Chapter 4 exercise answers: Another Peek Beyond the Point

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(a) 6 × 4 tenths = 24 tenths

1. 6 × 4 tenths means 6 groups of 4 tenths.
2. 6 × 4 = 24, so the product is 24 tenths.

Answer24 tenths = 2.4

(b) 7 × 0.3

1. 7 × 0.3 means 7 groups of 3 tenths.
2. 7 × 3 = 21, so the product is 21 tenths.

Answer21 tenths = 2.1

(c) 9 × 5 hundredths

1. 9 × 5 hundredths means 9 groups of 5 hundredths.
2. 9 × 5 = 45, so the product is 45 hundredths.

Answer45 hundredths = 0.45

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(a) 27.34 × 6

1. Multiply the digits as whole numbers: 2734 × 6 = 16404.
2. 27.34 has 2 decimal places and 6 has 0, so the answer has 2 decimal places.

Answer164.04

(b) 4.23 × 3.7

1. Multiply the digits as whole numbers: 423 × 37 = 15651.
2. 4.23 has 2 decimal places and 3.7 has 1, so the answer has 3 decimal places.

Answer15.651

(c) 0.432 × 0.23

1. Multiply the digits as whole numbers: 432 × 23 = 9936.
2. 0.432 has 3 decimal places and 0.23 has 2, so the answer has 5 decimal places.

Answer0.09936

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1. Cloth for 1 shirt is 1.65 m.
2. For 3 shirts, multiply: 1.65 × 3.
3. 165 × 3 = 495, and 1.65 has 2 decimal places, so the answer also has 2 decimal places.
4. 1.65 × 3 = 4.95 m.

Answer4.95 m

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1. Cost of 4 notebooks = 4 × ₹15.50 = ₹62.
2. Cost of 3 erasers = 3 × ₹2.75 = ₹8.25.
3. Total spent = ₹62 + ₹8.25 = ₹70.25.

Answer₹70.25

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1. Total height in mm = 1.45 × 36.
2. 145 × 36 = 5220, and 1.45 has 2 decimal places, so 1.45 × 36 = 52.20 mm.
3. Convert to centimetres by dividing by 10 (1 cm = 10 mm): 52.2 ÷ 10 = 5.22 cm.

Answer5.22 cm

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1. Price of 2.250 kg = ₹56.50 × 2.250.
2. Multiply as whole numbers: 5650 × 2250 = 12712500.
3. 56.50 has 2 decimal places and 2.250 has 3, so the answer has 5 decimal places: ₹127.12500, which is ₹127.125.
4. A zero written after the last non-zero digit of a decimal does not change its value, so 56.50 = 56.5 and 2.250 = 2.25.
5. 56.5 × 2.25 also works out to 127.125, so yes, both ways give the same product.

AnswerPrice = ₹127.125; yes, 56.50 × 2.250 and 56.5 × 2.25 give the same product, because trailing zeros after the decimal point do not change a number's value.

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1. Profit on each notebook = ₹30 − ₹23.6 = ₹6.4.
2. Profit on 50 notebooks = ₹6.4 × 50.
3. 64 × 50 = 3200, and 6.4 has 1 decimal place, so 6.4 × 50 = 320.

Answer₹320

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(a) 18 × 1.2

1. The digits come from 18 × 12 = 216.
2. 18 has 0 decimal places and 1.2 has 1, so the product has 0 + 1 = 1 decimal place: 21.6.

Answer21.6 (not less than 1)

(b) 18 × 0.12

1. The digits come from 18 × 12 = 216.
2. 18 has 0 decimal places and 0.12 has 2, so the product has 2 decimal places: 2.16.

Answer2.16 (not less than 1)

(c) 1.8 × 1.2

1. The digits come from 18 × 12 = 216.
2. 1.8 has 1 decimal place and 1.2 has 1, so the product has 2 decimal places: 2.16.

Answer2.16 (not less than 1)

(d) 0.18 × 0.12

1. The digits come from 18 × 12 = 216.
2. 0.18 has 2 decimal places and 0.12 has 2, so the product needs 4 decimal places.
3. 216 has only 3 digits, so 1 zero is written in front of it to fill 4 places: 0.0216.

Answer0.0216 (less than 1)

(e) 0.018 × 0.012

1. The digits come from 18 × 12 = 216.
2. 0.018 has 3 decimal places and 0.012 has 3, so the product needs 6 decimal places.
3. 216 has only 3 digits, so 3 zeros are written in front of it to fill 6 places: 0.000216.

Answer0.000216 (less than 1)

(f) 1.8 × 12

1. The digits come from 18 × 12 = 216.
2. 1.8 has 1 decimal place and 12 has 0, so the product has 1 decimal place: 21.6.

Answer21.6 (not less than 1)

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(a) 7 × 0.6

1. 7 is above 1 and 0.6 is below 1, so the product lands between them — bigger than 0.6.
2. 7 × 0.6 = 4.2, which is greater than 1.

Answer4.2 (not less than 1)

(b) 0.7 × 0.6

1. Both 0.7 and 0.6 are below 1, so their product must be below both of them, and so below 1.
2. 0.7 × 0.6 = 0.42.

Answer0.42 (less than 1)

(c) 0.7 × 6

1. 0.7 is below 1 and 6 is above 1, so the product lands between them — bigger than 0.7.
2. 0.7 × 6 = 4.2, which is greater than 1.

Answer4.2 (not less than 1)

(d) 0.07 × 0.06

1. Both 0.07 and 0.06 are below 1, so their product must be below 1 too.
2. 0.07 × 0.06 = 0.0042.

Answer0.0042 (less than 1)

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1. Number	× 10	× 100	× 1000
   5.7	57	570	5700
   23.02	230.2	2302	23020
   0.92	9.2	92	920
   0.306	3.06	30.6	306
   24.67	246.7	2467	24670

2. Multiplying by 10, 100 or 1000 moves the decimal point 1, 2 or 3 places to the right, adding a zero at the end whenever there is no digit left to shift into.

AnswerSee the completed table above.

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(a) 18/5

1. Multiply the numerator and denominator by 2 so the denominator becomes 10: 18/5 = 36/10.
2. 36/10 = 3.6.
3. Check by long division: 18 ÷ 5 = 3, remainder 3; 3 ones become 30 tenths, and 30 ÷ 5 = 6 tenths.

Answer3.6

(b) 415/4

1. Multiply the numerator and denominator by 25 so the denominator becomes 100: 415/4 = 10375/100.
2. 10375/100 = 103.75.
3. Check by long division: 415 ÷ 4 = 103, remainder 3; 3 ones become 30 tenths, giving 7 tenths, remainder 2; 2 tenths become 20 hundredths, giving 5 hundredths.

Answer103.75

(c) 1217/2

1. Multiply the numerator and denominator by 5 so the denominator becomes 10: 1217/2 = 6085/10.
2. 6085/10 = 608.5.
3. Check by long division: 1217 ÷ 2 = 608, remainder 1; 1 one becomes 10 tenths, giving 5 tenths.

Answer608.5

(d) 4827/8

1. Multiply the numerator and denominator by 125 so the denominator becomes 1000: 4827/8 = 603375/1000.
2. 603375/1000 = 603.375.
3. Check by long division: 4827 ÷ 8 = 603, remainder 3; 3 ones become 30 tenths, giving 3 tenths, remainder 6; 6 tenths become 60 hundredths, giving 7 hundredths, remainder 4; 4 hundredths become 40 thousandths, giving 5 thousandths.

Answer603.375

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(a) 1526/4 = (i) 38.15 (ii) 380.15 (iii) 381.5 (iv) 381.05

1. Divide 1526 by 4: 1526 ÷ 4 = 381, remainder 2.
2. 2 ones become 20 tenths; 20 ÷ 4 = 5 tenths, remainder 0.
3. 1526/4 = 381.5, which matches option (iii).

Answer381.5, option (iii)

(b) 3567/8 = (i) 4458.75 (ii) 44.5875 (iii) 445.875 (iv) 4458.75

1. Divide 3567 by 8: 3567 ÷ 8 = 445, remainder 7.
2. 7 ones become 70 tenths; 70 ÷ 8 = 8 tenths, remainder 6.
3. 6 tenths become 60 hundredths; 60 ÷ 8 = 7 hundredths, remainder 4.
4. 4 hundredths become 40 thousandths; 40 ÷ 8 = 5 thousandths.
5. 3567/8 = 445.875, which matches option (iii).

Answer445.875, option (iii)

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(a) 132 ÷ 4 =

1. 132 ÷ 4 = 33.

Answer33

(b) 13.2 ÷ 4 =

1. 13.2 has the same digits as 132, but is one place smaller, so 13.2 ÷ 4 is one-tenth of 132 ÷ 4.
2. 13.2 ÷ 4 = 3.3.

Answer3.3

(c) 1.32 ÷ 4 =

1. 1.32 is one-tenth of 13.2, so 1.32 ÷ 4 is one-tenth of 3.3.
2. 1.32 ÷ 4 = 0.33.

Answer0.33

(d) 0.132 ÷ 4 =

1. 0.132 is one-tenth of 1.32, so 0.132 ÷ 4 is one-tenth of 0.33.
2. 0.132 ÷ 4 = 0.033.

Answer0.033

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(a) 126 ÷ 8 =

1. 126 ÷ 8 = 15, remainder 6; 6 ones become 60 tenths; 60 ÷ 8 = 7 tenths, remainder 4; 4 tenths become 40 hundredths; 40 ÷ 8 = 5 hundredths.
2. 126 ÷ 8 = 15.75.

Answer15.75

(b) 12.6 ÷ 8 =

1. 12.6 is one-tenth of 126, so 12.6 ÷ 8 is one-tenth of 15.75.
2. 12.6 ÷ 8 = 1.575.

Answer1.575

(c) 1.26 ÷ 8 =

1. 1.26 is one-tenth of 12.6, so 1.26 ÷ 8 is one-tenth of 1.575.
2. 1.26 ÷ 8 = 0.1575.

Answer0.1575

(d) 0.126 ÷ 8 =

1. 0.126 is one-tenth of 1.26, so 0.126 ÷ 8 is one-tenth of 0.1575.
2. 0.126 ÷ 8 = 0.01575.

Answer0.01575

(e) 0.0126 ÷ 8 =

1. 0.0126 is one-tenth of 0.126, so 0.0126 ÷ 8 is one-tenth of 0.01575.
2. 0.0126 ÷ 8 = 0.001575.

Answer0.001575

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(a) 2/5

1. Multiply the numerator and denominator by 2 to get a denominator of 10: 2/5 = 4/10.
2. 4/10 = 0.4.

Answer0.4

(b) 13/4

1. Divide 13 by 4: 13 ÷ 4 = 3, remainder 1; 1 one becomes 10 tenths, giving 2 tenths, remainder 2; 2 tenths become 20 hundredths, giving 5 hundredths.
2. 13/4 = 3.25.

Answer3.25

(c) 4/50

1. 4/50 simplifies to 2/25.
2. Multiply the numerator and denominator by 4 to get a denominator of 100: 2/25 = 8/100.
3. 8/100 = 0.08.

Answer0.08

(d) 5/8

1. Divide 5 by 8: 5 ÷ 8 = 0, remainder 5; 5 ones become 50 tenths, giving 6 tenths, remainder 2; 2 tenths become 20 hundredths, giving 2 hundredths, remainder 4; 4 hundredths become 40 thousandths, giving 5 thousandths.
2. 5/8 = 0.625.

Answer0.625

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(a) 24.86 ÷ 1.2

1. Multiply both numbers by 10 so the divisor becomes a whole number: 24.86 ÷ 1.2 = 248.6 ÷ 12.
2. 248 ÷ 12 = 20, remainder 8.
3. 8 ones and 6 tenths make 86 tenths; 86 ÷ 12 = 7 tenths, remainder 2.
4. 2 tenths become 20 hundredths; 20 ÷ 12 = 1 hundredth, remainder 8.
5. 8 hundredths become 80 thousandths; 80 ÷ 12 = 6 thousandths, remainder 8.
6. The remainder 8 has come back, so the digit 6 repeats for ever: the division never ends.
7. As a fraction, 24.86 ÷ 1.2 = 2486/120 = 1243/60.

Answer1243/60 = 20.71666… (the 6 repeats for ever), about 20.72

(b) 5.728 ÷ 1.52

1. Multiply both numbers by 100 so the divisor becomes a whole number: 5.728 ÷ 1.52 = 572.8 ÷ 152.
2. 572 ÷ 152 = 3, remainder 116.
3. 116 ones and 8 tenths make 1168 tenths; 1168 ÷ 152 = 7 tenths, remainder 104.
4. 104 tenths become 1040 hundredths; 1040 ÷ 152 = 6 hundredths, remainder 128.
5. 1280 thousandths ÷ 152 = 8, remainder 64; then 640 ÷ 152 = 4, remainder 32; and so on.
6. As a fraction, 5.728 ÷ 1.52 = 5728/1520 = 358/95, and 95 = 5 × 19. Because 19 is not made of 2s and 5s, the division never ends.
7. Carrying on, the digits after 3.7 repeat in a block of 18: 684210526315789473.

Answer358/95 = 3.7684210526315789473… (the block 684210526315789473 repeats for ever), about 3.77

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(a) 15.6 × 1.2 =

1. 156 × 12 = 1872 gives the digits.
2. 15.6 has 1 decimal place and 1.2 has 1, so the product has 2 decimal places: 18.72.

Answer18.72

(b) 187.2 ÷ 1.2 =

1. 187.2 ÷ 1.2 is the same division as 1872 ÷ 12, with both numbers scaled up 10 times.
2. 1872 ÷ 12 = 156, so 187.2 ÷ 1.2 = 156.

Answer156

(c) 18.72 ÷ 15.6 =

1. Multiply both numbers by 10 to clear the decimals: 18.72 ÷ 15.6 becomes 187.2 ÷ 156.
2. Since 156 × 12 = 1872, dividing 1872 by 156 gives 12, so 187.2 ÷ 156 = 1.2.

Answer1.2

(d) 0.156 × 0.12 =

1. 156 × 12 = 1872 gives the digits.
2. 0.156 has 3 decimal places and 0.12 has 2, so the product has 5 decimal places: 0.01872.

Answer0.01872

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(a) 25 ÷ ___ = 0.025

1. 25 ÷ ____ = 0.025
2. To find the divisor: ____ = 25 ÷ 0.025
3. = 25 ÷ (25/1000) = 1000

Answer1000

(b) 25 ÷ ___ = 250

1. 25 ÷ ____ = 250
2. To find the divisor: ____ = 25 ÷ 250
3. = 1/10 or 0.1

Answer1/10

(c) 25 ÷ ___ = 2.5

1. 25 ÷ ____ = 2.5
2. To find the divisor: ____ = 25 ÷ 2.5
3. = 10

Answer10

(d) 25 ÷ 10 = 25 × ___

1. 25 ÷ 10 = 2.5
2. So 2.5 = 25 × ____
3. Therefore ____ = 2.5 ÷ 25 = 1/10

Answer1/10

(e) 25 ÷ 0.10 = 25 × ___

1. 25 ÷ 0.10 = 25 ÷ (1/10) = 25 × 10 = 250
2. So ____ = 10

Answer10

(f) 25 ÷ 0.01 = 25 × ___

1. 25 ÷ 0.01 = 25 ÷ (1/100) = 25 × 100 = 2500
2. So ____ = 100

Answer100

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(a) 2.46 ÷ 1.5 =

1. Write both numbers as fractions: 2.46 = 246/100 and 1.5 = 15/10.
2. 2.46 ÷ 1.5 = 246/100 × 10/15 = 2460/1500 = 41/25.
3. 41/25 = 164/100 = 1.64.

Answer1.64

(b) 2.46 ÷ 0.15 =

1. Multiply both numbers by 100 so that the divisor becomes a whole number: 2.46 ÷ 0.15 = 246 ÷ 15. Multiplying both by the same number does not change the quotient.
2. 246 ÷ 15: 15 × 16 = 240, so 16 ones each with 6 ones left over.
3. 6 ones = 60 tenths, and 60 ÷ 15 = 4 tenths. So 246 ÷ 15 = 16.4.

Answer16.4

(c) 2.46 ÷ 0.015 =

1. Multiply both numbers by 1000: 2.46 ÷ 0.015 = 2460 ÷ 15.
2. 2460 ÷ 15 = 164.

Answer164

same? Is the quotient obtained in 24.6 ÷ 1.5 the same …?

1. Multiply both numbers in 24.6 ÷ 1.5 by 10: it becomes 246 ÷ 15.
2. In part (b), 2.46 ÷ 0.15 also became 246 ÷ 15.
3. So both are the same division, 246 ÷ 15 = 16.4.

AnswerYes — both quotients are 16.4.

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1. Share 4 m equally among 5 pieces: 4 ÷ 5.
2. 4 ones cannot be split into 5 whole parts, so regroup 4 as 40 tenths.
3. 40 tenths ÷ 5 = 8 tenths, which is 0.8 m.

Answer0.8 m

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1. A regular polygon with 12 sides has 12 equal sides, so each side = 208.8 ÷ 12.
2. Share 208 ones among 12: 12 × 17 = 204, so 17 ones each and 4 ones left over.
3. The 4 leftover ones are 40 tenths; with the 8 tenths already there, that makes 48 tenths.
4. 48 tenths ÷ 12 = 4 tenths, with nothing left over.
5. So each side is 17.4 cm. Check: 17.4 × 12 = 208.8.

Answer17.4 cm

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1. 3 litres = 3000 millilitres.
2. Share 3000 ml among 8 friends: 3000 ÷ 8.
3. 3000 ÷ 8 = 375.

Answer375 ml

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1. Distance per litre = total distance ÷ litres used = 234.45 ÷ 12.6.
2. Multiply both numbers by 10 so that the divisor is a whole number: 2344.5 ÷ 126. The quotient does not change.
3. 2344 ones ÷ 126: 126 × 18 = 2268, so 18 ones each and 76 ones left over.
4. 76 ones and 5 tenths make 765 tenths; 765 ÷ 126 = 6 tenths each, 9 tenths left over.
5. 9 tenths = 90 hundredths, and 90 ÷ 126 = 0 hundredths. 900 thousandths ÷ 126 = 7 thousandths, 18 left over.
6. The division never ends: 18.6071… As a fraction the answer is exactly 521/28, which is about 18.61.

AnswerAbout 18.61 km per litre (exactly 521/28 km)

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1. Share 13.5 kg among 15 students: 13.5 ÷ 15.
2. 13 ones among 15 gives 0 each with 13 left, so regroup: 135 tenths ÷ 15 = 9 tenths.
3. 13.5 ÷ 15 = 0.9 kg.

Answer0.9 kg

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1. Find the price per gram for each packet: price ÷ weight.
2. Chikki: ₹70.5 ÷ 210 g = ₹0.3357… per gram.
3. Chips: ₹33.25 ÷ 110 g = ₹0.3023… per gram.
4. 0.3023 is less than 0.3357, so the chips cost less per gram.

AnswerThe potato chips are cheaper (per gram).

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line 1 3.1 … 3.2

1. The line from 3.1 to 3.2 is split into 10 equal gaps, so each gap is 0.01.
2. Counting the tick marks from 3.1, the arrow sits on the 6th tick after 3.1.
3. 3.1 + 6 × 0.01 = 3.16.

Answer3.16

line 2 2.15 … 2.17

1. The line from 2.15 to 2.17 is split into 10 equal gaps, so each gap is 0.002.
2. Counting the tick marks from 2.15, the arrow sits on the 6th tick after 2.15.
3. 2.15 + 6 × 0.002 = 2.162.

Answer2.162

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1. Cost price = 3 kg × ₹30 per kg = ₹90.
2. Selling price = 35 bananas × ₹5 each = ₹175.
3. Profit = selling price − cost price = 175 − 90 = ₹85.

Answer₹85

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1. Each book takes 2.5 cm, and the shelf is 160 cm long.
2. Number of books that fit = 160 ÷ 2.5 = 64.
3. Space used = 64 × 2.5 cm = 160 cm, which is exactly the shelf length.
4. So all 64 books fit exactly, with no space left over (the teacher cannot fit all 80).

Answer64 books fit, and the space left over is 0 cm: the shelf is exactly full.

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1. Conversion	Multiply/divide by	Result
   5.5 km → m	× 1000	5500 m
   35 cm → m	÷ 100	0.35 m
   14.5 cm → mm	× 10	145 mm
   68 g → kg	÷ 1000	0.068 kg
   9.02 m → mm	× 1000	9020 mm
   125.5 ml → l	÷ 1000	0.1255 l

2. Multiplying by 10, 100 or 1000 moves the decimal point right; dividing moves it left.

Answer5500 m; 0.35 m; 145 mm; 0.068 kg; 9020 mm; 0.1255 l

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6 1/4 ÷ 2 1/2

1. 6 1/4 = 6.25 and 2 1/2 = 2.5.
2. 6.25 ÷ 2.5: scale both by 10 to get 62.5 ÷ 25.
3. 62.5 ÷ 25 = 2.5.

Answer2.5

60 1/4 ÷ 3 1/2

1. 60 1/4 = 60.25 and 3 1/2 = 3.5.
2. 60.25 ÷ 3.5: scale both by 10 to get 602.5 ÷ 35.
3. 602.5 ÷ 35 = 17.2142857…, a repeating decimal, equal to 241/14.

Answer17.214285… (241/14)

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Open NCERT p. 94Checked by computerAnswers can differ: one example

(a) [ ] × [ ] = 2.4

1. Many pairs work: any two numbers whose product is 2.4. Here are two.
2. 1.2 × 2 = 2.4, because 12 × 2 = 24 and there is one decimal place.
3. 0.6 × 4 = 2.4, because 6 × 4 = 24 and there is one decimal place.

Answer1.2 × 2 = 2.4 and 0.6 × 4 = 2.4 (other pairs also work)

(b) [ ] × [ ] = 14.5

1. Many pairs work here too. Here are two.
2. 2.9 × 5 = 14.5, because 29 × 5 = 145 and there is one decimal place.
3. 1.45 × 10 = 14.5, because multiplying by 10 moves the point one place right.

Answer2.9 × 5 = 14.5 and 1.45 × 10 = 14.5 (other pairs also work)

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(a) 75.6 ÷ 3.6

1. 75.6 is 756 ÷ 10 and 3.6 is 36 ÷ 10. Both were divided by 10, so the quotient does not change.

Answer21

(b) 7.56 ÷ 0.36

1. 7.56 is 756 ÷ 100 and 0.36 is 36 ÷ 100. Both were divided by 100, so the quotient does not change.

Answer21

(c) 756 ÷ 0.36

1. 756 is unchanged, but 0.36 is 36 ÷ 100. Dividing by a number 100 times smaller gives a quotient 100 times bigger: 21 × 100 = 2100.

Answer2100

(d) 75.6 ÷ 360

1. 75.6 is 756 ÷ 10, which makes the quotient 10 times smaller; 360 is 36 × 10, which makes it 10 times smaller again. 21 ÷ 100 = 0.21.

Answer0.21

(e) 7560 ÷ 3.6

1. 7560 is 756 × 10, which makes the quotient 10 times bigger; 3.6 is 36 ÷ 10, which makes it 10 times bigger again. 21 × 100 = 2100.

Answer2100

(f) 7.56 ÷ 0.36

1. This is the same division as (b), 7.56 ÷ 0.36, so the answer is again 21.

Answer21

Watch this explained “One known division”, 8:35 into Dividing when the divisor has a decimal · हिंदी में देखें

Open NCERT p. 94Checked by computer

1. Start from the one given fact: 1517 ÷ 37 = 41.
2. Each column to the right divides a by 10 (151.7, 15.17, 1.517) or multiplies it by 10 (15170); each row down divides b by 10 (3.7, 0.37, 0.037) or multiplies it by 10 (370).
3. Shrinking a by 10 shrinks the quotient by 10; shrinking b by 10 grows the quotient by 10 — the two effects combine.

4. b \ a	1517	151.7	15.17	1.517	15170
   37	41	4.1	0.41	0.041	410
   3.7	410	41	4.1	0.41	4100
   0.37	4100	410	41	4.1	41000
   0.037	41000	4100	410	41	410000
   370	4.1	0.41	0.041	0.0041	41

AnswerEvery cell is 41 with its decimal point shifted by how far a and b moved from 1517 and 37 (see table).

Watch this explained “A whole grid from one”, 9:37 into Dividing when the divisor has a decimal · हिंदी में देखें

Open NCERT p. 94Checked by computerAnswers can differ: one example

(a) maximum product

1. The tens digit of the first number is worth the most, so put the biggest digit, 8, there.
2. Put the next biggest digit, 5, in the ones place of the second number, so the first number is multiplied by more than 5. (85._ × 4._ is smaller: at best 85.0 × 4.2 = 357.)
3. Place 0, 2 and 4 in the other three boxes and compare: 84.2 × 5.0 = 421, 84.0 × 5.2 = 436.8, 82.4 × 5.0 = 412, 82.0 × 5.4 = 442.8, 80.4 × 5.2 = 418.08, 80.2 × 5.4 = 433.08.
4. The largest is 82.0 × 5.4 = 442.8. Checking every arrangement of the five digits confirms nothing is larger.

Answer82.0 × 5.4 = 442.8

(b) minimum product

1. To make the product small, put 0 in the ones place of the second number, so it is less than 1. Multiplying by a number less than 1 gives less than the first number.
2. Try each digit after the point, with the other three making the smallest first number: 45.8 × 0.2 = 9.16, 25.8 × 0.4 = 10.32, 24.8 × 0.5 = 12.4, 24.5 × 0.8 = 19.6.
3. Every arrangement where the second number is more than 1 gives a product above 10, so the smallest is 9.16.

Answer45.8 × 0.2 = 9.16

(c) product greater than 150

1. Many arrangements work; here is one.
2. 84.0 × 2.5 = 210, which is greater than 150.

Answer84.0 × 2.5 = 210 (other arrangements also work)

(d) product nearest to 100

1. 20 × 5 = 100, so try a first number near 20 and a second number near 5.
2. 20.5 × 4.8 = 98.4, which is 1.6 away from 100.
3. The next closest are 40.8 × 2.5 = 102 (2 away) and 48.5 × 2.0 = 97 (3 away), so 98.4 is the nearest.

Answer20.5 × 4.8 = 98.4

(e) product nearest to 5

1. From part (b), the smallest product any arrangement can give is 45.8 × 0.2 = 9.16.
2. Every other product is bigger than 9.16, so it is further from 5. The nearest to 5 is 9.16.

Answer45.8 × 0.2 = 9.16

Watch this explained “Which side are you on”, 9:11 into Why multiplying by a decimal below 1 shrinks a number · हिंदी में देखें

Open NCERT p. 95Checked by computer

1. (e) is 245.05 itself. Compare the others with it by asking whether the second number is above or below 1.
2. 0.942368 is a little below 1: multiplying by it (a) gives a little less than 245.05 (about 231); dividing by it (d) gives a little more (about 260).
3. 7.9682 is about 8, well above 1: multiplying by it (b) gives much more (about 245 × 8 ≈ 1960); dividing by it (c) gives much less (about 245 ÷ 8 ≈ 31).
4. (f) is 7.9682, about 8, which is smaller than (c), about 31.
5. So from smallest to largest: (f) about 8, (c) about 31, (a) about 231, (e) 245.05, (d) about 260, (b) about 1960.

Answerf, c, a, e, d, b

Watch this explained “Six expressions, no arithmetic”, 9:16 into Estimating first, so a wrong answer is visible · हिंदी में देखें