Class 7 New Mathematics Chapter 8 Question Answer | দশমিক | English Medium | ASSEB

Decimals — Questions and Answers

Welcome to HSLC Guru. This page gives the complete solutions to ASSEB Class 7 New Mathematics Chapter 8, Decimals — tenths, hundredths and thousandths, the decimal point and place value, converting between decimals and fractions, comparing and ordering decimals, addition and subtraction, decimals on the number line, decimal sequences and estimation — together with every “Work it Out” box and all of Exercise 8, worked in full.

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Summary

When a unit is divided into 10 equal parts, each part is one-tenth ($\frac{1}{10}$); dividing each tenth again into 10 equal parts gives one-hundredth ($\frac{1}{100}$); and dividing a hundredth into 10 gives one-thousandth ($\frac{1}{1000}$). These fractional parts let us measure small lengths and quantities accurately.

To separate the whole-number part from the fractional part we use the symbol ‘.’, called the decimal point. Using it we write $\frac{1}{10}$ as 0.1, $\frac{1}{100}$ as 0.01 and $\frac{1}{1000}$ as 0.001. Every decimal number can be shown as a point on the number line, and two decimals are compared by matching their place values from left to right.

Decimals are added and subtracted just like whole numbers, keeping the place values (and decimal points) aligned. Adding zeroes to the right of a decimal number does not change its value (0.5 = 0.50 = 0.500). Decimals are widely used when converting units of length, weight, volume and money.

Summary: This ASSEB Class 7 New Mathematics Chapter 8 (Decimals) guide explains tenths, hundredths and thousandths, the decimal point and place value, converting between decimals and fractions, comparing and ordering decimals, addition and subtraction of decimals, decimals on a number line, decimal sequences and estimation. It solves every “Work it Out” box (8.1 to 8.9) and all twenty questions of Exercise 8 with full worked steps.

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Place Value and the Decimal Point

In a decimal place value table the whole-number places (ones, tens, hundreds, thousands) lie to the left of the decimal point and the fractional places (tenths, hundredths, thousandths) lie to the right. Each place is 10 times smaller than the place to its left.

Thousands	Hundreds	Tens	Ones	Point	Tenths ($\frac{1}{10}$)	Hundredths ($\frac{1}{100}$)	Thousandths ($\frac{1}{1000}$)
1000	100	10	1	.	0.1	0.01	0.001

For example, 9 tens and 5 tenths = $9 \times 10 + 0 \times 1 + 5 \times \frac{1}{10}$ = 90.5 (read “ninety point five”), while 9 ones and 5 hundredths = $9 \times 1 + 0 \times \frac{1}{10} + 5 \times \frac{1}{100}$ = 9.05 (read “nine point zero five”). Similarly 0.905 is read “zero point nine zero five”.

Decimal–fraction links: $0.3 = \frac{3}{10}$, $0.07 = \frac{7}{100}$, $0.25 = \frac{25}{100} = \frac{1}{4}$, $0.5 = \frac{5}{10} = \frac{1}{2}$. A number with three decimal places can be written as a fraction with denominator 1000, e.g. $0.127 = \frac{127}{1000}$.

On the number line, dividing the unit between 0 and 1 into 10 equal parts places 0.5 at the fifth mark; dividing the unit between 1 and 2 places 1.6 at the sixth mark.

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Textbook Questions and Answers

Work it Out 8.1

1. Which pen had more length?

Answer: Pinak’s pen is longer. Catherine’s pen is $14\frac{4}{10}$ cm and Pinak’s pen is $14\frac{8}{10}$ cm; the whole parts are equal (14), and comparing the tenths, $\frac{8}{10} > \frac{4}{10}$, so Pinak’s pen is longer.

2. Why do we divide a unit into smaller parts?

Answer: To measure lengths that lie between two whole units (like a length between 14 cm and 15 cm) accurately. Dividing one unit into 10 equal parts makes each part $\frac{1}{10}$ of a unit, so we can record both the whole part and the fractional part.

3. Write the length of the following measures in whole and parts — (a), (b), (c).

Answer: Count how many whole centimetre marks the object crosses (the whole part), then count the small tenth-divisions after it (the fractional part). Each length is written as (whole cm)$\frac{\text{tenth divisions}}{10}$ cm — for example $14\frac{4}{10}$ cm and $14\frac{8}{10}$ cm.

Work it Out 8.2.1

1. Determine the length of Pinak’s pen in terms of tenth parts.

Answer: Pinak’s pen = $14\frac{8}{10}$ cm = $\frac{(14 \times 10) + 8}{10} = \frac{148}{10}$ cm. So the pen is made up of 148 tenth-parts.

2. Find the length of the objects shown (sketch pen, painting brush, crayon, candle) and write it in terms of tenth parts.

Answer: Place each object on the scale and count the total tenth-divisions up to its end. Each length comes out as $(\text{whole cm}) + \frac{\text{tenth divisions}}{10}$ cm, which can also be written as $\frac{\text{total tenths}}{10}$ cm (for example $3\frac{5}{10}$ cm = $\frac{35}{10}$ cm). These readings are reused for addition and subtraction in Work it Out 8.2.2 and 8.2.3.

3. Arrange the following lengths in increasing order — $\frac{7}{10}$ cm, $(2+\frac{7}{10})$ cm, $\frac{23}{10}$ cm, $\frac{145}{10}$ cm, $(14+\frac{6}{10})$ cm, $(8+\frac{9}{10})$ cm.

Answer: As decimals these are 0.7, 2.7, 2.3, 14.5, 14.6, 8.9. In increasing order: $\frac{7}{10} < \frac{23}{10} < (2+\frac{7}{10}) < (8+\frac{9}{10}) < \frac{145}{10} < (14+\frac{6}{10})$, i.e. 0.7 < 2.3 < 2.7 < 8.9 < 14.5 < 14.6.

Work it Out 8.2.2

1. Determine the units and tenth parts of the fractions — (a) $\frac{49}{10}$, (b) $\frac{137}{10}$, (c) $\frac{245}{10}$, (d) $\frac{392}{10}$.

Answer: Dividing by 10, the quotient is the units and the remainder is the tenths —

• (a) $\frac{49}{10} = 4\frac{9}{10}$ → 4 units and 9 tenths (4.9).
• (b) $\frac{137}{10} = 13\frac{7}{10}$ → 13 units and 7 tenths (13.7).
• (c) $\frac{245}{10} = 24\frac{5}{10}$ → 24 units and 5 tenths (24.5).
• (d) $\frac{392}{10} = 39\frac{2}{10}$ → 39 units and 2 tenths (39.2).

2. Add the fractions — (a) $2\frac{7}{10} + 8\frac{3}{10}$, (b) $6\frac{4}{10} + 13\frac{2}{10}$.

Answer: (a) $2\frac{7}{10} + 8\frac{3}{10} = (2+8) + \frac{7+3}{10} = 10 + \frac{10}{10} = 10 + 1 = 11$. (b) $6\frac{4}{10} + 13\frac{2}{10} = (6+13) + \frac{4+2}{10} = 19\frac{6}{10}$ (that is 19.6).

3. From question 2 of Work it Out 8.2.1, find the total length of — (a) sketch pen and painting brush, (b) sketch pen and crayon, (c) sketch pen and candle, (d) painting brush and crayon, (e) painting brush and candle, (f) crayon and candle.

Answer: Add the two lengths read off the scale in Work it Out 8.2.1 (Q2): add whole to whole and tenths to tenths; if the tenths total 10 or more, carry $\frac{10}{10} = 1$ into the whole part. For example, $3\frac{5}{10}$ cm and $2\frac{8}{10}$ cm give $(3+2) + \frac{5+8}{10} = 5 + \frac{13}{10} = 6\frac{3}{10}$ cm.

Work it Out 8.2.3

1. Find the difference in lengths for the pairs given in question 3 of Work it Out 8.2.2.

Answer: Subtract the shorter length from the longer in each pair: subtract whole from whole and tenths from tenths; if the tenths cannot be subtracted, borrow $1 = \frac{10}{10}$ from the whole part. For example, $6\frac{3}{10} – 2\frac{8}{10} = 5\frac{13}{10} – 2\frac{8}{10} = 3\frac{5}{10}$ cm.

2. The head part of a Kawoi fish is $(1+\frac{6}{10})$ cm and the remaining part is $(4+\frac{6}{10})$ cm. Find the difference between the remaining part and the head part.

Answer: Difference = $(4+\frac{6}{10}) – (1+\frac{6}{10}) = (4-1) + \frac{6-6}{10} = 3 + 0 = 3$ cm.

3. Observe the number patterns and fill up the blanks.

• (a) $6\frac{5}{10}, 6\frac{8}{10}, 7\frac{1}{10}, \dots$ → increases by 0.3; next terms: 7.4, 7.7, 8.0, 8.3, 8.6.
• (b) $2\frac{7}{10}, 3\frac{6}{10}, 4\frac{5}{10}, \dots$ → increases by 0.9; next terms: 5.4, 6.3, 7.2, 8.1, 9.0.
• (c) $3\frac{6}{10}, 4, 4\frac{4}{10}, \dots$ → increases by 0.4; next terms: 4.8, 5.2, 5.6, 6.0, 6.4.
• (d) $7\frac{5}{10}, 6\frac{9}{10}, 6\frac{3}{10}, \dots$ → decreases by 0.6; next terms: 5.7, 5.1, 4.5, 3.9, 3.3.
• (e) $5\frac{7}{10}, 5, 4\frac{3}{10}, \dots$ → decreases by 0.7; next terms: 3.6, 2.9, 2.2, 1.5, 0.8.

Work it Out 8.3.1

1. Find the measurement of each pencil in the figure and express it in words.

Answer: Count the whole centimetres, then the tenth-divisions, then the hundredth-divisions. For example, a pencil of length $(6 + \frac{3}{10} + \frac{4}{100})$ cm = $6\frac{34}{100}$ cm is read as “six and three-tenth and four-hundredths”, or “six and thirty-four-hundredths”.

2. Plot the following on the scale and arrange them in ascending and descending order — (a) $3\frac{7}{10}$, $3\frac{7}{100}$, $3\frac{7}{10}\frac{7}{100}$; (b) $\frac{67}{100}$, $\frac{76}{100}$, $\frac{6}{10}$, $\frac{7}{10}$; (c) $\frac{8}{10}$, $\frac{8}{100}$, $\frac{88}{100}$.

Answer:

• (a) As decimals: 3.7, 3.07, 3.77. Ascending: 3.07 < 3.7 < 3.77; Descending: 3.77 > 3.7 > 3.07.
• (b) As decimals: 0.67, 0.76, 0.6, 0.7. Ascending: 0.6 < 0.67 < 0.7 < 0.76; Descending: 0.76 > 0.7 > 0.67 > 0.6.
• (c) As decimals: 0.8, 0.08, 0.88. Ascending: 0.08 < 0.8 < 0.88; Descending: 0.88 > 0.8 > 0.08.

Work it Out 8.3.2

1. Simplify the following.

• (a) $4\frac{4}{10} + 4\frac{4}{100} = 4.4 + 4.04 = 8.44 = 8\frac{44}{100}$.
• (b) $2\frac{6}{10}\frac{5}{100} + 7\frac{9}{10}\frac{2}{100} = 2.65 + 7.92 = 10.57 = 10\frac{57}{100}$.
• (c) $8\frac{1}{10}\frac{9}{100} + 3\frac{8}{10}\frac{1}{100} = 8.19 + 3.81 = 12.00 = 12$.
• (d) $4\frac{9}{10}\frac{9}{100} + 5\frac{1}{100} = 4.99 + 5.01 = 10.00 = 10$.
• (e) $12\frac{2}{10} + \frac{3}{10}\frac{8}{100} – 4\frac{5}{100} = 12.2 + 0.38 – 4.05 = 8.53 = 8\frac{53}{100}$.
• (f) $8\frac{3}{100} – 5\frac{7}{100} = 8.03 – 5.07 = 2.96 = 2\frac{96}{100}$.
• (g) $10\frac{55}{100} – 6\frac{5}{10}\frac{5}{100} = 10.55 – 6.55 = 4.00 = 4$.
• (h) $19\frac{4}{10}\frac{5}{100} – 12\frac{5}{10}\frac{4}{100} = 19.45 – 12.54 = 6.91 = 6\frac{91}{100}$.

Work it Out 8.4

1. Write each number in decimal form using the place value table and read it — (a) 5 ones, 4 tenths and 6 hundredths; (b) 5 ones and 6 hundredths; (c) 3 hundred, 2 ones and 2 hundredths.

Answer: (a) $5 + \frac{4}{10} + \frac{6}{100} = 5.46$ — read “five point four six”. (b) $5 + \frac{6}{100} = 5.06$ — “five point zero six”. (c) $300 + 2 + \frac{2}{100} = 302.02$ — “three hundred two point zero two”.

2. Write the numbers in decimal form — (a) 25 tenths, (b) 257 tenths, (c) 257 hundredths.

Answer: (a) $\frac{25}{10} = 2.5$. (b) $\frac{257}{10} = 25.7$. (c) $\frac{257}{100} = 2.57$.

Let us think

We know that 0.3, 0.03 and 0.003 are different decimal numbers. But what about 0.3, 0.30 and 0.300 — are they the same or different?

Answer: They are the same number, because $0.3 = \frac{3}{10}$, $0.30 = \frac{30}{100} = \frac{3}{10}$ and $0.300 = \frac{300}{1000} = \frac{3}{10}$ — all equal to 0.3. So zeroes at the right end of a decimal number do not change its value (whereas in 0.3, 0.03, 0.003 the zeroes shift the digit’s place after the decimal point, so those are different).

Work it Out 8.5.1

1. Which of these represent the same value? 2.07, 0.207, 2.7, 2.70, 2.007, 2.700, 02.70, 002.7.

Answer: Dropping unnecessary zeroes, the five numbers 2.7, 2.70, 2.700, 02.70 and 002.7 all represent the same value 2.7 (extra zeroes on the right or left do not change the value). The remaining 2.07, 0.207 and 2.007 each represent a different value.

2. Place the following decimals on the number line — (a) 2.3, (b) 2.37, (c) 3.40, (d) 3.52, (e) 0.402, (f) 1.05.

Answer: Divide the unit after the whole part into 10 equal parts (tenths), then divide again for hundredths, and locate each point —

• (a) 2.3 → the 3rd tenth-mark between 2 and 3.
• (b) 2.37 → the 7th hundredth-mark between 2.3 and 2.4.
• (c) 3.40 = 3.4 → the 4th tenth-mark between 3 and 4.
• (d) 3.52 → the 2nd hundredth-mark between 3.5 and 3.6.
• (e) 0.402 ≈ 0.40 → just after the 4th tenth-mark between 0 and 1.
• (f) 1.05 → the 5th hundredth-mark between 1 and 1.1.

3. Write the appropriate decimal numbers in the empty boxes on the number lines — (a) to (e).

Answer: First decide what each small division of the number line stands for (the scale), then read the box positions. For instance, if two marked points 3 and 3.1 have 10 small divisions between them, each division is 0.01, so the boxes are 3.01, 3.02, 3.03 …; if the divisions between 4.4 and 4.9 are tenths, the boxes are 4.5, 4.6, 4.7, 4.8; and between 2.4 and 2.5 the divisions are hundredths, giving 2.41, 2.42 ….

Work it Out 8.5.2

1. Find the largest number from each pair — (a) 1.67 and 1.76, (b) 5.79 and 57.9, (c) 2.005 and 2.050, (d) 3.569 and 3.509.

Answer: Comparing place values from the left —

• (a) whole parts equal (1); tenths 6 < 7, so 1.76 is larger.
• (b) whole parts 5 < 57, so 57.9 is larger.
• (c) hundredths 0 < 5, so 2.050 is larger.
• (d) hundredths 6 > 0, so 3.569 is larger.

2. Arrange in ascending order — (a) 1.02, 1.002, 1.31, 1.15, 0.1; (b) 0.9, 0.75, 0.98, 0.3; (c) 5.077, 5.15, 5.51, 4.97.

Answer: (a) 0.1 < 1.002 < 1.02 < 1.15 < 1.31. (b) 0.3 < 0.75 < 0.9 < 0.98. (c) 4.97 < 5.077 < 5.15 < 5.51.

Work it Out 8.5.3

1. Find out — (a) which of 0.8, 1.03, 0.76, 1.12 is nearest to 1? (b) which of 5.37, 4.9, 4.87, 5.29 is nearest to 5?

Answer: (a) distances from 1: 0.8 → 0.2, 1.03 → 0.03, 0.76 → 0.24, 1.12 → 0.12; the smallest is 1.03, so 1.03 is nearest. (b) distances from 5: 5.37 → 0.37, 4.9 → 0.1, 4.87 → 0.13, 5.29 → 0.29; the smallest is 4.9, so 4.9 is nearest.

2. Using the digits 0, 1, 2, 3, 8 (each once) make — (a) the decimal number nearest to 23, (b) the smallest possible decimal number between 100 and 1000.

Answer: (a) Take 23 as the whole part and place the remaining digits 0, 1, 8 as the smallest fraction (in ascending order) — 23.018 (only 0.018 away from 23). (b) The whole part must be a 3-digit number; the smallest is 102 (1, 0, 2), and placing 3, 8 smallest-first after the point gives 102.38.

Work it Out 8.6

1. Add — (a) 4.7 and 2.2, (b) 8.3 and 11.9, (c) 7.48 and 3.93, (d) 13.58 and 27.3, (e) 1.279 and 5.72, (f) 0.47 and 0.047, (g) 4.937 and 9.285.

Answer: Aligning the decimal points — (a) 4.7 + 2.2 = 6.9; (b) 8.3 + 11.9 = 20.2; (c) 7.48 + 3.93 = 11.41; (d) 13.58 + 27.3 = 40.88; (e) 1.279 + 5.72 = 6.999; (f) 0.47 + 0.047 = 0.517; (g) 4.937 + 9.285 = 14.222.

2. Subtract — (a) 9.7 from 11.3, (b) 17 from 24.3, (c) 0.06 from 7.4, (d) 39.3 from 47, (e) 16.714 from 29.235, (f) 0.006 from 0.16, (g) 5.05 from 5.5.

Answer: (a) 11.3 − 9.7 = 1.6; (b) 24.3 − 17 = 7.3; (c) 7.4 − 0.06 = 7.34; (d) 47 − 39.3 = 7.7; (e) 29.235 − 16.714 = 12.521; (f) 0.16 − 0.006 = 0.154; (g) 5.5 − 5.05 = 0.45.

Work it Out 8.7.1

1. Measure the following in millimetre and convert to centimetre — (a) height of a notebook, (b) length of an eraser, (c) radius of a pencil battery, (d) length of a chocolate.

Answer: This is a practical (project) task. Measure each object in millimetres, then use $1\text{ mm} = \frac{1}{10}\text{ cm} = 0.1$ cm — divide the millimetre reading by 10 to get centimetres (e.g. 45 mm = 4.5 cm).

2. Express the following lengths in cm — (a) 3 mm, (b) 13 mm, (c) 135 mm.

Answer: (a) $3\text{ mm} = \frac{3}{10}$ cm = 0.3 cm; (b) $13\text{ mm} = \frac{13}{10}$ cm = 1.3 cm; (c) $135\text{ mm} = \frac{135}{10}$ cm = 13.5 cm.

3. Find the following in metre — (a) height of a small plant, (b) length of an A4 sheet, (c) length of a pen, (d) length of a pencil.

Answer: This is also a practical task. Measure each in centimetres, then use $1\text{ cm} = \frac{1}{100}$ m = 0.01 m — divide the centimetre reading by 100 to get metres (e.g. an A4 sheet ≈ 29.7 cm = 0.297 m).

Work it Out 8.7.2

1. Convert and fill in the blanks — 58g, 734g, 1405g; ___g = 0.58kg; 750g; ___g = 4.6kg.

Answer: $1\text{ g} = \frac{1}{1000}$ kg = 0.001 kg (divide grams by 1000) —

• 58 g = 0.058 kg
• 734 g = 0.734 kg
• 1405 g = 1.405 kg
• 580 g = 0.58 kg
• 750 g = 0.75 kg
• 4600 g = 4.6 kg

2. From the figures (Atta packets of 10g, 100g, 500g, 1kg, 2kg, 5kg), determine the total weight of Atta in kg.

Answer: Converting all to kg and adding — $5 + 2 + 1 + 0.5 + 0.1 + 0.01 = $ 8.61 kg.

Work it Out 8.7.3

1. Match Column A with Column B. A: (A) 250 L, (B) 2500 mL, (C) 25 mL, (D) 2.5 mL. B: (a) 0.025 L, (b) 0.0025 L, (c) 0.25 kL, (d) 0.025 kL, (e) 2.5 L, (f) 0.25 L.

Answer: Using $1\text{ mL} = 0.001$ L and $1\text{ L} = 0.001$ kL — (A) 250 L = 0.25 kL → (c); (B) 2500 mL = 2.5 L → (e); (C) 25 mL = 0.025 L → (a); (D) 2.5 mL = 0.0025 L → (b).

2. From the figures (containers of 20 L, 5 L, 2 L, 1 L, 500 mL, 200 mL), determine the total water in L.

Answer: Converting all to litres and adding — $20 + 5 + 2 + 1 + 0.5 + 0.2 = $ 28.7 L.

Work it Out 8.7.4

1. Fill in the blanks — ₹0.08 = ___ paisa; ₹___ = 500 paisa; ₹___ = 25 paisa; ₹0.6 = ___ paisa.

Answer: Using ₹1 = 100 paisa and 1 paisa = ₹0.01 — ₹0.08 = 8 paisa; ₹5 = 500 paisa; ₹0.25 = 25 paisa; ₹0.6 = 60 paisa.

Work it Out 8.8

1. Observe each decimal sequence. Find how much each succeeding term changes from the preceding term, and find the next 5 terms.

• (a) 2.9, 3.2, 3.5, … → +0.3; next: 3.8, 4.1, 4.4, 4.7, 5.0.
• (b) 17.70, 17.78, 17.86, … → +0.08; next: 17.94, 18.02, 18.10, 18.18, 18.26.
• (c) 365.25, 365.50, 365.75, … → +0.25; next: 366.00, 366.25, 366.50, 366.75, 367.00.
• (d) 5.5, 5.1, 4.7, … → −0.4; next: 4.3, 3.9, 3.5, 3.1, 2.7.
• (e) 24, 23.93, 23.86, … → −0.07; next: 23.79, 23.72, 23.65, 23.58, 23.51.
• (f) 1729.153, 1729.045, 1728.937, … → −0.108; next: 1728.829, 1728.721, 1728.613, 1728.505, 1728.397.

2. Try to make your own decimal sequences.

Answer: Start with any number and add or subtract the same decimal each time; e.g. 0.5, 0.9, 1.3, 1.7, 2.1, … (adding 0.4) or 10.0, 9.75, 9.50, 9.25, … (subtracting 0.25).

3. Write the missing number in each number line.

• (a) 3, ___, 5.7, 8.4, ___, 13.8 → each step +2.7; the sequence is 3, 5.7, 8.4, 11.1, 13.8 (missing = 11.1).
• (b) 10, ___, 11 → each step +0.5; missing = 10.5.
• (c) ___, 2.4, 3, 3.6 → each step +0.6; missing = 1.8.

Work it Out 8.9

1. Estimate the following by rounding up to tenths and hundredths place — (a) 27.89436 + 4.73214, (b) 1.29475 + 48.71743, (c) 13.2945 − 8.4756, (d) 5.9999 − 2.00423.

Answer: First round each number to one decimal place (tenths) or two (hundredths), then add or subtract —

• (a) tenths: 27.9 + 4.7 = 32.6; hundredths: 27.89 + 4.73 = 32.62 (actual 32.6265).
• (b) tenths: 1.3 + 48.7 = 50.0; hundredths: 1.29 + 48.72 = 50.01 (actual 50.01218).
• (c) tenths: 13.3 − 8.5 = 4.8; hundredths: 13.29 − 8.48 = 4.81 (actual 4.8189).
• (d) tenths: 6.0 − 2.0 = 4.0; hundredths: 6.00 − 2.00 = 4.00 (actual 3.99567).

Notice that an estimate does not always match the actual value exactly; it only gives a close (approximate) answer.

Exercise 8

1. Which of the following is greatest? A. $\frac{1}{10}$ B. $\frac{10}{1000}$ C. $\frac{9}{10}$ D. $\frac{9}{100}$

Answer: C. $\frac{9}{10}$ (= 0.9; the others are 0.1, 0.01, 0.09).

2. Which of the following is the smallest? A. One-hundredth B. 20 thousandths C. Ten-hundredth D. 12 tenths

Answer: A. One-hundredth (= 0.01; the others are 0.020, 0.1, 1.2).

3. Which is equal to 9 tens, 9 ones, 9 tenths and 9 hundredths? A. 9.999 B. 99.99 C. 999.9 D. 9999

Answer: B. 99.99 ($90 + 9 + 0.9 + 0.09 = 99.99$).

4. Which is equal to 37 ones, 5 tenths and 10 hundredths? A. 37.51 B. 37.560 C. 37.6 D. 38.51

Answer: C. 37.6 ($37 + 0.5 + 0.10 = 37.60 = 37.6$).

5. Which decimal number is closest to 5? A. 5.05 B. 5.45 C. 5.1 D. 4.98

Answer: D. 4.98 (only 0.02 away from 5).

6. What is the value of 0.61 + 0.7? A. 0.617 B. 1.31 C. 0.68 D. None

Answer: B. 1.31 ($0.61 + 0.70 = 1.31$).

7. What is the value of $\frac{1}{5}$? A. 0.5 B. 0.25 C. 1.5 D. 0.2

Answer: D. 0.2 ($\frac{1}{5} = \frac{2}{10} = 0.2$).

8. What is the fractional form of 0.3? A. $\frac{3}{100}$ B. $\frac{3}{10}$ C. $\frac{3}{5}$ D. $\frac{1}{3}$

Answer: B. $\frac{3}{10}$.

9. Which is the decimal expansion of $\frac{45}{100}$? A. 0.045 B. 4.5 C. 0.40 D. 0.45

Answer: D. 0.45.

10. What is the decimal expansion of 96 hundredths? A. 0.96 B. 9.6 C. 0.096 D. 96

Answer: A. 0.96 ($\frac{96}{100} = 0.96$).

11. Identify true or false.

• (a) 0.554 is a decimal number with digits at both the hundredths and thousandths place — True (5 at hundredths, 4 at thousandths).
• (b) 7.69 is a decimal number between 7 and 8 with 6 at the tenth place and 9 at the hundredths place — True.
• (c) 3.03 is a decimal number with 3 at both the tenth and the hundredth place — False (0 at tenth, 3 at hundredth).
• (d) 0.127 is a decimal number with 1 at the tenth place and 7 at the thousandths place — True.
• (e) Every decimal number can be represented by a point on the number line — True.
• (f) A decimal number with 3 decimal places can be written as a fraction with denominator 1000 — True.

12. Choose the correct symbol ‘<’, ‘>’ or ‘=’.

• (a) 25.05 < 25.50
• (b) 0.65 > 0.55
• (c) 3.03 + 0.3 = 3.33 (3.33 = 3.33)
• (d) 9.09 + 0.005 < 9.1 − 0.004 (9.095 < 9.096)
• (e) 62.1 − 6.21 > 0.621 (55.89 > 0.621)
• (f) 25.83 − 0.38 > 25.38 (25.45 > 25.38)
• (g) 54.94 − 37.08 = 17.86 (17.86 = 17.86)

13. Your parents give you ₹300 and a grocery list. The shopkeeper’s receipt is — Masoor Dal 500 g (₹95/kg), Mustard Oil 300 mL (₹210/L), Rice 2 kg (₹50/kg), Sugar 250 g (₹51/kg), Onion 500 g (₹35/kg). (a) Find the total cost of all the items. (b) How much money can you return to your parents?

Answer: Cost of each item — Masoor Dal $0.5 \times 95 = ₹47.5$; Mustard Oil $0.3 \times 210 = ₹63$; Rice $2 \times 50 = ₹100$; Sugar $0.25 \times 51 = ₹12.75$; Onion $0.5 \times 35 = ₹17.5$. (a) Total = $47.5 + 63 + 100 + 12.75 + 17.5 = $ ₹240.75. (b) Money returned = $300 − 240.75 = $ ₹59.25.

14. The nutrition of a biscuit packet (per biscuit) is — Protein 0.63g, Carbohydrates 5.56g, Saturated Sugars 1.38g, Saturated Fatty Acids 0.87g, Fat 1.73g, Cholesterol 0.00mg, Dietary Fibre 0.50g; and per 100g — 7.37g, 66.48g, 16.30g, 10.64g, 21.14g, 0.00mg, 6.08g. (a) Total nutrition per biscuit in grams (excluding energy)? (b) Total nutrition per 100g in grams? (c) Which nutrient is highest in each biscuit?

Answer: (a) Per biscuit (excluding energy) = $0.63 + 5.56 + 1.38 + 0.87 + 1.73 + 0.00 + 0.50 = $ 10.67 g. (b) Per 100g = $7.37 + 66.48 + 16.30 + 10.64 + 21.14 + 0.00 + 6.08 = $ 128.01 g. (c) The highest nutrient per biscuit is Carbohydrates — 5.56 g.

15. The gaseous composition of the atmosphere (per unit) is — Argon 0.0093, Carbon-di-oxide 0.0003, Oxygen 0.21, Nitrogen 0.78, All others 0.0004. (a) Which gas has the highest contribution? (b) Which gas has the lowest contribution?

Answer: Comparing place values — (a) highest is Nitrogen (0.78); (b) lowest is Carbon-di-oxide (0.0003).

16. Riya saw two juice bottles priced ₹45.75 and ₹45.07. She said, “the second one is cheaper because 07 is smaller than 75.” Do you agree? Justify.

Answer: The conclusion is correct — the second bottle is cheaper — but Riya’s reasoning is not quite right; we should not compare ‘07’ and ‘75’ as whole numbers, we compare by place value. Both have the same whole part (45); at the tenths place 45.75 has 7 and 45.07 has 0, and 0 < 7. So 45.07 < 45.75, i.e. the ₹45.07 bottle is indeed cheaper.

17. Form numbers up to 3 decimal places using the digits 2, 3, 7, 9 (each used once). What are the largest and smallest, and how did you arrange them?

Answer: Each number has one digit before the decimal point and three after it (X.YZW). For the largest, put the biggest digit in the ones place and the rest in descending order — 9.732. For the smallest, put the smallest digit in the ones place and the rest in ascending order — 2.379.

18. Assertion (A): 0.50 and 0.5 represent the same value. Reason (R): Adding zeroes to the right of a decimal number does not change its value.

Answer: (i) Both A and R are true and R is the correct explanation of A.

19. Assertion (A): Adding 0.9 and 0.09 gives 0.99. Reason (R): Estimating the sums of decimal numbers may not always result in the actual values.

Answer: (ii) Both A and R are true, but R is not the correct explanation of A ($0.9 + 0.09 = 0.99$ is true, but estimation is not relevant here).

20. Assertion (A): The difference between 5.6 and 2.48 is 3.12. Reason (R): While subtracting decimals, it is not necessary to align the place values after the decimal point.

Answer: (iii) A is true, but R is false ($5.60 − 2.48 = 3.12$ is correct, but the place values must always be aligned).

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Additional Questions and Answers

Multiple Choice Questions (MCQ)

1. Which two parts of a decimal number does the decimal point separate? (a) two whole numbers (b) the whole part and the fractional part (c) tenths and hundredths (d) none

Answer: (b) the whole part and the fractional part.

2. The decimal form of $\frac{7}{100}$ is — (a) 0.7 (b) 0.07 (c) 7.0 (d) 0.007

Answer: (b) 0.07.

3. The simplest fraction form of 2.5 is — (a) $\frac{25}{10}$ (b) $\frac{5}{2}$ (c) $\frac{2}{5}$ (d) $\frac{1}{2}$

Answer: (b) $\frac{5}{2}$ ($2.5 = \frac{25}{10} = \frac{5}{2}$).

4. Which is the greatest of 0.7, 0.07, 0.707, 0.77? (a) 0.7 (b) 0.07 (c) 0.707 (d) 0.77

Answer: (d) 0.77.

5. What is the value of 3.6 + 0.44? (a) 3.10 (b) 4.04 (c) 4.4 (d) 8.0

Answer: (b) 4.04 ($3.60 + 0.44 = 4.04$).

6. 10 mm is equal to how many cm? (a) 0.1 cm (b) 1 cm (c) 10 cm (d) 100 cm

Answer: (b) 1 cm.

7. 250 g is equal to how many kg? (a) 2.5 (b) 0.25 (c) 0.025 (d) 25

Answer: (b) 0.25 kg ($\frac{250}{1000} = 0.25$).

8. Which is true about 0.3, 0.30 and 0.300? (a) all are different (b) all are equal (c) 0.3 is the greatest (d) 0.300 is the greatest

Answer: (b) all are equal.

9. The number form of “four point zero five” is — (a) 4.5 (b) 4.05 (c) 4.50 (d) 45.0

Answer: (b) 4.05.

10. What is the next term of the sequence 1.5, 1.8, 2.1, …? (a) 2.2 (b) 2.3 (c) 2.4 (d) 2.5

Answer: (c) 2.4 (each step +0.3).

Fill in the Blanks

1. The ‘.’ between the whole part and the fractional part is called the ______.

Answer: decimal point.

2. The decimal form of $\frac{1}{100}$ is ______.

Answer: 0.01.

3. 1 kg = ______ g.

Answer: 1000.

4. The simplest fraction form of 0.25 is ______.

Answer: $\frac{1}{4}$.

5. Adding zeroes to the right of a decimal number does not ______ its value.

Answer: change.

True or False

1. 0.5 and 0.50 represent the same value.

Answer: True.

2. 0.03 is greater than 0.3.

Answer: False (0.03 < 0.3).

3. 1 metre = 100 centimetres.

Answer: True.

4. An estimate always matches the actual value exactly.

Answer: False.

5. In 0.127 the digit in the thousandths place is 7.

Answer: True.

Short Answer Questions

1. What do tenth, hundredth and thousandth mean?

Answer: Dividing a unit into 10 equal parts gives one-tenth ($\frac{1}{10}$); dividing that again into 10 gives one-hundredth ($\frac{1}{100}$); and dividing a hundredth into 10 gives one-thousandth ($\frac{1}{1000}$).

2. Write the place value of each digit in 14.702.

Answer: 1 → tens (10), 4 → ones (4), 7 → tenths (0.7), 0 → hundredths (0.00), 2 → thousandths (0.002). So $14.702 = 10 + 4 + 0.7 + 0.002$.

3. Which is greater, 6.35 or 6.4? Show using place value.

Answer: Writing 6.4 as 6.40, both have whole part 6, and the tenths are 3 versus 4. Since 3 < 4, 6.35 < 6.40, so 6.4 is greater.

4. Express 347 cm in metres.

Answer: $1\text{ cm} = \frac{1}{100}$ m = 0.01 m, so $347\text{ cm} = \frac{347}{100}$ m = 3.47 m.

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Key Terms

Term	Meaning
Decimal	A number with a whole part and a fractional part written using a decimal point
Decimal point	The ‘.’ that separates the whole part from the fractional part
Tenth ($\frac{1}{10}$)	One of ten equal parts of a unit
Hundredth ($\frac{1}{100}$)	One of a hundred equal parts of a unit
Thousandth ($\frac{1}{1000}$)	One of a thousand equal parts of a unit
Place value	The value of a digit according to its position
Number line	A straight line used to represent numbers
Fraction	A number showing part of a whole ($\frac{a}{b}$)
Sequence	A row of numbers arranged by a fixed rule
Estimated value	A close (approximate) value obtained by rounding
Ascending order	Arranged from smallest to largest
Descending order	Arranged from largest to smallest