Multiplication and Division of Decimals — Questions and Answers
Welcome to HSLC Guru. In this chapter you will find the complete, worked solutions for ASSEB Class 7 New Mathematics Chapter 12, Multiplication and Division of Decimals — multiplying decimals by whole numbers, by 10/100/1000 and by other decimals, dividing decimals in every way, cyclic numbers and the magic of 9 — including each “Work it Out” box (12.1–12.7), all of Exercise 12 and both Puzzles.
Summary
To multiply two decimals, first convert them into fractions, multiply, and express the product as a decimal. A quick rule follows from this: the number of digits after the decimal point in the product equals the sum of the digits after the decimal points in the multiplier and the multiplicand. For example, $4.5 × 98.20 = 441.900$, where $(1+2)=3$ digits stand after the decimal point.
The product of two decimals may be greater or smaller than the numbers themselves. If both are greater than 1, the product is greater than each; if both lie between 0 and 1, the product is smaller than each; if one is greater than 1 and the other lies between 0 and 1, the product lies between the two numbers.
To divide decimals, the divisor is first changed to 10, 100, 1000, etc. (or the number is multiplied by the reciprocal). Division can also be done using place value and the long division method. Dividing by 2, 4 or 8 is the same as multiplying by 0.5, 0.25 or 0.125, and dividing by 5, 25 or 125 is the same as multiplying by 0.2, 0.04 or 0.008. A quotient may be terminating, non-terminating repeating, or non-terminating non-repeating.
Summary: This ASSEB Class 7 New Mathematics Chapter 12 (Multiplication and Division of Decimals) guide shows how to multiply decimals by whole numbers, by 10/100/1000 and by decimals, where to place the decimal point in a product, how a product compares with the two numbers, and every method of dividing decimals — by 10/100/1000, by converting the denominator, using place value, long division, and with decimal divisors. It solves each “Work it Out” box (12.1–12.7), all 22 questions of Exercise 12, both Puzzles, and covers cyclic numbers, leap years and the magic of 9.
Textbook Questions and Answers
Work it Out 12.1
1. Find the product— (a) 5.97 × 8 (b) 9.6 × 2.74 (c) 3.47 × 3.14 (d) 2.45 × 4.732
Answer: Multiply as whole numbers, then place the decimal point after (sum of decimal digits) places from the right. (a) $597 × 8 = 4776$; $2$ decimal digits → 47.76. (b) $96 × 274 = 26304$; $1+2=3$ digits → 26.304. (c) $347 × 314 = 108958$; $2+2=4$ digits → 10.8958. (d) $245 × 4732 = 1159340$; $2+3=5$ digits → 11.5934.
2. How much distance does a four-wheeler cover using 13 litres of petrol, if it covers 16.8 km per litre?
Answer: Distance $= 16.8 × 13 = 218.4$ km.
3. If the price of 1 kg of masoor dal is ₹98.50, what is the price of 2.250 kg of masoor dal?
Answer: Price $= 98.50 × 2.250 = $ ₹221.625 (about ₹221.63).
Work it Out 12.2
1. Find the product— (a) 7 × 7.8 (b) 5.7 × 6 (c) 11 × 4.79 (d) 9.2 × 11.56 (e) 15.38 × 3.572 (f) 0.25 × 0.015
Answer: (a) $7 × 7.8 = 54.6$. (b) $5.7 × 6 = 34.2$. (c) $11 × 4.79 = 52.69$. (d) $9.2 × 11.56 = 106.352$. (e) $15.38 × 3.572 = 54.93736$. (f) $0.25 × 0.015 = 0.00375$.
2. Nayan bought 8 eggs at ₹8.25 each and 9 pencils at ₹5.50 each. How much money did he spend?
Answer: Eggs $= 8 × 8.25 = ₹66$; pencils $= 9 × 5.50 = ₹49.50$. Total $= 66 + 49.50 = $ ₹115.50.
3. If the thickness of a ₹5 coin is 1.85 mm, what will be the height, in centimetres, of a cylindrical stack of 23 such coins?
Answer: Height $= 23 × 1.85 = 42.55$ mm. Since $10$ mm $= 1$ cm, $42.55$ mm $= 42.55 ÷ 10 = $ 4.255 cm.
4. If the price of 1 kg of onions is ₹38.50, what is the price of 5.250 kg? Can we write 38.50 and 5.250 as 38.5 and 5.25? Will the value stay the same?
Answer: Price $= 38.50 × 5.250 = $ ₹202.125 (about ₹202.13). Yes — a trailing zero after the decimal point does not change the value, so $38.50 = 38.5$ and $5.250 = 5.25$, and the value remains the same.
5. If 24 × 17 = 408, find— (a) 24 × 1.7 (b) 24 × 0.17 (c) 2.4 × 1.7 (d) 0.24 × 0.17 (e) 0.024 × 0.017 (f) 2.4 × 17
Answer: Only place the decimal point in $408$ according to the total decimal digits. (a) $24 × 1.7 = 40.8$. (b) $24 × 0.17 = 4.08$. (c) $2.4 × 1.7 = 4.08$. (d) $0.24 × 0.17 = 0.0408$. (e) $0.024 × 0.017 = 0.000408$. (f) $2.4 × 17 = 40.8$.
6. Fill up the table by multiplying by 10, 100 and 1000.
Answer: Multiplying by 10, 100, 1000 shifts the decimal point one, two and three places to the right.
| Decimal number | × 10 | × 100 | × 1000 |
|---|---|---|---|
| 3.8 | 38 | 380 | 3800 |
| 64.03 | 640.3 | 6403 | 64030 |
| 0.75 | 7.5 | 75 | 750 |
| 0.209 | 2.09 | 20.9 | 209 |
| 0.023 | 0.23 | 2.3 | 23 |
Work it Out 12.3
1. Identify whether correct or incorrect— (a) The value of 4.7 × 3.21 is smaller than both numbers. (b) The value of 4.7 × 0.321 lies between the numbers. (c) The value of 0.47 × 0.321 is smaller than both numbers.
Answer: (a) Incorrect — both numbers are greater than 1, so $4.7 × 3.21 = 15.087$ is greater than both. (b) Correct — one is greater than 1 and the other lies between 0 and 1, so $4.7 × 0.321 = 1.5087$ lies between $0.321$ and $4.7$. (c) Correct — both lie between 0 and 1, so $0.47 × 0.321 = 0.15087$ is smaller than both.
2. Which of the following products is smaller than 1? Can you tell without finding the product? (a) 8 × 0.9 (b) 0.8 × 0.9 (c) 0.8 × 9 (d) 0.08 × 0.09
Answer: A product is smaller than 1 only when both numbers lie between 0 and 1. That is true for (b) $0.8 × 0.9 = 0.72$ and (d) $0.08 × 0.09 = 0.0072$, so (b) and (d) are smaller than 1. In (a) $8 × 0.9 = 7.2$ and (c) $0.8 × 9 = 7.2$, one number is greater than 1, so the product exceeds 1.
Work it Out 12.4
1. Find the value by converting the denominator to 10, 100, 1000, etc.— (a) 29 ÷ 4 (b) 178 ÷ 8 (c) 4765 ÷ 25 (d) 78.61 ÷ 2 (e) 4218 ÷ 8
Answer: (a) $\frac{29}{4}=\frac{29×25}{100}=\frac{725}{100}=7.25$. (b) $\frac{178}{8}=\frac{178×125}{1000}=\frac{22250}{1000}=22.25$. (c) $\frac{4765}{25}=\frac{4765×4}{100}=\frac{19060}{100}=190.6$. (d) $\frac{78.61}{2}=\frac{78.61×5}{10}=\frac{393.05}{10}=39.305$. (e) $\frac{4218}{8}=\frac{4218×125}{1000}=\frac{527250}{1000}=527.25$.
2. Convert the following fractions into decimals— (a) $\frac{4}{5}$ (b) $\frac{16}{5}$ (c) $\frac{23}{5}$ (d) $\frac{3}{25}$ (e) $\frac{12}{25}$ (f) $\frac{127}{25}$ (g) $\frac{465}{125}$ (h) $\frac{615}{125}$ (i) $\frac{474}{125}$
Answer: (a) $\frac{4×2}{10}=0.8$. (b) $\frac{16×2}{10}=3.2$. (c) $\frac{23×2}{10}=4.6$. (d) $\frac{3×4}{100}=0.12$. (e) $\frac{12×4}{100}=0.48$. (f) $\frac{127×4}{100}=5.08$. (g) $\frac{465×8}{1000}=3.72$. (h) $\frac{615×8}{1000}=4.92$. (i) $\frac{474×8}{1000}=3.792$.
3. Find the quotient— (a) $\frac{17}{4}$ (b) $\frac{37}{4}$ (c) $\frac{175}{2}$ (d) $\frac{9}{8}$ (e) $\frac{13}{8}$ (f) $\frac{67}{4}$ (g) $\frac{143}{4}$ (h) $\frac{5}{8}$ (i) $\frac{39}{8}$ (j) $\frac{371}{125}$
Answer: (a) $4.25$. (b) $9.25$. (c) $87.5$. (d) $1.125$. (e) $1.625$. (f) $16.75$. (g) $35.75$. (h) $0.625$. (i) $4.875$. (j) $2.968$.
Work it Out 12.5
1. Divide using place value— (a) 4214 ÷ 4 (b) 9172 ÷ 8 (c) 5689 ÷ 2 (d) 6418 ÷ 4 (e) 6006 ÷ 8
Answer: Writing each number in expanded form and dividing every place value by the divisor, then adding— (a) $4214 ÷ 4 = 1053.5$. (b) $9172 ÷ 8 = 1146.5$. (c) $5689 ÷ 2 = 2844.5$. (d) $6418 ÷ 4 = 1604.5$. (e) $6006 ÷ 8 = 750.75$.
2. Divide using the long division method— (a) 4525 ÷ 2 (b) 8436 ÷ 8 (c) 6278 ÷ 4 (d) 4975 ÷ 2
Answer: Converting the remainder into tenths, hundredths, etc. and continuing— (a) $4525 ÷ 2 = 2262.5$. (b) $8436 ÷ 8 = 1054.5$. (c) $6278 ÷ 4 = 1569.5$. (d) $4975 ÷ 2 = 2487.5$.
Work it Out 12.6
1. Find the division— (a) 24.03 ÷ 5 (b) 457 ÷ 25 (c) 729 ÷ 2.5 (d) 5789 ÷ 2.5 (e) 321.7 ÷ 1.25 (f) 47.83 ÷ 0.5
Answer: Dividing by 5, 25 and 125 is the same as multiplying by 0.2, 0.04 and 0.008. (a) $24.03 ÷ 5 = 4.806$. (b) $457 ÷ 25 = 18.28$. (c) $729 ÷ 2.5 = 291.6$. (d) $5789 ÷ 2.5 = 2315.6$. (e) $321.7 ÷ 1.25 = 257.36$. (f) $47.83 ÷ 0.5 = 95.66$.
2. Find the division— (a) $\frac{27}{5}$ (b) $\frac{87}{5}$ (c) $\frac{2.97}{5}$ (d) $\frac{136}{5}$ (e) $\frac{30.2}{25}$ (f) $\frac{83}{25}$ (g) $\frac{407}{25}$ (h) $\frac{539}{25}$ (i) $\frac{73}{125}$ (j) $\frac{192}{125}$ (k) $\frac{78.9}{125}$ (l) $\frac{93.04}{125}$
Answer: (a) $5.4$. (b) $17.4$. (c) $0.594$. (d) $27.2$. (e) $1.208$. (f) $3.32$. (g) $16.28$. (h) $21.56$. (i) $0.584$. (j) $1.536$. (k) $0.6312$. (l) $0.74432$.
Work it Out 12.7
1. Fill in the boxes— (a) ⬚ × ⬚ = 4.8 (b) ⬚ × ⬚ = 17.2
Answer: Several answers are possible. (a) e.g. $1.2 × 4 = 4.8$ (or $0.6 × 8$, $2.4 × 2$). (b) e.g. $4.3 × 4 = 17.2$ (or $8.6 × 2$, $1.72 × 10$).
2. Fill in the boxes, where each box represents the value of a ÷ b.
Answer: Shifting both the dividend ($a$) and the divisor ($b$) by the same number of places keeps the quotient unchanged, so each box ($a ÷ b$) is—
| a → / b ↓ | 1316 | 131.6 | 13.16 | 1.316 | 13160 |
|---|---|---|---|---|---|
| 28 | 47 | 4.7 | 0.47 | 0.047 | 470 |
| 2.8 | 470 | 47 | 4.7 | 0.47 | 4700 |
| 0.28 | 4700 | 470 | 47 | 4.7 | 47000 |
| 0.028 | 47000 | 4700 | 470 | 47 | 470000 |
| 280 | 4.7 | 0.47 | 0.047 | 0.0047 | 47 |
3. Using the digits 0, 2, 3, 7, 9 without repeating, fill the boxes so that (a) the product is maximum (Tens Ones . Tenths × Ones . Tenths), (b) the product is minimum.
Answer: (a) Maximum: put the largest digits in the highest place values, so $92.0 × 7.3 = 671.6$ (the same value is given by $73.0 × 9.2 = 671.6$). (b) Minimum: make the second factor as small as possible by placing 0 in its ones place, so $37.9 × 0.2 = 7.58$.
4. Find the product— (a) 2.8 × 10 (b) 5.7 × 100 (c) 3.79 × 100 (d) 4.286 × 100 (e) 12.54 × 100 (f) 6.824 × 1000
Answer: (a) $28$. (b) $570$. (c) $379$. (d) $428.6$. (e) $1254$. (f) $6824$.
5. Find the product— (a) 0.01 × 8 (b) 3.89 × 4.2 (c) 7.21 × 9.04 (d) 0.8 × 11.7 (e) 101.01 × 1.01
Answer: (a) $0.08$. (b) $16.338$. (c) $65.1784$. (d) $9.36$. (e) $102.0201$.
6. What length of wall is needed to enclose a garden measuring 18.25 m × 15.75 m?
Answer: The wall equals the perimeter $= 2 × (18.25 + 15.75) = 2 × 34 = $ 68 m.
7. What is the area of a square whose side is 2.4 cm?
Answer: Area $= $ side $×$ side $= 2.4 × 2.4 = $ 5.76 sq. cm.
8. How far will a vehicle travel with 100 litres of petrol if it covers 15.5 km per litre?
Answer: Distance $= 15.5 × 100 = $ 1550 km.
Cyclic Numbers and the Magic of 9 (in-text explorations)
1. In $1 ÷ 7 = 0.142857142857…$, multiply the repeating number 142857 by 1 to 7 and look for a pattern.
Answer: $1 × 142857 = 142857$; $2 × 142857 = 285714$; $3 × 142857 = 428571$; $4 × 142857 = 571428$; $5 × 142857 = 714285$; $6 × 142857 = 857142$; $7 × 142857 = 999999$. For 1 to 6 the same six digits (1, 4, 2, 8, 5, 7) simply rotate; multiplying by 7 gives $999999$. Such a number is called a cyclic number.
2. Following the pattern of $1 ÷ 9$, $12 ÷ 99$, $123 ÷ 999$, find the other quotients and state their type.
Answer: $1 ÷ 9 = 0.111…$, $2 ÷ 9 = 0.222…$, and so on up to $8 ÷ 9 = 0.888…$ (that is, $n ÷ 9 = 0.\overline{n}$). Also $12 ÷ 99 = 0.121212…$, $123 ÷ 999 = 0.123123…$, $1234 ÷ 9999 = 0.12341234…$, $12345 ÷ 99999 = 0.1234512345…$, and so on. All of these are non-terminating repeating decimals.
Exercise 12
1. 0.18 × 0.12 = ? (A) 0.0216 (B) 0.216 (C) 2.16 (D) 21.6
Answer: (A) 0.0216 [$18 × 12 = 216$, $2+2=4$ decimal digits].
2. 0.01 × 0.001 = ? (A) 1 (B) 0.01 (C) 0.001 (D) 0.00001
Answer: (D) 0.00001 [$1 × 1 = 1$, $2+3=5$ decimal digits].
3. The length of a rectangle is 3.67 cm and its breadth 2.5 cm. What is its area in square centimetres? (A) 0.9175 (B) 9.175 (C) 91.75 (D) 917.5
Answer: (B) 9.175 [$3.67 × 2.5 = 9.175$].
4. 52.7 ÷ 100 = ? (A) 0.0527 (B) 0.527 (C) 527 (D) 52700
Answer: (B) 0.527 [dividing by 100 shifts the point two places left].
5. 52.5 ÷ 0.25 = ? (A) 0.210 (B) 2.10 (C) 21.0 (D) 210
Answer: (D) 210 [$\frac{52.5}{0.25}=\frac{5250}{25}=210$].
6. If 48 × 25 = 1200, then 4.8 × 2.5 = ? (A) 0.12 (B) 1.2 (C) 12 (D) 120
Answer: (C) 12 [$1200$ with $1+1=2$ decimal digits → $12.00$].
7. If 1651 ÷ 13 = 127, then 16.51 ÷ 13 = ? (A) 0.127 (B) 1.27 (C) 12.7 (D) 1270
Answer: (B) 1.27.
8. 52.5 ÷ 5.25 = ? (A) 100 (B) 10 (C) 1 (D) 0.1
Answer: (B) 10.
9. Find the quotient— (a) 21.4 ÷ 10 (b) 0.52 ÷ 10 (c) 521.1 ÷ 10 (d) 236.75 ÷ 100 (e) 527.33 ÷ 100 (f) 0.01 ÷ 100 (g) 1.482 ÷ 1000 (h) 0.7 ÷ 1000
Answer: (a) $2.14$. (b) $0.052$. (c) $52.11$. (d) $2.3675$. (e) $5.2733$. (f) $0.0001$. (g) $0.001482$. (h) $0.0007$.
10. Find the quotient— (a) 3.96 ÷ 4 (b) 14.49 ÷ 7 (c) 86.1 ÷ 3 (d) 107.52 ÷ 7 (e) 66.33 ÷ 11
Answer: (a) $0.99$. (b) $2.07$. (c) $28.7$. (d) $15.36$. (e) $6.03$.
11. Find the quotient— (a) 0.5 ÷ 0.25 (b) 8.64 ÷ 0.2 (c) 329.4 ÷ 0.04 (d) 76.5 ÷ 0.125 (e) 48.56 ÷ 3.2
Answer: (a) $2$. (b) $43.2$. (c) $8235$. (d) $612$. (e) $15.175$.
12. If the price of 5.5 metres of cloth is ₹547.25, what is the price of 1 metre?
Answer: Price of 1 m $= 547.25 ÷ 5.5 = $ ₹99.50.
13. If a car covers 150.5 km in 3.5 hours, how far will it go in 1 hour?
Answer: Distance in 1 hour $= 150.5 ÷ 3.5 = $ 43 km.
14. The perimeter of a square vegetable garden is 76.8 m. Find its area.
Answer: Side $= 76.8 ÷ 4 = 19.2$ m. Area $= 19.2 × 19.2 = $ 368.64 sq. m.
15. Using 137 × 16 = 2192, find— (a) 1.37 × 16 (b) 13.7 × 1.6 (c) 1.37 × 0.16 (d) 0.137 × 1.6 (e) 0.137 × 0.16 (f) 2192 ÷ 16 (g) 21.92 ÷ 1.6 (h) 2.192 ÷ 1.37 (i) 2.192 ÷ 0.16 (j) 0.2192 ÷ 0.016
Answer: (a) $21.92$. (b) $21.92$. (c) $0.2192$. (d) $0.2192$. (e) $0.02192$. (f) $137$. (g) $13.7$. (h) $1.6$. (i) $13.7$. (j) $13.7$.
16. Using 1476 ÷ 12 = 123, find— (a) 147.6 ÷ 12 (b) 14.76 ÷ 1.2 (c) 1.476 ÷ 0.12 (d) 0.1476 ÷ 0.12 (e) 14.76 ÷ 0.012
Answer: (a) $12.3$. (b) $12.3$. (c) $12.3$. (d) $1.23$. (e) $1230$.
17. Fill in the boxes— (a) 46 ÷ ⬚ = 460 (b) 46 ÷ ⬚ = 4600 (c) 46 ÷ 100 = ⬚ (d) 46 ÷ 0.01 = ⬚ (e) 46 × 0.001 = ⬚ (f) 46 × ⬚ = 0.0046 (g) 46 × ⬚ = 46 ÷ 0.10 (h) 46 ÷ 0.01 = 46 × ⬚
Answer: (a) $0.1$. (b) $0.01$. (c) $0.46$. (d) $4600$. (e) $0.046$. (f) $0.0001$. (g) $10$. (h) $100$.
18. Find the quotient— (a) 425 ÷ 25 (b) 0.425 ÷ 0.25 (c) 4.25 ÷ 0.25 (d) 42.5 ÷ 2.5 (e) 4.25 ÷ 2.5 (f) 0.425 ÷ 0.025. Which quotients are the same and why?
Answer: (a) $17$. (b) $1.7$. (c) $17$. (d) $17$. (e) $1.7$. (f) $17$. (a), (c), (d) and (f) are all 17, and (b) and (e) are both 1.7. This is because multiplying both the dividend and divisor by the same power of 10 leaves the quotient unchanged; each of these reduces to $\frac{425}{25}=17$ or $\frac{42.5}{25}=1.7$.
19. If 5 litres of milk are shared equally among 8 players, how much does each get?
Answer: Each player gets $= 5 ÷ 8 = $ 0.625 litre.
20. A car covers 459 km using 25.50 litres of petrol. How much distance does it cover per litre?
Answer: Distance per litre $= 459 ÷ 25.50 = $ 18 km.
21. A 250 g biscuit packet costs ₹62.50 and a 200 g cake packet costs ₹54. Which has the lower price per gram?
Answer: Biscuit per gram $= 62.50 ÷ 250 = ₹0.25$; cake per gram $= 54 ÷ 200 = ₹0.27$. So the biscuit has the lower price per gram (₹0.25).
22. Arrange the following products and quotients in ascending order— (a) 325.05 × 0.45 (b) 325.05 × 4.5 (c) 325.05 × 0.045 (d) 325.05 ÷ 0.024 (e) 325.05 ÷ 0.24
Answer: The values are (a) $146.2725$, (b) $1462.725$, (c) $14.62725$, (d) $13543.75$, (e) $1354.375$. Ascending order: (c) < (a) < (e) < (b) < (d).
Puzzle 1
Step 1: Consider a number. Step 2: Add 2.5. Step 3: Multiply the sum by 0.5. Step 4: Subtract 0.75. Step 5: Divide the difference by 0.5. Step 6: Subtract 1. Step 7: What do you get?
Answer: Let the number be $x$. Adding gives $x + 2.5$; multiplying by $0.5$ gives $0.5x + 1.25$; subtracting $0.75$ gives $0.5x + 0.5$; dividing by $0.5$ gives $x + 1$; subtracting $1$ gives $x$. So whatever number you start with, you always get back the original number. For example, with $x = 6$: $6 → 8.5 → 4.25 → 3.5 → 7 → 6$.
Puzzle 2
Colour the path by identifying the correct route from A to B (a maze of decimal multiplications and divisions).
Answer: This is a path-finding activity. Starting at A, apply the operation (× or ÷ and + or −) in each cell in order; the correct route is the one whose every intermediate result matches the number printed in the next cell and finally gives 12.5 at B — that route is the one to colour. For example, from the starting $4$: $4 × 2.5 = 10$, then $10 + 2.5 = 12.5$, reaching B with the value $12.5$.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. The product 2.5 × 0.4 is — (a) 0.1 (b) 1.0 (c) 10 (d) 0.01
Answer: (b) 1.0 [$25 × 4 = 100$, $2$ decimal digits → $1.00$].
2. Dividing a decimal number by 1000 shifts the decimal point how many places to the left? (a) one (b) two (c) three (d) four
Answer: (c) three.
3. 4.56 × 10 = ? (a) 0.456 (b) 4.56 (c) 45.6 (d) 456
Answer: (c) 45.6.
4. Multiplying a number by 0.125 is the same as dividing it by — (a) 2 (b) 4 (c) 8 (d) 5
Answer: (c) 8 [since $\frac{1}{8}=0.125$].
5. If the multiplier and multiplicand have 2 and 3 decimal digits, the product has how many decimal digits? (a) 2 (b) 3 (c) 5 (d) 6
Answer: (c) 5 [$2 + 3 = 5$].
6. $0.333…$ is which kind of decimal? (a) terminating (b) non-terminating repeating (c) non-terminating non-repeating (d) none
Answer: (b) non-terminating repeating decimal.
7. Which product lies between the two numbers? (a) 2.5 × 3.6 (b) 0.5 × 0.6 (c) 4.2 × 0.5 (d) 0.1 × 0.2
Answer: (c) 4.2 × 0.5 [one is greater than 1 and the other lies between 0 and 1, so the product lies between them].
8. The repeating block in the decimal form of $\frac{1}{7}$ is — (a) 142857 (b) 123456 (c) 111111 (d) 987654
Answer: (a) 142857.
9. The value of 87 ÷ 2.5 is — (a) 3.48 (b) 34.8 (c) 348 (d) 0.348
Answer: (b) 34.8 [$\frac{87×10}{25}=\frac{870}{25}=34.8$].
10. Which of these years is NOT a leap year? (a) 2000 (b) 2004 (c) 2100 (d) 2400
Answer: (c) 2100 [a century year divisible by 4 and 100 but not by 400 is not a leap year].
Fill in the Blanks
1. To multiply two decimals, they are first converted into ______.
Answer: fractions.
2. Dividing a number by 4 is the same as multiplying it by ______.
Answer: 0.25.
3. Dividing by 10 is the same as multiplying by the reciprocal of 10, that is ______.
Answer: $\frac{1}{10}$ (i.e. 0.1).
4. A decimal whose expansion ends at a fixed place is called a ______ decimal.
Answer: terminating.
5. The repeating number 142857 in the quotient of $1 ÷ 7$ is called a ______ number.
Answer: cyclic.
True or False
1. If both numbers lie between 0 and 1, the product is smaller than both.
Answer: True.
2. $37.5 × 100 = 375$.
Answer: False ($37.5 × 100 = 3750$).
3. A trailing zero after the decimal point does not change the value of a decimal number.
Answer: True ($2.50 = 2.5$).
4. The quotient of a decimal division always terminates.
Answer: False (it may be non-terminating repeating, e.g. $1 ÷ 3 = 0.333…$).
5. Dividing a number by a number smaller than 1 gives a quotient larger than the dividend.
Answer: True.
Short Answer Questions
1. Where is the decimal point placed in the product of two decimals?
Answer: Counting from the rightmost digit of the product, place the decimal point after as many digits as the sum of the decimal digits in the multiplier and the multiplicand. For example, in $4.5 × 98.20$, $(1 + 2) = 3$ digits are counted to give $441.900$.
2. Find $31 ÷ 8$ using multiplication.
Answer: Dividing by 8 is the same as multiplying by 0.125, so $31 ÷ 8 = 31 × 0.125 = 3.875$.
3. Give one example each of a terminating and a non-terminating repeating decimal.
Answer: Terminating — $\frac{17}{4}=4.25$ (ends at a fixed place). Non-terminating repeating — $\frac{1}{3}=0.333…$ (never ends but the digit repeats).
4. When is a year a leap year?
Answer: A year is a leap year if it is divisible by 4; but a century year (like 1900 or 2100) is a leap year only when it is also divisible by 400. So 2000 and 2400 are leap years, while 2100, 2200 and 2300 are not.
Key Terms
| Term | Meaning |
|---|---|
| Decimal number | A number that contains a decimal point |
| Multiplier | The number by which we multiply |
| Multiplicand | The number that is multiplied |
| Product | The result of multiplication |
| Dividend | The number that is divided |
| Divisor | The number by which we divide |
| Quotient | The result of division |
| Reciprocal | The inverted fraction; the reciprocal of 10 is 1/10 |
| Terminating decimal | A decimal whose expansion ends at a fixed place |
| Non-terminating repeating decimal | A decimal that never ends but whose digits repeat |
| Cyclic number | A number whose digits rotate on multiplication, e.g. 142857 |
| Leap year | A year of 366 days |