Decimal Fractions

Convert the following into fractions in their lowest terms:

(i) 3.75

(ii) 0.5

(iii) 2.04

(iv) 0.65

(v) 2.405

(vi) 0.085

(vii) 8.025

Answer

(i) $3.75 = \dfrac{375}{100} = \dfrac{375 \div 25}{100 \div 25} = \dfrac{15}{4}$

Hence, $3.75 = \dfrac{15}{4}$

(ii) $0.5 = \dfrac{5}{10} = \dfrac{5 \div 5}{10 \div 5} = \dfrac{1}{2}$

Hence, $0.5 = \dfrac{1}{2}$

(iii) $2.04 = \dfrac{204}{100} = \dfrac{204 \div 4}{100 \div 4} = \dfrac{51}{25}$

Hence, $2.04 = \dfrac{51}{25}$

(iv) $0.65 = \dfrac{65}{100} = \dfrac{65 \div 5}{100 \div 5} = \dfrac{13}{20}$

Hence, $0.65 = \dfrac{13}{20}$

(v) $2.405 = \dfrac{2405}{1000} = \dfrac{2405 \div 5}{1000 \div 5} = \dfrac{481}{200}$

Hence, $2.405 = \dfrac{481}{200}$

(vi) $0.085 = \dfrac{85}{1000} = \dfrac{85 \div 5}{1000 \div 5} = \dfrac{17}{200}$

Hence, $0.085 = \dfrac{17}{200}$

(vii) $8.025 = \dfrac{8025}{1000} = \dfrac{8025 \div 25}{1000 \div 25} = \dfrac{321}{40}$

Hence, $8.025 = \dfrac{321}{40}$

Convert into decimal fractions:

(i) $2\dfrac{4}{5}$

(ii) $\dfrac{79}{100}$

(iii) $\dfrac{37}{10,000}$

(iv) $\dfrac{7543}{10^4}$

(v) $\dfrac{3}{4}$

(vi) $9\dfrac{3}{5}$

(vii) $8\dfrac{5}{8}$

(viii) $5\dfrac{7}{8}$

Answer

(i) $2\dfrac{4}{5} = \dfrac{2 \times 5 + 4}{5} = \dfrac{14}{5} = \dfrac{14 \times 2}{5 \times 2} = \dfrac{28}{10} = 2.8$

Hence, $2\dfrac{4}{5} = 2.8$

(ii) $\dfrac{79}{100} = 0.79$

Hence, $\dfrac{79}{100} = 0.79$

(iii) $\dfrac{37}{10,000} = 0.0037$

Hence, $\dfrac{37}{10,000} = 0.0037$

(iv) $\dfrac{7543}{10^4} = \dfrac{7543}{10000} = 0.7543$

Hence, $\dfrac{7543}{10^4} = 0.7543$

(v) $\dfrac{3}{4} = \dfrac{3 \times 25}{4 \times 25} = \dfrac{75}{100} = 0.75$

Hence, $\dfrac{3}{4} = 0.75$

(vi) $9\dfrac{3}{5} = \dfrac{9 \times 5 + 3}{5} = \dfrac{48}{5} = \dfrac{48 \times 2}{5 \times 2} = \dfrac{96}{10} = 9.6$

Hence, $9\dfrac{3}{5} = 9.6$

(vii) $8\dfrac{5}{8} = \dfrac{8 \times 8 + 5}{8} = \dfrac{69}{8} = \dfrac{69 \times 125}{8 \times 125} = \dfrac{8625}{1000} = 8.625$

Hence, $8\dfrac{5}{8} = 8.625$

(viii) $5\dfrac{7}{8} = \dfrac{5 \times 8 + 7}{8} = \dfrac{47}{8} = \dfrac{47 \times 125}{8 \times 125} = \dfrac{5875}{1000} = 5.875$

Hence, $5\dfrac{7}{8} = 5.875$

Write the number of decimal places in:

(i) 0.4762

(ii) 7.00349

(iii) 8235.403

(iv) 35.4

(v) 2.608

(vi) 0.000879

Answer

The number of figures that follow the decimal point is called the number of decimal places.

(i) 0.4762 has 4 decimal places.

(ii) 7.00349 has 5 decimal places.

(iii) 8235.403 has 3 decimal places.

(iv) 35.4 has 1 decimal place.

(v) 2.608 has 3 decimal places.

(vi) 0.000879 has 6 decimal places.

Write the following decimals as word statements:

(i) 0.4, 0.9, 0.1

(ii) 1.9, 4.4, 7.5

(iii) 0.02, 0.56, 13.06

(iv) 0.005, 0.207, 111.519

(v) 0.8, 0.08, 0.008, 0.0008

(vi) 256.1, 10.22, 0.634

Answer

(i) 0.4 : zero-point-four

0.9 : zero-point-nine

0.1 : zero-point-one

(ii) 1.9 : one-point-nine

4.4 : four-point-four

7.5 : seven-point-five

(iii) 0.02 : zero-point-zero-two

0.56 : zero-point-five-six

13.06 : thirteen-point-zero-six

(iv) 0.005 : zero-point-zero-zero-five

0.207 : zero-point-two-zero-seven

111.519 : one hundred eleven-point-five-one-nine

(v) 0.8 : zero-point-eight

0.08 : zero-point-zero-eight

0.008 : zero-point-zero-zero-eight

0.0008 : zero-point-zero-zero-zero-eight

(vi) 256.1 : two hundred fifty six-point-one

10.22 : ten-point-two-two

0.634 : zero-point-six-three-four

Convert the given decimals into like decimals:

(i) 0.5, 3.62, 43.981 and 232.0037

(ii) 215.78, 33.0006, 530.3 and 0.03569

Answer

Like decimals have the same number of decimal places. We make the number of decimal places equal to the greatest number of decimal places among the given decimals by adding the required number of zeros at the right end (which does not change the value).

(i) The greatest number of decimal places among the given decimals is 4 (in 232.0037). Making each decimal have 4 decimal places:

0.5 = 0.5000
3.62 = 3.6200
43.981 = 43.9810
232.0037 = 232.0037

Hence, the like decimals are 0.5000, 3.6200, 43.9810 and 232.0037.

(ii) The greatest number of decimal places among the given decimals is 5 (in 0.03569). Making each decimal have 5 decimal places:

215.78 = 215.78000
33.0006 = 33.00060
530.3 = 530.30000
0.03569 = 0.03569

Hence, the like decimals are 215.78000, 33.00060, 530.30000 and 0.03569.

Add:

(i) 0.5 and 0.37

(ii) 3.8 and 8.7

(iii) 0.02, 0.008 and 0.309

(iv) 0.4136, 0.3195 and 0.52

(v) 9.25, 3.4 and 6.666

(vi) 3.007, 0.587 and 18.341

(vii) 0.2, 0.02 and 2.0002

(viii) 6.08, 60.8, 0.608 and 0.0608

(ix) 29.03, 0.0003, 0.3 and 7.2

(x) 3.4, 2.025, 9.36 and 3.6221

Answer

While adding decimals, the decimal points are placed in the same vertical line and empty places are filled with zeros to make like decimals.

(i) Solving,

$\begin{array}{r} 0.50 \\ +0.37 \\ \hline 0.87 \end{array}$

Hence, 0.5 + 0.37 = 0.87

(ii) Solving,

$\begin{array}{r} 3.8 \\ +8.7 \\ \hline 12.5 \end{array}$

Hence, 3.8 + 8.7 = 12.5

(iii) Solving,

$\begin{array}{r} 0.020 \\ 0.008 \\ +0.309 \\ \hline 0.337 \end{array}$

Hence, 0.02 + 0.008 + 0.309 = 0.337

(iv) Solving,

$\begin{array}{r} 0.4136 \\ 0.3195 \\ +0.5200 \\ \hline 1.2531 \end{array}$

Hence, 0.4136 + 0.3195 + 0.52 = 1.2531

(v) Solving,

$\begin{array}{r} 9.250 \\ 3.400 \\ +6.666 \\ \hline 19.316 \end{array}$

Hence, 9.25 + 3.4 + 6.666 = 19.316

(vi) Solving,

$\begin{array}{r} 3.007 \\ 0.587 \\ +18.341 \\ \hline 21.935 \end{array}$

Hence, 3.007 + 0.587 + 18.341 = 21.935

(vii) Solving,

$\begin{array}{r} 0.2000 \\ 0.0200 \\ +2.0002 \\ \hline 2.2202 \end{array}$

Hence, 0.2 + 0.02 + 2.0002 = 2.2202

(viii) Solving,

$\begin{array}{r} 6.0800 \\ 60.8000 \\ 0.6080 \\ +0.0608 \\ \hline 67.5488 \end{array}$

Hence, 6.08 + 60.8 + 0.608 + 0.0608 = 67.5488

(ix) Solving,

$\begin{array}{r} 29.0300 \\ 0.0003 \\ 0.3000 \\ +7.2000 \\ \hline 36.5303 \end{array}$

Hence, 29.03 + 0.0003 + 0.3 + 7.2 = 36.5303

(x) Solving,

$\begin{array}{r} 3.4000 \\ 2.0250 \\ 9.3600 \\ +3.6221 \\ \hline 18.4071 \end{array}$

Hence, 3.4 + 2.025 + 9.36 + 3.6221 = 18.4071

Subtract the first number from the second:

(i) 5.4, 9.8

(ii) 0.16, 4.3

(iii) 0.82, 8.6

(iv) 0.07, 8.43

(v) 2.237, 9.425

(vi) 41.03, 59.46

(vii) 3.92, 26.86

(viii) 4.73, 8.5

(ix) 12.63, 36.2

(x) 0.845, 3.71

Answer

(i) Solving,

$\begin{array}{r} 9.8 \\ -5.4 \\ \hline 4.4 \end{array}$

Hence, 9.8 - 5.4 = 4.4

(ii) Solving,

$\begin{array}{r} 4.30 \\ -0.16 \\ \hline 4.14 \end{array}$

Hence, 4.3 - 0.16 = 4.14

(iii) Solving,

$\begin{array}{r} 8.60 \\ -0.82 \\ \hline 7.78 \end{array}$

Hence, 8.6 - 0.82 = 7.78

(iv) Solving,

$\begin{array}{r} 8.43 \\ -0.07 \\ \hline 8.36 \end{array}$

Hence, 8.43 - 0.07 = 8.36

(v) Solving,

$\begin{array}{r} 9.425 \\ -2.237 \\ \hline 7.188 \end{array}$

Hence, 9.425 - 2.237 = 7.188

(vi) Solving,

$\begin{array}{r} 59.46 \\ -41.03 \\ \hline 18.43 \end{array}$

Hence, 59.46 - 41.03 = 18.43

(vii) Solving,

$\begin{array}{r} 26.86 \\ -3.92 \\ \hline 22.94 \end{array}$

Hence, 26.86 - 3.92 = 22.94

(viii) Solving,

$\begin{array}{r} 8.50 \\ -4.73 \\ \hline 3.77 \end{array}$

Hence, 8.5 - 4.73 = 3.77

(ix) Solving,

$\begin{array}{r} 36.20 \\ -12.63 \\ \hline 23.57 \end{array}$

Hence, 36.2 - 12.63 = 23.57

(x) Solving,

$\begin{array}{r} 3.710 \\ -0.845 \\ \hline 2.865 \end{array}$

Hence, 3.71 - 0.845 = 2.865

Simplify:

(i) 28.796 − 13.42 − 2.555

(ii) 93.354 − 62.82 − 13.045

(iii) 36 − 18.59 − 3.2

(iv) 86 + 16.95 − 3.0042

(v) 32.8 − 13 − 10.725 + 3.517

(vi) 4000 − 30.51 − 753.101 − 69.43

(vii) 0.1835 + 163.2005 − 25.9 − 100

(viii) 38.00 − 30 + 200.200 − 0.230

(ix) 555.555 + 55.555 − 5.55 − 0.555

Answer

(i) Solving,

28.796 - 13.42 - 2.555 = 28.796 - (13.420 + 2.555)

= 28.796 - 15.975 = 12.821

Hence, 28.796 - 13.42 - 2.555 = 12.821

(ii) Solving,

93.354 - 62.82 - 13.045 = 93.354 - (62.820 + 13.045)

= 93.354 - 75.865 = 17.489

Hence, 93.354 - 62.82 - 13.045 = 17.489

(iii) Solving,

36 - 18.59 - 3.2 = 36.00 - (18.59 + 3.20)

= 36.00 - 21.79 = 14.21

Hence, 36 - 18.59 - 3.2 = 14.21

(iv) Solving,

86 + 16.95 - 3.0042 = (86.0000 + 16.9500) - 3.0042

= 102.9500 - 3.0042 = 99.9458

Hence, 86 + 16.95 - 3.0042 = 99.9458

(v) Solving,

32.8 - 13 - 10.725 + 3.517 = (32.800 + 3.517) - (13.000 + 10.725)

= 36.317 - 23.725 = 12.592

Hence, 32.8 - 13 - 10.725 + 3.517 = 12.592

(vi) Solving,

4000 - 30.51 - 753.101 - 69.43 = 4000.000 - (30.510 + 753.101 + 69.430)

= 4000.000 - 853.041 = 3146.959

Hence, 4000 - 30.51 - 753.101 - 69.43 = 3146.959

(vii) Solving,

0.1835 + 163.2005 - 25.9 - 100 = (0.1835 + 163.2005) - (25.9000 + 100.0000)

= 163.3840 - 125.9000 = 37.484

Hence, 0.1835 + 163.2005 - 25.9 - 100 = 37.484

(viii) Solving,

38.00 - 30 + 200.200 - 0.230 = (38.000 + 200.200) - (30.000 + 0.230)

= 238.200 - 30.230 = 207.97

Hence, 38.00 - 30 + 200.200 - 0.230 = 207.97

(ix) Solving,

555.555 + 55.555 - 5.55 - 0.555 = (555.555 + 55.555) - (5.550 + 0.555)

= 611.110 - 6.105 = 605.005

Hence, 555.555 + 55.555 - 5.55 - 0.555 = 605.005

Find the difference between 6.85 and 0.685.

Answer

Difference = 6.850 - 0.685 = 6.165

Hence, the difference between 6.85 and 0.685 is 6.165.

Take out the sum of 19.38 and 56.025 from 200.111.

Answer

Sum of 19.38 and 56.025:

19.380 + 56.025 = 75.405

Taking out this sum from 200.111:

200.111 - 75.405 = 124.706

Hence, the required result is 124.706.

Add 13.95 and 1.003, and from the result, subtract the sum of 2.794 and 6.2.

Answer

Adding 13.95 and 1.003:

13.950 + 1.003 = 14.953

Sum of 2.794 and 6.2:

2.794 + 6.200 = 8.994

Subtracting this sum from 14.953:

14.953 - 8.994 = 5.959

Hence, the required result is 5.959.

What should be added to 39.587 to give 80.375?

Answer

Let x be the number to be added.

⇒ 39.587 + x = 80.375

⇒ x = 80.375 - 39.587

⇒ x = 40.788

Hence, 40.788 should be added to 39.587 to give 80.375.

What should be subtracted from 100 to give 19.29?

Answer

Required number = 100.00 - 19.29 = 80.71

Hence, 80.71 should be subtracted from 100 to give 19.29.

What is the excess of 584.29 over 213.95?

Answer

Excess = 584.29 - 213.95 = 370.34

Hence, the excess of 584.29 over 213.95 is 370.34.

Evaluate:

(i) (5.4 − 0.8) + (2.97 − 1.462)

(ii) (6.25 + 0.36) − (17.2 − 8.97)

(iii) 9.004 + (3 − 2.462)

(iv) 879.4 − (87.94 − 8.794)

Answer

(i) Solving,

(5.4 - 0.8) + (2.97 - 1.462) = 4.6 + (2.970 - 1.462)

= 4.600 + 1.508 = 6.108

Hence, (5.4 - 0.8) + (2.97 - 1.462) = 6.108

(ii) Solving,

(6.25 + 0.36) - (17.2 - 8.97) = 6.61 - (17.20 - 8.97)

= 6.61 - 8.23 = -1.62

Hence, (6.25 + 0.36) - (17.2 - 8.97) = -1.62

(iii) Solving,

9.004 + (3 - 2.462) = 9.004 + (3.000 - 2.462)

= 9.004 + 0.538 = 9.542

Hence, 9.004 + (3 - 2.462) = 9.542

(iv) Solving,

879.4 - (87.94 - 8.794) = 879.4 - (87.940 - 8.794)

= 879.400 - 79.146 = 800.254

Hence, 879.4 - (87.94 - 8.794) = 800.254

What is the excess of 75 over 48.29?

Answer

Excess = 75.00 - 48.29 = 26.71

Hence, the excess of 75 over 48.29 is 26.71.

If A = 237.98 and B = 83.47.

Find: (i) A − B (ii) B − A.

Answer

(i) A - B = 237.98 - 83.47 = 154.51

Hence, A - B = 154.51

(ii) B - A = 83.47 - 237.98 = -154.51

Hence, B - A = -154.51

The cost of one kg of sugar increases from ₹ 28.47 to ₹ 32.65. Find the increase in cost.

Answer

Increase in cost = New cost - Old cost

= ₹ 32.65 - ₹ 28.47 = ₹ 4.18

Hence, the increase in cost is ₹ 4.18.

Multiply:

(i) 0.87 by 10

(ii) 2.948 by 100

(iii) 6.4 by 1000

(iv) 5.8 by 4

(v) 16.32 by 28

(vi) 5.037 by 8

(vii) 4.6 by 2.1

(viii) 0.568 by 6.4

Answer

(i) On multiplying by 10, shift the decimal point one place to the right.

⇒ 0.87 × 10 = 8.7

Hence, 0.87 × 10 = 8.7

(ii) On multiplying by 100, shift the decimal point two places to the right.

⇒ 2.948 × 100 = 294.8

Hence, 2.948 × 100 = 294.8

(iii) On multiplying by 1000, shift the decimal point three places to the right.

⇒ 6.4 × 1000 = 6400

Hence, 6.4 × 1000 = 6400

(iv) 58 × 4 = 232.

The multiplicand 5.8 has 1 decimal place, so the product has 1 decimal place.

⇒ 5.8 × 4 = 23.2

Hence, 5.8 × 4 = 23.2

(v) 1632 × 28 = 45696.

The multiplicand 16.32 has 2 decimal places, so the product has 2 decimal places.

⇒ 16.32 × 28 = 456.96

Hence, 16.32 × 28 = 456.96

(vi) 5037 × 8 = 40296. The multiplicand 5.037 has 3 decimal places, so the product has 3 decimal places.

⇒ 5.037 × 8 = 40.296

Hence, 5.037 × 8 = 40.296

(vii) 46 × 21 = 966.

The sum of decimal places is 1 + 1 = 2 (4.6 and 2.1 have 1 decimal place each), so the product has 2 decimal places.

⇒ 4.6 × 2.1 = 9.66

Hence, 4.6 × 2.1 = 9.66

(viii) 568 × 64 = 36352. The sum of decimal places is 3 + 1 = 4, so the product has 4 decimal places.

⇒ 0.568 × 6.4 = 3.6352

Hence, 0.568 × 6.4 = 3.6352

Multiply each number by 10, 100 and 1000:

(i) 0.5

(ii) 0.112

(iii) 4.8

(iv) 0.0359

(v) 16.27

(vi) 234.8

Answer

On multiplying a decimal number by 10, 100 and 1000, the decimal point shifts to the right, by 1, 2 and 3 places respectively.

(i) Solving,

⇒ 0.5 × 10 = 5

⇒ 0.5 × 100 = 50

⇒ 0.5 × 1000 = 500

Hence, the products are 5, 50 and 500.

(ii) Solving,

⇒ 0.112 × 10 = 1.12

⇒ 0.112 × 100 = 11.2

⇒ 0.112 × 1000 = 112

Hence, the products are 1.12, 11.2 and 112.

(iii) Solving,

⇒ 4.8 × 10 = 48

⇒ 4.8 × 100 = 480

⇒ 4.8 × 1000 = 4800

Hence, the products are 48, 480 and 4800.

(iv) Solving,

⇒ 0.0359 × 10 = 0.359

⇒ 0.0359 × 100 = 3.59

⇒ 0.0359 × 1000 = 35.9

Hence, the products are 0.359, 3.59 and 35.9.

(v) Solving,

⇒ 16.27 × 10 = 162.7

⇒ 16.27 × 100 = 1627

⇒ 16.27 × 1000 = 16270

Hence, the products are 162.7, 1627 and 16270.

(vi) Solving,

⇒ 234.8 × 10 = 2348

⇒ 234.8 × 100 = 23480

⇒ 234.8 × 1000 = 234800

Hence, the products are 2348, 23480 and 234800.

Evaluate:

(i) 5.897 × 2.3

(ii) 0.894 × 87

(iii) 0.01 × 0.001

(iv) 0.84 × 2.2 × 4

(v) 4.75 × 0.08 × 3

(vi) 2.4 × 3.5 × 4.8

(vii) 0.8 × 1.2 × 0.25

(viii) 0.3 × 0.03 × 0.003

Answer

(i) 5897 × 23 = 135631.

Sum of decimal places = 3 + 1 = 4 (3 decimal places in 5.897 and 1 decimal place in 2.3)

⇒ 5.897 × 2.3 = 13.5631

Hence, 5.897 × 2.3 = 13.5631

(ii) 894 × 87 = 77778.

Decimal places = 3.

⇒ 0.894 × 87 = 77.778

Hence, 0.894 × 87 = 77.778

(iii) 1 × 1 = 1.

Sum of decimal places = 2 + 3 = 5.

⇒ 0.01 × 0.001 = 0.00001

Hence, 0.01 × 0.001 = 0.00001

(iv) Solving,

⇒ 0.84 × 2.2 × 4 = (0.84 × 2.2) × 4

= 1.848 × 4 = 7.392

Hence, 0.84 × 2.2 × 4 = 7.392

(v) Solving,

⇒ 4.75 × 0.08 × 3 = (4.75 × 0.08) × 3

= 0.38 × 3 = 1.14

Hence, 4.75 × 0.08 × 3 = 1.14

(vi) Solving,

⇒ 2.4 × 3.5 × 4.8 = (2.4 × 3.5) × 4.8

= 8.4 × 4.8 = 40.32

Hence, 2.4 × 3.5 × 4.8 = 40.32

(vii) Solving,

⇒ 0.8 × 1.2 × 0.25 = (0.8 × 1.2) × 0.25

= 0.96 × 0.25 = 0.24

Hence, 0.8 × 1.2 × 0.25 = 0.24

(viii) Solving,

⇒ 0.3 × 0.03 × 0.003 = (0.3 × 0.03) × 0.003

= 0.009 × 0.003 = 0.000027

Hence, 0.3 × 0.03 × 0.003 = 0.000027

Divide:

54.9 by 10

Answer

On dividing by 10, shift the decimal point one place to the left.

⇒ 54.9 ÷ 10 = 5.49

Hence, 54.9 ÷ 10 = 5.49

Divide:

7.8 by 100

Answer

On dividing by 100, shift the decimal point two places to the left.

⇒ 7.8 ÷ 100 = 0.078

Hence, 7.8 ÷ 100 = 0.078

Divide:

324.76 by 1000

Answer

On dividing by 1000, shift the decimal point three places to the left.

⇒ 324.76 ÷ 1000 = 0.32476

Hence, 324.76 ÷ 1000 = 0.32476

Divide:

12.8 by 4

Answer

By long division,

$\begin{array}{l} 4\overline{\smash{\big)}\phantom{11}12.8\smash{\big(}}3.2 \\ \phantom{4\smash{\big)},}\underline{-12} \\ \phantom{4\smash{\big)},}\phantom{000..}8 \\ \phantom{4\smash{\big)},}\phantom{00.}\underline{-8} \\ \phantom{4\smash{\big)},}\phantom{000.}\times \end{array}$

Hence, 12.8 ÷ 4 = 3.2

Divide:

27.918 by 9

Answer

By long division,

$\begin{array}{l} 9\overline{\smash{\big)}\phantom{11}27.918\smash{\big(}}3.102 \\ \phantom{9\smash{\big)},}\underline{-27} \\ \phantom{9\smash{\big)},}\phantom{0000}9 \\ \phantom{9\smash{\big)},}\phantom{00.}\underline{-9} \\ \phantom{9\smash{\big)},}\phantom{000.11}18 \\ \phantom{9\smash{\big)},}\phantom{0000}\underline{-18} \\ \phantom{9\smash{\big)},}\phantom{00000.}\times \end{array}$

Hence, 27.918 ÷ 9 = 3.102

Divide:

4.672 by 8

Answer

By long division,

$\begin{array}{l} 8\overline{\smash{\big)}\phantom{000}4.672\smash{\big(}}0.584 \\ \phantom{8\smash{\big)},}\phantom{}\underline{-40} \\ \phantom{8\smash{\big)},}\phantom{000.}67 \\ \phantom{8\smash{\big)},}\phantom{00}\underline{-64} \\ \phantom{8\smash{\big)},}\phantom{0000..}32 \\ \phantom{8\smash{\big)},}\phantom{000.}\underline{-32} \\ \phantom{8\smash{\big)},}\phantom{00000}\times \end{array}$

Hence, 4.672 ÷ 8 = 0.584

Divide:

4.32 by 1.2

Answer

Shift the decimal point of both dividend and divisor by 1 place to make the divisor a whole number.

$4.32 \div 1.2 = \dfrac{43.2}{12}$

By long division,

$\begin{array}{l} 12\overline{\smash{\big)}\phantom{0.}43.2\smash{\big(}}3.6 \\ \phantom{12\smash{\big)},}\underline{-36} \\ \phantom{12\smash{\big)},}\phantom{00.}72 \\ \phantom{12\smash{\big)},}\phantom{0}\underline{-72} \\ \phantom{12\smash{\big)},}\phantom{00.}\times \end{array}$

Hence, 4.32 ÷ 1.2 = 3.6

Divide:

7.644 by 1.4

Answer

Shift the decimal point of both dividend and divisor by 1 place to make the divisor a whole number.

$7.644 \div 1.4 = \dfrac{76.44}{14}$

By long division,

$\begin{array}{l} 14\overline{\smash{\big)}\phantom{00}76.44\smash{\big(}}5.46 \\ \phantom{14\smash{\big)},}\underline{-70} \\ \phantom{14\smash{\big)},}\phantom{000}64 \\ \phantom{14\smash{\big)},}\phantom{0.}\underline{-56} \\ \phantom{14\smash{\big)},}\phantom{0000.}84 \\ \phantom{14\smash{\big)},}\phantom{000}\underline{-84} \\ \phantom{14\smash{\big)},}\phantom{0000.}\times \end{array}$

Hence, 7.644 ÷ 1.4 = 5.46

Divide:

4.8432 by 0.08

Answer

Shift the decimal point of both dividend and divisor by 2 places to make the divisor a whole number.

$4.8432 \div 0.08 = \dfrac{484.32}{8}$

By long division,

$\begin{array}{l} 8\overline{\smash{\big)}\phantom{00.}484.32\smash{\big(}}60.54 \\ \phantom{8\smash{\big)},}\underline{-48} \\ \phantom{8\smash{\big)},}\phantom{000..}43 \\ \phantom{8\smash{\big)},}\phantom{00.}\underline{-40} \\ \phantom{8\smash{\big)},}\phantom{0000..}32 \\ \phantom{8\smash{\big)},}\phantom{000.}\underline{-32} \\ \phantom{8\smash{\big)},}\phantom{00000}\times \end{array}$

Hence, 4.8432 ÷ 0.08 = 60.54

Divide each of the given numbers by 10, 100, 1000 and 10000:

2.1

Answer

On dividing a decimal number by 10, 100, 1000, 10000, the decimal point shifts to the left, by 1, 2, 3 and 4 places respectively.

Solving,

⇒ 2.1 ÷ 10 = 0.21

⇒ 2.1 ÷ 100 = 0.021

⇒ 2.1 ÷ 1000 = 0.0021

⇒ 2.1 ÷ 10000 = 0.00021

Hence, the quotients are 0.21, 0.021, 0.0021 and 0.00021.

Divide each of the given numbers by 10, 100, 1000 and 10000:

8.64

Answer

On dividing a decimal number by 10, 100, 1000, 10000, the decimal point shifts to the left, by 1, 2, 3 and 4 places respectively.

Solving,

⇒ 8.64 ÷ 10 = 0.864

⇒ 8.64 ÷ 100 = 0.0864

⇒ 8.64 ÷ 1000 = 0.00864

⇒ 8.64 ÷ 10000 = 0.000864

Hence, the quotients are 0.864, 0.0864, 0.00864 and 0.000864.

Divide each of the given numbers by 10, 100, 1000 and 10000:

5.01

Answer

On dividing a decimal number by 10, 100, 1000, 10000, the decimal point shifts to the left, by 1, 2, 3 and 4 places respectively.

Solving,

⇒ 5.01 ÷ 10 = 0.501

⇒ 5.01 ÷ 100 = 0.0501

⇒ 5.01 ÷ 1000 = 0.00501

⇒ 5.01 ÷ 10000 = 0.000501

Hence, the quotients are 0.501, 0.0501, 0.00501 and 0.000501.

Divide each of the given numbers by 10, 100, 1000 and 10000:

0.0906

Answer

On dividing a decimal number by 10, 100, 1000, 10000, the decimal point shifts to the left, by 1, 2, 3 and 4 places respectively.

Solving,

⇒ 0.0906 ÷ 10 = 0.00906

⇒ 0.0906 ÷ 100 = 0.000906

⇒ 0.0906 ÷ 1000 = 0.0000906

⇒ 0.0906 ÷ 10000 = 0.00000906

Hence, the quotients are 0.00906, 0.000906, 0.0000906 and 0.00000906.

Divide each of the given numbers by 10, 100, 1000 and 10000:

0.125

Answer

On dividing a decimal number by 10, 100, 1000, 10000, the decimal point shifts to the left, by 1, 2, 3 and 4 places respectively.

Solving,

⇒ 0.125 ÷ 10 = 0.0125

⇒ 0.125 ÷ 100 = 0.00125

⇒ 0.125 ÷ 1000 = 0.000125

⇒ 0.125 ÷ 10000 = 0.0000125

Hence, the quotients are 0.0125, 0.00125, 0.000125 and 0.0000125.

Divide each of the given numbers by 10, 100, 1000 and 10000:

111.11

Answer

On dividing a decimal number by 10, 100, 1000, 10000, the decimal point shifts to the left, by 1, 2, 3 and 4 places respectively.

Solving,

⇒ 111.11 ÷ 10 = 11.111

⇒ 111.11 ÷ 100 = 1.1111

⇒ 111.11 ÷ 1000 = 0.11111

⇒ 111.11 ÷ 10000 = 0.011111

Hence, the quotients are 11.111, 1.1111, 0.11111 and 0.011111.

Evaluate:

9.75 ÷ 5

Answer

By long division,

$\begin{array}{l} 5\overline{\smash{\big)}\phantom{00.}9.75\smash{\big(}}1.95 \\ \phantom{5\smash{\big)},}\underline{-5} \\ \phantom{5\smash{\big)},}\phantom{00}47 \\ \phantom{5\smash{\big)},}\phantom{.}\underline{-45} \\ \phantom{5\smash{\big)},}\phantom{000.}25 \\ \phantom{5\smash{\big)},}\phantom{00}\underline{-25} \\ \phantom{5\smash{\big)},}\phantom{0000}\times \end{array}$

Hence, 9.75 ÷ 5 = 1.95

Evaluate:

4.4064 ÷ 4

Answer

By long division,

$\begin{array}{l} 4\overline{\smash{\big)}\phantom{00}4.4064\smash{\big(}}1.1016 \\ \phantom{4\smash{\big)},}\underline{-4} \\ \phantom{4\smash{\big)},}\phantom{000}4 \\ \phantom{4\smash{\big)},}\phantom{0.}\underline{-4} \\ \phantom{000000.}06 \\ \phantom{+))}\phantom{00..}\underline{-4} \\ \phantom{+))}\phantom{0000.}24 \\ \phantom{+))}\phantom{00..}\underline{-24} \\ \phantom{4\smash{\big)},}\phantom{00000.}\times \end{array}$

Hence, 4.4064 ÷ 4 = 1.1016

Evaluate:

27.69 ÷ 30

Answer

By long division,

$\begin{array}{l} 30\overline{\smash{\big)}\phantom{00.}27.690\smash{\big(}}0.923 \\ \phantom{((()}\phantom{.}\underline{-270} \\ \phantom{30\smash{\big)},}\phantom{0000.}69 \\ \phantom{30\smash{\big)},}\phantom{000}\underline{-60} \\ \phantom{30\smash{\big)},}\phantom{00000.}90 \\ \phantom{30\smash{\big)},}\phantom{000..}\underline{-90} \\ \phantom{30\smash{\big)},}\phantom{000000}\times \end{array}$

Hence, 27.69 ÷ 30 = 0.923

Evaluate:

19.25 ÷ 25

Answer

By long division,

$\begin{array}{l} 25\overline{\smash{\big)}\phantom{0...}19.25\smash{\big(}}0.77 \\ \phantom{25\smash{\big)},}\phantom{}\underline{-175} \\ \phantom{25\smash{\big)},}\phantom{000.}175 \\ \phantom{25\smash{\big)},}\phantom{00}\underline{-175} \\ \phantom{25\smash{\big)},}\phantom{0000.}\times \end{array}$

Hence, 19.25 ÷ 25 = 0.77

Evaluate:

20.64 ÷ 16

Answer

By long division,

$\begin{array}{l} 16\overline{\smash{\big)}\phantom{00.}20.64\smash{\big(}}1.29 \\ \phantom{16\smash{\big)},}\underline{-16} \\ \phantom{16\smash{\big)},}\phantom{000}46 \\ \phantom{16\smash{\big)},}\phantom{0.}\underline{-32} \\ \phantom{16\smash{\big)},}\phantom{000}144 \\ \phantom{16\smash{\big)},}\phantom{0.}\underline{-144} \\ \phantom{16\smash{\big)},}\phantom{0000}\times \end{array}$

Hence, 20.64 ÷ 16 = 1.29

Evaluate:

3.204 ÷ 9

Answer

By long division,

$\begin{array}{l} 9\overline{\smash{\big)}\phantom{00.}3.204\smash{\big(}}0.356 \\ \phantom{9\smash{\big)},}\phantom{..}\underline{-27} \\ \phantom{9\smash{\big)},}\phantom{000.}50 \\ \phantom{9\smash{\big)},}\phantom{00}\underline{-45} \\ \phantom{9\smash{\big)},}\phantom{0000.}54 \\ \phantom{9\smash{\big)},}\phantom{000}\underline{-54} \\ \phantom{9\smash{\big)},}\phantom{0000.}\times \end{array}$

Hence, 3.204 ÷ 9 = 0.356

Evaluate:

0.125 ÷ 25

Answer

By long division,

$\begin{array}{l} 25\overline{\smash{\big)}\phantom{00}0.125\smash{\big(}}0.005 \\ \phantom{25\smash{\big)},}\phantom{0.}\underline{-125} \\ \phantom{25\smash{\big)},}\phantom{0000}\times \end{array}$

Hence, 0.125 ÷ 25 = 0.005

Evaluate:

0.14616 ÷ 72

Answer

By long division,

$\begin{array}{l} 72\overline{\smash{\big)}\phantom{00}0.14616\smash{\big(}}0.00203 \\ \phantom{72\smash{\big)},}\phantom{0.}\underline{-144} \\ \phantom{72\smash{\big)},}\phantom{0000..}216 \\ \phantom{72\smash{\big)},}\phantom{000.}\underline{-216} \\ \phantom{72\smash{\big)},}\phantom{00000..}\times \end{array}$

Hence, 0.14616 ÷ 72 = 0.00203

Evaluate:

0.6227 ÷ 1300

Answer

By long division,

$\begin{array}{l} 1300\overline{\smash{\big)}\phantom{000..}0.622700\smash{\big(}}0.000479 \\ \phantom{1300\smash{\big)},}\phantom{000.}\underline{-5200} \\ \phantom{1300\smash{\big)},}\phantom{00000}10270 \\ \phantom{1300\smash{\big)},}\phantom{0000.}\underline{-9100} \\ \phantom{1300\smash{\big)},}\phantom{00000..}11700 \\ \phantom{1300\smash{\big)},}\phantom{0000.}\underline{-11700} \\ \phantom{1300\smash{\big)},}\phantom{00000000}\times \end{array}$

Hence, 0.6227 ÷ 1300 = 0.000479

Evaluate:

257.894 ÷ 0.169

Answer

Shift the decimal point of both dividend and divisor by 3 places to make the divisor a whole number.

$257.894 \div 0.169 = \dfrac{257894}{169}$

By long division,

$\begin{array}{l} 169\overline{\smash{\big)}\phantom{00.}257894\smash{\big(}}1526 \\ \phantom{169\smash{\big)},}\underline{-169} \\ \phantom{169\smash{\big)},}\phantom{00.}888 \\ \phantom{169\smash{\big)},}\phantom{0}\underline{-845} \\ \phantom{169\smash{\big)},}\phantom{0.00}439 \\ \phantom{169\smash{\big)},}\phantom{00}\underline{-338} \\ \phantom{169\smash{\big)},}\phantom{000.}1014 \\ \phantom{169\smash{\big)},}\phantom{00}\underline{-1014} \\ \phantom{169\smash{\big)},}\phantom{00000}\times \end{array}$

Hence, 257.894 ÷ 0.169 = 1526

Evaluate:

6.3 ÷ (0.3)²

Answer

$6.3 \div (0.3)^2 = 6.3 \div 0.09 = \dfrac{630}{9}$

By long division,

$\begin{array}{l} 9\overline{\smash{\big)}\phantom{00}630\smash{\big(}}70 \\ \phantom{9\smash{\big)},}\underline{-63} \\ \phantom{9\smash{\big)},}\phantom{0.}\times \end{array}$

Hence, 6.3 ÷ (0.3)² = 70

Evaluate:

(i) 4.3 × 0.52 × 0.3

(ii) 3.2 × 2.5 × 0.7

(iii) 0.8 × 1.5 × 0.6

(iv) 0.3 × 0.3 × 0.3

(v) 1.2 × 1.2 × 0.4

(vi) 0.4 × 0.04 × 0.004

(vii) 0.5 × 0.6 × 0.7

(viii) 0.5 × 0.06 × 0.007

Answer

(i) Solving,

⇒ 4.3 × 0.52 × 0.3 = (4.3 × 0.52) × 0.3

= 2.236 × 0.3 = 0.6708

Hence, 4.3 × 0.52 × 0.3 = 0.6708

(ii) Solving,

⇒ 3.2 × 2.5 × 0.7 = (3.2 × 2.5) × 0.7

= 8 × 0.7 = 5.6

Hence, 3.2 × 2.5 × 0.7 = 5.6

(iii) Solving,

⇒ 0.8 × 1.5 × 0.6 = (0.8 × 1.5) × 0.6

= 1.2 × 0.6 = 0.72

Hence, 0.8 × 1.5 × 0.6 = 0.72

(iv) Solving,

⇒ 0.3 × 0.3 × 0.3 = (0.3 × 0.3) × 0.3

= 0.09 × 0.3 = 0.027

Hence, 0.3 × 0.3 × 0.3 = 0.027

(v) Solving,

⇒ 1.2 × 1.2 × 0.4 = (1.2 × 1.2) × 0.4

= 1.44 × 0.4 = 0.576

Hence, 1.2 × 1.2 × 0.4 = 0.576

(vi) Solving,

⇒ 0.4 × 0.04 × 0.004 = (0.4 × 0.04) × 0.004

= 0.016 × 0.004 = 0.000064

Hence, 0.4 × 0.04 × 0.004 = 0.000064

(vii) Solving,

⇒ 0.5 × 0.6 × 0.7 = (0.5 × 0.6) × 0.7

= 0.3 × 0.7 = 0.21

Hence, 0.5 × 0.6 × 0.7 = 0.21

(viii) Solving,

⇒ 0.5 × 0.06 × 0.007 = (0.5 × 0.06) × 0.007

= 0.03 × 0.007 = 0.00021

Hence, 0.5 × 0.06 × 0.007 = 0.00021

Evaluate:

(i) (0.9)²

(ii) (0.6)² × 0.5

(iii) 0.3 × (0.5)²

(iv) (0.4)³

(v) (0.2)³ × 5

(vi) (0.2)³ × 0.05

Answer

(i) (0.9)² = 0.9 × 0.9 = 0.81

Hence, (0.9)² = 0.81

(ii) Solving,

⇒ (0.6)² × 0.5 = (0.6 × 0.6) × 0.5

= 0.36 × 0.5 = 0.18

Hence, (0.6)² × 0.5 = 0.18

(iii) Solving,

⇒ 0.3 × (0.5)² = 0.3 × (0.5 × 0.5)

= 0.3 × 0.25 = 0.075

Hence, 0.3 × (0.5)² = 0.075

(iv) (0.4)³ = 0.4 × 0.4 × 0.4 = 0.064

Hence, (0.4)³ = 0.064

(v) Solving,

⇒ (0.2)³ × 5 = (0.2 × 0.2 × 0.2) × 5

= 0.008 × 5 = 0.04

Hence, (0.2)³ × 5 = 0.04

(vi) Solving,

⇒ (0.2)³ × 0.05 = (0.2 × 0.2 × 0.2) × 0.05

= 0.008 × 0.05 = 0.0004

Hence, (0.2)³ × 0.05 = 0.0004

Find the cost of 36.75 kg wheat at the rate of ₹ 12.80 per kg.

Answer

Cost of wheat = Rate per kg × Weight

= ₹ 12.80 × 36.75 = ₹ 470.40

Hence, the cost of 36.75 kg wheat is ₹ 470.40.

The cost of a pen is ₹ 56.15. Find the cost of 16 such pens.

Answer

Cost of 16 pens = Cost of one pen × 16

= ₹ 56.15 × 16 = ₹ 898.40

Hence, the cost of 16 pens is ₹ 898.40.

Evaluate:

0.0072 ÷ 0.06

Answer

Shift the decimal point of both numbers by 2 places.

$0.0072 \div 0.06 = \dfrac{0.72}{6}$

By long division,

$\begin{array}{l} 6\overline{\smash{\big)}\phantom{0..}0.72\smash{\big(}}0.12 \\ \phantom{6\smash{\big)},}\phantom{0.}\underline{-6} \\ \phantom{6\smash{\big)},}\phantom{000}12 \\ \phantom{6\smash{\big)},}\phantom{0.}\underline{-12} \\ \phantom{6\smash{\big)},}\phantom{00.0}\times \end{array}$

Hence, 0.0072 ÷ 0.06 = 0.12

Evaluate:

0.621 ÷ 0.3

Answer

Shift the decimal point of both numbers by 1 place.

$0.621 \div 0.3 = \dfrac{6.21}{3}$

By long division,

$\begin{array}{l} 3\overline{\smash{\big)}\phantom{00}6.21\smash{\big(}}2.07 \\ \phantom{3\smash{\big)},}\underline{-6} \\ \phantom{3\smash{\big)},}\phantom{000}21 \\ \phantom{3\smash{\big)},}\phantom{0.}\underline{-21} \\ \phantom{3\smash{\big)},}\phantom{000}\times \end{array}$

Hence, 0.621 ÷ 0.3 = 2.07

Evaluate:

0.0532 ÷ 0.005

Answer

Shift the decimal point of both numbers by 3 places.

$0.0532 \div 0.005 = \dfrac{53.2}{5}$

By long division,

$\begin{array}{l} 5\overline{\smash{\big)}\phantom{00}53.20\smash{\big(}}10.64 \\ \phantom{5\smash{\big)},}\underline{-5} \\ \phantom{+}\phantom{000.}32 \\ \phantom{+}\phantom{00}\underline{-30} \\ \phantom{+}\phantom{0000.}20 \\ \phantom{+}\phantom{000}\underline{-20} \\ \phantom{+}\phantom{0000.}\times \end{array}$

Hence, 0.0532 ÷ 0.005 = 10.64

Evaluate:

0.01162 ÷ 0.14

Answer

Shift the decimal point of both numbers by 2 places.

$0.01162 \div 0.14 = \dfrac{1.162}{14}$

By long division,

$\begin{array}{l} 14\overline{\smash{\big)}\phantom{00.}1.162\smash{\big(}}0.083 \\ \phantom{14\smash{\big)},}\phantom{.}\underline{-112} \\ \phantom{14\smash{\big)},}\phantom{0000.}42 \\ \phantom{14\smash{\big)},}\phantom{000}\underline{-42} \\ \phantom{14\smash{\big)},}\phantom{00000}\times \end{array}$

Hence, 0.01162 ÷ 0.14 = 0.083

Evaluate:

(7.5 × 40.4) ÷ 25

Answer

(7.5 × 40.4) ÷ 25 = 303 ÷ 25

By long division,

$\begin{array}{l} 25\overline{\smash{\big)}\phantom{00.}303.00\smash{\big(}}12.12 \\ \phantom{25\smash{\big)},}\underline{-25} \\ \phantom{25\smash{\big)},}\phantom{00.}53 \\ \phantom{25\smash{\big)},}\phantom{0}\underline{-50} \\ \phantom{25\smash{\big)},}\phantom{0000.}30 \\ \phantom{25\smash{\big)},}\phantom{00..}\underline{-25} \\ \phantom{25\smash{\big)},}\phantom{00000.}50 \\ \phantom{25\smash{\big)},}\phantom{0000}\underline{-50} \\ \phantom{25\smash{\big)},}\phantom{00000.}\times \end{array}$

Hence, (7.5 × 40.4) ÷ 25 = 12.12

Evaluate:

2.1 ÷ (0.1 × 0.1)

Answer

$2.1 \div (0.1 \times 0.1) = 2.1 \div 0.01 = \dfrac{210}{1} = 210$

Hence, 2.1 ÷ (0.1 × 0.1) = 210

Fifteen identical articles weigh 31.50 kg. Find the weight of each article.

Answer

Weight of each article = Total weight ÷ Number of articles

= 31.50 ÷ 15 = 2.1 kg

Hence, the weight of each article is 2.1 kg.

The product of two numbers is 211.2. If one of these two numbers is 16.5, find the other number.

Answer

Other number = Product ÷ One number

$= 211.2 \div 16.5 = \dfrac{2112}{165} = 12.8$

Hence, the other number is 12.8.

One dozen identical articles cost ₹ 45.96. Find the cost of each article.

Answer

One dozen = 12 articles.

Cost of each article = Total cost ÷ Number of articles

= ₹ 45.96 ÷ 12 = ₹ 3.83

Hence, the cost of each article is ₹ 3.83.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

3 ÷ 8

Answer

By long division,

$\begin{array}{l} 8\overline{\smash{\big)}\phantom{00.}3.000\smash{\big(}}0.375 \\ \phantom{8\smash{\big)},}\phantom{..}\underline{-24} \\ \phantom{8\smash{\big)},}\phantom{000.}60 \\ \phantom{8\smash{\big)},}\phantom{00}\underline{-56} \\ \phantom{8\smash{\big)},}\phantom{0000.}40 \\ \phantom{8\smash{\big)},}\phantom{000}\underline{-40} \\ \phantom{8\smash{\big)},}\phantom{00000}\times \end{array}$

The remainder becomes zero, so the division ends after a finite number of steps and 3 ÷ 8 = 0.375.

Hence, it is a Terminating decimal.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

8 ÷ 3

Answer

By long division,

$\begin{array}{l} 3\overline{\smash{\big)}\phantom{00}8.000\smash{\big(}}2.666\ldots \\ \phantom{3\smash{\big)},}\underline{-6} \\ \phantom{+}\phantom{00.}20 \\ \phantom{+}\phantom{..}\underline{-18} \\ \phantom{+}\phantom{000.}20 \\ \phantom{+}\phantom{0..}\underline{-18} \\ \phantom{+}\phantom{0000.}20 \\ \phantom{+}\phantom{000}\underline{-18} \\ \phantom{+}\phantom{00000.}2 \\ \phantom{+}\phantom{00000}\ \vdots \end{array}$

The remainder never becomes zero, so the division never ends and 8 ÷ 3 = 2.666….

Hence, it is a Non-terminating decimal.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

6 ÷ 5

Answer

By long division,

$\begin{array}{l} 5\overline{\smash{\big)}\phantom{0..}6.0\smash{\big(}}1.2 \\ \phantom{5\smash{\big)}.}\underline{-5} \\ \phantom{5\smash{\big)},}\phantom{00}10 \\ \phantom{5\smash{\big)},}\phantom{.}\underline{-10} \\ \phantom{5\smash{\big)},}\phantom{00}\times \end{array}$

The remainder becomes zero, so the division ends after a finite number of steps and 6 ÷ 5 = 1.2.

Hence, it is a Terminating decimal.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

5 ÷ 6

Answer

By long division,

$\begin{array}{l} 6\overline{\smash{\big)}\phantom{000}5.0000\smash{\big(}}0.8333\ldots \\ \phantom{6\smash{\big)},}\phantom{0.}\underline{-48} \\ \phantom{6\smash{\big)},}\phantom{0000}20 \\ \phantom{6\smash{\big)},}\phantom{00.}\underline{-18} \\ \phantom{6\smash{\big)},}\phantom{0000..}20 \\ \phantom{6\smash{\big)},}\phantom{000.}\underline{-18} \\ \phantom{6\smash{\big)},}\phantom{0000.0.}20 \\ \phantom{6\smash{\big)},}\phantom{0000.}\underline{-18} \\ \phantom{6\smash{\big)},}\phantom{000000..}2 \\ \phantom{6\smash{\big)},}\phantom{000000.}\ \vdots \end{array}$

The remainder never becomes zero, so the division never ends and 5 ÷ 6 = 0.8333….

Hence, it is a Non-terminating decimal.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

12.5 ÷ 4

Answer

By long division,

$\begin{array}{l} 4\overline{\smash{\big)}\phantom{00}12.500\smash{\big(}}3.125 \\ \phantom{4\smash{\big)},}\underline{-12} \\ \phantom{4\smash{\big)},}\phantom{000.}5 \\ \phantom{4\smash{\big)},}\phantom{00}\underline{-4} \\ \phantom{4\smash{\big)},}\phantom{0000}10 \\ \phantom{4\smash{\big)},}\phantom{000.}\underline{-8} \\ \phantom{4\smash{\big)},}\phantom{00000.}20 \\ \phantom{4\smash{\big)},}\phantom{0000}\underline{-20} \\ \phantom{4\smash{\big)},}\phantom{000000}\times \end{array}$

The remainder becomes zero, so the division ends after a finite number of steps and 12.5 ÷ 4 = 3.125.

Hence, it is a Terminating decimal.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

23 ÷ 0.7

Answer

$23 \div 0.7 = \dfrac{230}{7}$

By long division,

$\begin{array}{l} 7\overline{\smash{\big)}\phantom{00}230.000\smash{\big(}}32.857\ldots \\ \phantom{7\smash{\big)},}\underline{-21} \\ \phantom{7\smash{\big)},}\phantom{00.}20 \\ \phantom{7\smash{\big)},}\phantom{0}\underline{-14} \\ \phantom{7\smash{\big)},}\phantom{000.}60 \\ \phantom{7\smash{\big)},}\phantom{00}\underline{-56} \\ \phantom{7\smash{\big)},}\phantom{0000.}40 \\ \phantom{7\smash{\big)},}\phantom{000}\underline{-35} \\ \phantom{7\smash{\big)},}\phantom{00000.}50 \\ \phantom{7\smash{\big)},}\phantom{000..}\underline{-49} \\ \phantom{7\smash{\big)},}\phantom{0000000}1 \\ \phantom{7\smash{\big)},}\phantom{000000}\ \vdots \end{array}$

The remainder never becomes zero, so the division never ends and 23 ÷ 0.7 = 32.857….

Hence, it is a Non-terminating decimal.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

42 ÷ 9

Answer

By long division,

$\begin{array}{l} 9\overline{\smash{\big)}\phantom{00}42.000\smash{\big(}}4.666\ldots \\ \phantom{9\smash{\big)},}\underline{-36} \\ \phantom{9\smash{\big)},}\phantom{00.}60 \\ \phantom{9\smash{\big)},}\phantom{0}\underline{-54} \\ \phantom{9\smash{\big)},}\phantom{0000}60 \\ \phantom{9\smash{\big)},}\phantom{00.}\underline{-54} \\ \phantom{9\smash{\big)},}\phantom{00000.}60 \\ \phantom{9\smash{\big)},}\phantom{0000}\underline{-54} \\ \phantom{9\smash{\big)},}\phantom{000000.}6 \\ \phantom{9\smash{\big)},}\phantom{00000}\ \vdots \end{array}$

The remainder never becomes zero, so the division never ends and 42 ÷ 9 = 4.666….

Hence, it is a Non-terminating decimal.

Find whether the given division forms a terminating decimal or a non-terminating decimal:

0.56 ÷ 0.11

Answer

$0.56 \div 0.11 = \dfrac{56}{11}$

By long division,

$\begin{array}{l} 11\overline{\smash{\big)}\phantom{00}56.0000\smash{\big(}}5.0909\ldots \\ \phantom{11\smash{\big)},}\underline{-55} \\ \phantom{11\smash{\big)},}\phantom{000}100 \\ \phantom{11\smash{\big)},}\phantom{00.}\underline{-99} \\ \phantom{11\smash{\big)},}\phantom{00000.}100 \\ \phantom{11\smash{\big)},}\phantom{00000}\underline{-99} \\ \phantom{11\smash{\big)},}\phantom{0000000.}1 \\ \phantom{11\smash{\big)},}\phantom{000000.}\ \vdots \end{array}$

The remainder never becomes zero, so the division never ends and 0.56 ÷ 0.11 = 5.0909….

Hence, it is a Non-terminating decimal.

The weight of an object is 3.06 kg. Find the total weight of 48 similar objects.

Answer

Total weight = Weight of one object × Number of objects

= 3.06 × 48 = 146.88 kg

Hence, the total weight of 48 objects is 146.88 kg.

Find the cost of 17.5 m cloth at the rate of ₹ 112.50 per metre.

Answer

Cost of cloth = Rate per metre × Length

= ₹ 112.50 × 17.5 = ₹ 1968.75

Hence, the cost of 17.5 m cloth is ₹ 1968.75.

One kilogram of oil costs ₹ 73.40. Find the cost of 9.75 kilograms of the oil.

Answer

Cost of oil = Cost per kg × Weight

= ₹ 73.40 × 9.75 = ₹ 715.65

Hence, the cost of 9.75 kg of oil is ₹ 715.65.

Total weight of 8 identical objects is 51.2 kg. Find the weight of each object.

Answer

Weight of each object = Total weight ÷ Number of objects

= 51.2 ÷ 8

= 6.4 kg

Hence, the weight of each object is 6.4 kg.

18.5 m of cloth costs ₹ 666. Find the cost of 3.8 m cloth.

Answer

Given,

Cost of 18.5 m of cloth = ₹ 666

Cost of 1 m cloth = ₹ 666 ÷ 18.5 = ₹ 36

Cost of 3.8 m cloth = ₹ 36 × 3.8 = ₹ 136.80

Hence, the cost of 3.8 m cloth is ₹ 136.80.

Find the value of:

(i) 0.5 of ₹ 7.60 + 1.62 of ₹ 30

(ii) 2.3 of 7.3 kg + 0.9 of 0.48 kg

(iii) 6.25 of 8.4 − 4.7 of 3.24

(iv) 0.98 of 235 − 0.09 of 3.2

Answer

(i) Solving,

0.5 of ₹ 7.60 + 1.62 of ₹ 30 = (0.5 × 7.60) + (1.62 × 30)

= 3.80 + 48.60 = ₹ 52.40

Hence, the value is ₹ 52.40.

(ii) Solving,

2.3 of 7.3 kg + 0.9 of 0.48 kg = (2.3 × 7.3) + (0.9 × 0.48)

= 16.79 + 0.432 = 17.222 kg

Hence, the value is 17.222 kg.

(iii) Solving,

6.25 of 8.4 - 4.7 of 3.24 = (6.25 × 8.4) - (4.7 × 3.24)

= 52.5 - 15.228 = 37.272

Hence, the value is 37.272.

(iv) Solving,

0.98 of 235 - 0.09 of 3.2 = (0.98 × 235) - (0.09 × 3.2)

= 230.3 - 0.288 = 230.012

Hence, the value is 230.012.

Evaluate:

(i) 5.6 − 1.5 of 3.4

(ii) 4.8 ÷ 0.04 of 5

(iii) 0.72 of 80 ÷ 0.2

(iv) 0.72 ÷ 80 of 0.2

(v) 6.45 ÷ (3.9 − 1.75)

(vi) 0.12 of (0.104 − 0.02) + 0.36 × 0.5

Answer

Following the order of operations (here 'of' is performed before ÷ and ×):

(i) Solving,

5.6 - 1.5 of 3.4 = 5.6 - (1.5 × 3.4)

= 5.6 - 5.1 = 0.5

Hence, 5.6 - 1.5 of 3.4 = 0.5

(ii) Solving,

4.8 ÷ 0.04 of 5 = 4.8 ÷ (0.04 × 5)

$= 4.8 \div 0.2 = \dfrac{48}{2} = 24$

Hence, 4.8 ÷ 0.04 of 5 = 24

(iii) Solving,

0.72 of 80 ÷ 0.2 = (0.72 × 80) ÷ 0.2

$= 57.6 \div 0.2 = \dfrac{576}{2} = 288$

Hence, 0.72 of 80 ÷ 0.2 = 288

(iv) Solving,

0.72 ÷ 80 of 0.2 = 0.72 ÷ (80 × 0.2)

= 0.72 ÷ 16 = 0.045

Hence, 0.72 ÷ 80 of 0.2 = 0.045

(v) Solving,

6.45 ÷ (3.9 - 1.75) = 6.45 ÷ 2.15

$= \dfrac{645}{215} = 3$

Hence, 6.45 ÷ (3.9 - 1.75) = 3

(vi) Solving,

0.12 of (0.104 - 0.02) + 0.36 × 0.5 = 0.12 × 0.084 + 0.18

= 0.01008 + 0.18 = 0.19008

Hence, 0.12 of (0.104 - 0.02) + 0.36 × 0.5 = 0.19008

Which is greater: 5.038 or 5.3?

1. 5.038

2. 5.3

3. none of these

Answer

Converting to like decimals: 5.038 and 5.300.

Comparing the digits after the decimal point, 300 > 038, so 5.3 > 5.038.

∴ 5.3 is greater.

Hence, Option 2 is the correct option.

Shyama bought 5 kg 300 g apples and 3 kg 250 g mangoes. Sarla bought 4 kg 800 g oranges and 4 kg 150 g bananas. Who bought more fruits?

1. Shyama

2. Sarla

3. none of these

Answer

Shyama bought 5 kg 300 g = 5.300 kg apples

Shyama bought 3 kg 250 g = 3.250 kg mangoes

Sarla bought 4 kg 800 g = 4.800 kg oranges

Sarla bought 4 kg 150 g = 4.150 kg bananas

Total weight of fruits bought by Shyama = 5.300 + 3.250 = 8.550 kg

Total weight of fruits bought by Sarla = 4.800 + 4.150 = 8.950 kg

Since 8.950 > 8.550, Sarla bought more fruits.

Hence, Option 2 is the correct option.

Two kg of milk contains 0.315 kg of cream. The cream in 20 kg milk is:

1. 31.5 kg

2. 0.63 kg

3. 3.15 kg

4. 0.315 kg

Answer

Cream in 2 kg milk = 0.315 kg

Cream in 1 kg milk $= \dfrac{0.315}{2}$ kg

Cream in 20 kg milk $= \dfrac{0.315}{2} \times 20 = 0.315 \times 10 = 3.15$ kg

Hence, Option 3 is the correct option.

The distance walked by a boy is 86.4 km in 4.8 hours. The distance covered by him in one hour, with the same speed, is:

1. 86.4 × 4.8 km

2. $\dfrac{86.4}{4.8}$ km

3. $\dfrac{4.8}{86.4}$ km

4. none of these

Answer

Distance covered in 4.8 hours = 86.4 km

Distance covered in 1 hour $= \dfrac{86.4}{4.8}$ km

Hence, Option 2 is the correct option.

The number seven and 7 thousandth is:

1. 7.700

2. 7.7000

3. 7.07

4. 7.007

Answer

Seven and 7 thousandth $= 7 + \dfrac{7}{1000} = 7 + 0.007 = 7.007$

Hence, Option 4 is the correct option.

56.56 ÷ 1.4 is equal to:

1. 4.4

2. 40.4

3. 4.04

4. 40

Answer

$56.56 \div 1.4 = \dfrac{565.6}{14} = 40.4$

Hence, Option 2 is the correct option.

$\left(2 + \dfrac{1}{2}\right) \div \dfrac{3}{5}$ is equal to:

1. $4\dfrac{1}{6}$

2. $6\dfrac{1}{4}$

3. $5\dfrac{1}{6}$

4. $5\dfrac{1}{4}$

Answer

Solving,

$\Rightarrow \left(2 + \dfrac{1}{2}\right) \div \dfrac{3}{5} \\[1em] \Rightarrow \dfrac{5}{2} \div \dfrac{3}{5} \\[1em] \Rightarrow \dfrac{5}{2} \times \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{25}{6} \\[1em] \Rightarrow 4\dfrac{1}{6}$

Hence, Option 1 is the correct option.

Total cost of two pens at ₹ 5.30 each and four notebooks at ₹ 20.50 each is:

1. ₹ 92

2. ₹ 9.26

3. ₹ 92.60

4. ₹ 926

Answer

Cost of 2 pens = 2 × 5.30 = ₹ 10.60

Cost of 4 notebooks = 4 × 20.50 = ₹ 82.00

Total cost = 10.60 + 82.00 = ₹ 92.60

Hence, Option 3 is the correct option.

2.5 + 3.8 ÷ 0.02 is equal to:

1. 315

2. 192.5

3. 19.25

4. 26.5

Answer

Following the order of operations, division is performed before addition:

$\Rightarrow 2.5 + 3.8 \div 0.02 \\[1em] \Rightarrow 2.5 + \dfrac{3.8}{0.02} \\[1em] \Rightarrow 2.5 + \dfrac{380}{2} \\[1em] \Rightarrow 2.5 + 190 \\[1em] \Rightarrow 192.5$

Hence, Option 2 is the correct option.

By what decimal number should 0.0001 be divided to get 0.01?

1. 100

2. 10

3. 0.1

4. 0.01

Answer

Let the required number be $x$.

$\Rightarrow 0.0001 \div x = 0.01 \\[1em] \Rightarrow \dfrac{0.0001}{x} = 0.01 \\[1em] \Rightarrow x = \dfrac{0.0001}{0.01} \\[1em] \Rightarrow x =\dfrac{100}{10000} \\[1em] \Rightarrow x = \dfrac{1}{100} = 0.01$

Hence, Option 4 is the correct option.

$3\dfrac{1}{5} \times \left(\dfrac{1}{2} + \dfrac{3}{8}\right) \div \dfrac{21}{40}$ is equal to:

1. $\dfrac{3}{16}$

2. $3\dfrac{1}{5}$

3. $5\dfrac{1}{3}$

4. none of these

Answer

Solving,

$\Rightarrow 3\dfrac{1}{5} \times \left(\dfrac{1}{2} + \dfrac{3}{8}\right) \div \dfrac{21}{40} \\[1em] \Rightarrow \dfrac{16}{5} \times \left(\dfrac{4}{8} + \dfrac{3}{8}\right) \div \dfrac{21}{40} \\[1em] \Rightarrow \dfrac{16}{5} \times \dfrac{7}{8} \div \dfrac{21}{40} \\[1em] \Rightarrow \dfrac{14}{5} \times \dfrac{40}{21} \\[1em] \Rightarrow \dfrac{16}{3} = 5\dfrac{1}{3}$

Hence, Option 3 is the correct option.

5.80, 0.95, 1.87 and 1.92 in descending order are:

1. 0.95, 1.87, 1.92 and 5.80

2. 5.80, 1.92, 1.87 and 0.95

3. 5.80, 1.87, 1.92 and 0.95

4. 5.80, 0.95, 1.87 and 1.92

Answer

Comparing the numbers: 5.80 > 1.92 > 1.87 > 0.95.

So the descending order is 5.80, 1.92, 1.87 and 0.95.

Hence, Option 2 is the correct option.

$3 - \dfrac{1}{4}$ of (15.8 − 3) is equal to:

1. 35.2

2. 0.2

3. −0.2

4. 6.2

Answer

Solving,

$\Rightarrow 3 - \dfrac{1}{4} \text{ of } (15.8 - 3) \\[1em] \Rightarrow 3 - \dfrac{1}{4} \text{ of } 12.8 \\[1em] \Rightarrow 3 - \dfrac{1}{4} \times 12.8 \\[1em] \Rightarrow 3 - 3.2 = -0.2$

Hence, Option 3 is the correct option.

Statement 1: 0.05 = 0.050 = 0.005 = 0.00500

Statement 2: Any number of zeros put at the end (i.e. on the right side) of a decimal number does not change its value.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

In Statement 1, 0.05 = 0.050 = 0.0500, but 0.005 = 0.00500, and 0.05 ≠ 0.005 (since 0.05 is five hundredths while 0.005 is five thousandths). So Statement 1 is false.

Statement 2 correctly states that adding zeros at the right end of a decimal number does not change its value, so Statement 2 is true.

∴ Statement 1 is false, and statement 2 is true.

Hence, Option 4 is the correct option.

Assertion (A): Representation of 6.25 as a vulgar fraction is $6\dfrac{1}{4}$.

Reason (R): A fraction is said to be a vulgar fraction if the denominator of its fractional part is a natural number but not of the form 10ⁿ, n∈N (natural number).

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

For the Assertion:

$6.25 = 6 + 0.25 = 6 + \dfrac{25}{100} = 6 + \dfrac{1}{4} = 6\dfrac{1}{4}$

In the fraction $6\dfrac{1}{4}$, the denominator (4) is a natural number and cannot be expressed as 10ⁿ, n∈N. So the Assertion (A) is true.

The Reason correctly states the condition for a fraction to be a vulgar fraction, so the Reason (R) is true.

∴ Both A and R are true.

Hence, Option 3 is the correct option.

Assertion (A): If the product of two decimal numbers is 17.55 and one of them is 6.5, then the other one is 2.7.

Reason (R): In division of decimal numbers, the dividend is not always exactly divisible.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

For the Assertion, the other number $= 17.55 \div 6.5 = \dfrac{175.5}{65} = 2.7$. So the Assertion (A) is true.

The Reason states a true general fact that in division of decimal numbers, the dividend is not always exactly divisible. So the Reason (R) is true.

∴ Both A and R are true.

Hence, Option 3 is the correct option.

Assertion (A): 3.10 ÷ (0.1 × 0.1) = 3.1

Reason (R): In division of a decimal number by 10ⁿ, n∈N, shift the decimal point to the left by n places (digits).

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

For the Assertion:

$3.10 \div (0.1 \times 0.1) = 3.10 \div 0.01 = \dfrac{310}{1} = 310$

Since the result is 310 and not 3.1, the Assertion (A) is false.

The Reason correctly states that in division of a decimal number by 10ⁿ, the decimal point is shifted to the left by n places, so the Reason (R) is true.

∴ A is false, R is true.

Hence, Option 2 is the correct option.

Assertion (A): 9, 9.56, 9.2, 9.005 are all unlike decimals, hence addition operations can't be performed.

Reason (R): A whole number can also be expressed as a decimal number by putting a decimal after its unit's digit and after it as many zeroes as required.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

For the Assertion, 9, 9.56, 9.2 and 9.005 are indeed unlike decimals (they have different numbers of decimal places). However, addition can still be performed by converting them into like decimals (9.000, 9.560, 9.200, 9.005). So the claim that "addition can't be performed" is wrong, and the Assertion (A) is false.

The Reason correctly states that a whole number can be expressed as a decimal number by placing a decimal point after the units digit and adding as many zeros to its right as required. So, the Reason (R) is true.

∴ A is false, R is true.

Hence, Option 2 is the correct option.