Fill the boxes using each digit 0 – 9 at most once so that the sum is closest to 10.5.

0.643 + 9.857 = 10.500. The sum is exactly 10.5.

16. Write in decimal form.

(a) 12 = 510 = 0.5

(b) 32 = 1510 = 1.5

(d) 34 = 75100 = 0.75

(e) 15 = 210 = 0.2

(f) 45 = 810 = 0.8

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