Decimal Fractions

(i) In 7.03, the decimal part is .03, which contains two digits.

Hence, 7.03 has 2 decimal places.

(ii) In 0.509, the decimal part is .509, which contains three digits.

Hence, 0.509 has 3 decimal places.

(iii) In 146.2, the decimal part is .2, which contains one digit.

Hence, 146.2 has 1 decimal place.

(iv) In 0.0065, the decimal part is .0065, which contains four digits.

Hence, 0.0065 has 4 decimal places.

(v) In 8.03207, the decimal part is .03207, which contains five digits.

Hence, 8.03207 has 5 decimal places.

To make decimal fractions like, we make the number of decimal places equal by putting the required number of zeroes on the extreme right of the decimal part.

(i) The maximum number of decimal places is 3, so we make each number have 3 decimal places.

1.360, 239.800 and 47.008

(ii) The maximum number of decimal places is 5, so we make each number have 5 decimal places.

507.07520, 8.52073 and 0.80800

(iii) The maximum number of decimal places is 6, so we make each number have 6 decimal places.

459.220000, 7.030930 and 0.200037

Count the number of zeroes in the denominator of the given fraction and mark the decimal point in the numerator, after as many digits from the right as there are zeroes in the denominator.

(i) $\frac{7}{10}$ = 0.7

(ii) $\frac{47}{10}$ = 4.7

(iii) $\frac{343}{100}$ = 3.43

(iv) $\frac{3}{10^3} = \frac{3}{1000}$ = 0.003

(v) $\frac{7295}{10^5} = \frac{7295}{100000}$ = 0.07295

(vi) $\frac{289}{10^6} = \frac{289}{1000000}$ = 0.000289

(vii) Ninety five-hundredths = $\frac{95}{100}$ = 0.95

To convert into a decimal fraction, we multiply the numerator and the denominator by a number so that the denominator becomes 10 or a higher power of 10.

(i) $\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100}$ = 0.75

(ii) $\frac{3}{40} = \frac{3 \times 25}{40 \times 25} = \frac{75}{1000}$ = 0.075

(iii) $\frac{1}{125} = \frac{1 \times 8}{125 \times 8} = \frac{8}{1000}$ = 0.008

(iv) $\frac{7}{25} = \frac{7 \times 4}{25 \times 4} = \frac{28}{100}$ = 0.28

To change the given decimal fractions, remove the decimal point and write in the denominator as many zeroes to the right of 1 as there are digits in the decimal part.

0.05 = $\frac{5}{100}$ = $\frac{1}{20}$

To change the given decimal fractions, remove the decimal point and write in the denominator as many zeroes to the right of 1 as there are digits in the decimal part.

3.95 = $\frac{395}{100}$ = $\frac{79}{20} = 3\frac{19}{20}$

To change the given decimal fractions, remove the decimal point and write in the denominator as many zeroes to the right of 1 as there are digits in the decimal part.

4.005 = $\frac{4005}{1000}$ = $\frac{801}{200} = 4\frac{1}{200}$

To change the given decimal fractions, remove the decimal point and write in the denominator as many zeroes to the right of 1 as there are digits in the decimal part.

0.876 = $\frac{876}{1000}$ = $\frac{219}{250}$

To change the given decimal fractions, remove the decimal point and write in the denominator as many zeroes to the right of 1 as there are digits in the decimal part.

50.06 = $\frac{5006}{100}$ = $\frac{2503}{50} = 50\frac{3}{50}$

To change the given decimal fractions, remove the decimal point and write in the denominator as many zeroes to the right of 1 as there are digits in the decimal part.

0.01075 = $\frac{1075}{100000}$ = $\frac{43}{4000}$

To change the given decimal fractions, remove the decimal point and write in the denominator as many zeroes to the right of 1 as there are digits in the decimal part.

4.8806 = $\frac{48806}{10000}$ = $\frac{24403}{5000} = 4\frac{4403}{5000}$

Converting the decimals into like decimals :

The like decimal numbers are 0.243, 2.470 and 3.009

$$\begin{array}{r} & 0.243 \\ & 2.470 \\ + & 3.009 \\ \hline & 5.722 \end{array}$$

Hence, 0.243 + 2.47 + 3.009 = 5.722

Converting the decimals into like decimals :

The like decimal numbers are 0.0736, 0.6095 and 0.9107

$$\begin{array}{r} & 0.0736 \\ & 0.6095 \\ + & 0.9107 \\ \hline & 1.5938 \end{array}$$

Hence, 0.0736 + 0.6095 + 0.9107 = 1.5938

Converting the decimals into like decimals :

The like decimal numbers are 1.010, 257.000 and 0.200

$$\begin{array}{rr} & 1.010 \\ & 257.000 \\ + & 0.200 \\ \hline & 258.210 \end{array}$$

Hence, 1.01 + 257 + 0.200 = 258.21

Converting the decimals into like decimals :

The like decimal numbers are 18.00, 200.35, 11.72 and 2.30

$$\begin{array}{rr} & 18.00 \\ & 200.35 \\ & 11.72 \\ + & 2.30 \\ \hline & 232.37 \end{array}$$

Hence, 18 + 200.35 + 11.72 + 2.3 = 232.37

Converting the decimals into like decimals :

The like decimal numbers are 0.58600, 0.05860 and 0.00586

$$\begin{array}{r} & 0.58600 \\ & 0.05860 \\ + & 0.00586 \\ \hline & 0.65046 \end{array}$$

Hence, 0.586 + 0.0586 + 0.00586 = 0.65046

Converting the decimals into like decimals :

The like decimal numbers are 6.80 and 2.64

$$\begin{array}{r} & 6.80 \\ - & 2.64 \\ \hline & 4.16 \end{array}$$

Hence, 6.8 - 2.64 = 4.16

Converting the decimals into like decimals :

The like decimal numbers are 2.0000 and 1.0304

$$\begin{array}{r} & 2.0000 \\ - & 1.0304 \\ \hline & 0.9696 \end{array}$$

Hence, 2 - 1.0304 = 0.9696

Converting the decimals into like decimals :

The like decimal numbers are 0.10 and 0.08

$$\begin{array}{r} & 0.10 \\ - & 0.08 \\ \hline & 0.02 \end{array}$$

Hence, 0.1 - 0.08 = 0.02

Converting the decimals into like decimals :

The like decimal numbers are 0.830 and 0.342

$$\begin{array}{r} & 0.830 \\ - & 0.342 \\ \hline & 0.488 \end{array}$$

Hence, 0.83 - 0.342 = 0.488

Converting the decimals into like decimals :

The like decimal numbers are 0.97 and 0.43

$$\begin{array}{r} & 0.97 \\ - &0.43 \\ \hline & 0.54 \end{array}$$

Hence, 0.97 - 0.43 = 0.54

Converting the decimals into like decimals :

The like decimal numbers are 22.1058 and 2.0080

$$\begin{array}{rr} & 22.1058 \\ - & 2.0080 \\ \hline & 20.0978 \end{array}$$

Hence, 22.1058 - 2.008 = 20.0978

Converting the decimals into like decimals :

The like decimal numbers are 0.60 and 0.18

$$\begin{array}{r} & 0.60 \\ - & 0.18 \\ \hline & 0.42 \end{array}$$

Hence, 0.6 - 0.18 = 0.42

Converting the decimals into like decimals :

The like decimal numbers are 17.000 and 1.002

$$\begin{array}{rr} & 17.000 \\ - & 1.002 \\ \hline & 15.998 \end{array}$$

Hence, 17 - 1.002 = 15.998

Converting the decimals into like decimals :

The like decimal numbers are 92.05 and 83.00

$$\begin{array}{rr} & 92.05 \\ -& 83.00 \\ \hline & 9.05 \end{array}$$

Hence, 92.05 - 83 = 9.05

(i) The like decimal numbers are 3.500, 2.430 and 0.075

First, adding the positive numbers 3.500 and 0.075

$$\begin{array}{r} & 3.500 \\ + & 0.075 \\ \hline & 3.575 \end{array}$$

Now, subtracting 2.430 from 3.575

$$\begin{array}{r} & 3.575 \\ - & 2.430 \\ \hline & 1.145 \end{array}$$

Hence, 3.5 - 2.43 + 0.075 = 1.145

(ii) The like decimal numbers are 7.840, 0.300 and 4.016

First, adding the positive numbers 7.840 and 0.300

$$\begin{array}{r} & 7.840 \\ + & 0.300 \\ \hline & 8.140 \end{array}$$

Now, subtracting 4.016 from 8.140

$$\begin{array}{r} & 8.140 \\ - & 4.016 \\ \hline & 4.124 \end{array}$$

Hence, 7.84 + 0.3 - 4.016 = 4.124

(iii) The like decimal numbers are 2.987, 1.250 and 0.540

First, adding the numbers to be subtracted 1.250 and 0.540

$$\begin{array}{r} & 1.250 \\ + &0.540 \\ \hline & 1.790 \end{array}$$

Now, subtracting 1.790 from 2.987

$$\begin{array}{r} & 2.987 \\ - & 1.790 \\ \hline & 1.197 \end{array}$$

Hence, 2.987 - 1.25 - 0.54 = 1.197

(iv) The like decimal numbers are 52.900, 231.666 and 204.000

First, adding the positive numbers 52.900 and 204.000

$$\begin{array}{rr} & 52.900 \\ + &204.000 \\ \hline & 256.900 \end{array}$$

Now, subtracting 231.666 from 256.900

$$\begin{array}{r} & 256.900 \\ - & 231.666 \\ \hline & 25.234 \end{array}$$

Hence, 52.9 - 231.666 + 204 = 25.234

(v) The like decimal numbers are 8.5700, 6.4432, 1.7000 and 0.6830

First, adding the positive numbers 8.5700 and 0.6830

$$\begin{array}{r} & 8.5700 \\ + & 0.6830 \\ \hline & 9.2530 \end{array}$$

Adding the numbers to be subtracted 6.4432 and 1.7000

$$\begin{array}{r} & 6.4432 \\ + &1.7000 \\ \hline & 8.1432 \end{array}$$

Now, subtracting 8.1432 from 9.2530

$$\begin{array}{r} & 9.2530 \\ - & 8.1432 \\ \hline & 1.1098 \end{array}$$

Hence, 8.57 - 6.4432 - 1.70 + 0.683 = 1.1098

The like decimal numbers are 75.750, 4.900 and 28.465

First, adding 75.750 and 4.900

$$\begin{array}{rr} & 75.750 \\ + & 4.900 \\ \hline & 80.650 \end{array}$$

Now, subtracting 28.465 from 80.650

$$\begin{array}{r} & 80.650 \\ - & 28.465 \\ \hline & 52.185 \end{array}$$

Hence, the required answer = 52.185

The like decimal numbers are 8.14, 12.90 and 32.70

First, adding 8.14 and 12.90

$$\begin{array}{rr} & 8.14 \\ + & 12.90 \\ \hline & 21.04 \end{array}$$

Now, subtracting 21.04 from 32.70

$$\begin{array}{r} & 32.70 \\ - & 21.04 \\ \hline & 11.66 \end{array}$$

Hence, the required answer = 11.66

The like decimal numbers are 34.270, 159.800, 20.937 and 200.600

First, adding 34.270 and 159.800

$$\begin{array}{rr} & 34.270 \\ + & 159.800 \\ \hline & 194.070 \end{array}$$

Adding 20.937 and 200.600

$$\begin{array}{rr} & 20.937 \\ + & 200.600 \\ \hline & 221.537 \end{array}$$

Now, subtracting 194.070 from 221.537

$$\begin{array}{rr} & 221.537 \\ - & 194.070 \\ \hline & 27.467 \end{array}$$

Hence, the required answer = 27.467

The like decimal numbers are 2.430, 4.349, 0.800 and 3.150

First, adding 2.430 and 4.349

$$\begin{array}{r} &2.430 \\ + & 4.349 \\ \hline & 6.779 \end{array}$$

Adding 0.800 and 3.150

$$\begin{array}{r} & 0.800 \\ + & 3.150 \\ \hline & 3.950 \end{array}$$

Now, subtracting 3.950 from 6.779

$$\begin{array}{r} & 6.779 \\ - & 3.950 \\ \hline & 2.829 \end{array}$$

Hence, the required answer = 2.829

The like decimal numbers are 18.0495, 34.9644, 7.6752 and 24.8760

First, adding 18.0495 and 34.9644

$$\begin{array}{r} & 18.0495 \\ + & 34.9644 \\ \hline & 53.0139 \end{array}$$

Adding 7.6752 and 24.8760

$$\begin{array}{rr} & 7.6752 \\ + & 24.8760 \\ \hline & 32.5512 \end{array}$$

Now, subtracting 32.5512 from 53.0139

$$\begin{array}{r} & 53.0139 \\ - & 32.5512 \\ \hline & 20.4627 \end{array}$$

Hence, the first sum exceeds the second sum by 20.4627

Required least number = 1000 - 89.376

$$\begin{array}{rr} & 1000.000 \\ - & 89.376 \\ \hline & 910.624 \end{array}$$

Hence, 910.624 must be added to 89.376 to get 1000.

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

56 × 8 = 448

∴ 5.6 × 8 = 44.8

Hence, 5.6 × 8 = 44.8

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

3846 × 9 = 34614

∴ 38.46 × 9 = 346.14

Hence, 38.46 × 9 = 346.14

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

943 × 62 = 58466

∴ 0.943 × 62 = 58.466

Hence, 0.943 × 62 = 58.466

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

453 × 35 = 15855

∴ 0.0453 × 35 = 1.5855

Hence, 0.0453 × 35 = 1.5855

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

75 × 25 = 1875

∴ 7.5 × 2.5 = 18.75

Hence, 7.5 × 2.5 = 18.75

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

423 × 8 = 3384

∴ 4.23 × 0.8 = 3.384

Hence, 4.23 × 0.8 = 3.384

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

8354 × 7 = 58478

∴ 83.54 × 0.07 = 5.8478

Hence, 83.54 × 0.07 = 5.8478

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

636 × 183 = 116388

∴ 0.636 × 1.83 = 1.16388

Hence, 0.636 × 1.83 = 1.16388

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

8 × 26 = 208

∴ 0.0008 × 26 = 0.0208

Hence, 0.0008 × 26 = 0.0208

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

38 × 95 = 3610

∴ 0.038 × 95 = 3.610 = 3.61

Hence, 0.038 × 95 = 3.61

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

12 × 24 × 36 = (12 × 24) × 36 = 288 × 36 = 10368

∴ 1.2 × 2.4 × 3.6 = 10.368

Hence, 1.2 × 2.4 × 3.6 = 10.368

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

9 × 18 × 27 = (9 × 18) × 27 = 162 × 27 = 4374

∴ 0.9 × 1.8 × 0.27 = 0.4374

Hence, 0.9 × 1.8 × 0.27 = 0.4374

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

15 × 15 × 15 = (15 × 15) × 15 = 225 × 15 = 3375

∴ 1.5 × 1.5 × 1.5 = 3.375

Hence, 1.5 × 1.5 × 1.5 = 3.375

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

25 × 25 = 625

∴ 0.025 × 0.025 = 0.000625

Hence, 0.025 × 0.025 = 0.000625

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point in the product so that the number of decimal places in it equals the sum of the decimal places of the numbers multiplied.

2 × 2 × 1 = 4

∴ 0.2 × 0.002 × 0.001 = 0.0000004

Hence, 0.2 × 0.002 × 0.001 = 0.0000004

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

3.9 × 10 = 39,

3.9 × 100 = 390,

3.9 × 1000 = 3900.

Hence, 39, 390 and 3900

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

2.89 × 10 = 28.9,

2.89 × 100 = 289,

2.89 × 1000 = 2890.

Hence, 28.9, 289 and 2890

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

0.0829 × 10 = 0.829,

0.0829 × 100 = 8.29,

0.0829 × 1000 = 82.9.

Hence, 0.829, 8.29 and 82.9

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

40.3 × 10 = 403,

40.3 × 100 = 4030,

40.3 × 1000 = 40300.

Hence, 403, 4030 and 40300

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

0.3725 × 10 = 3.725,

0.3725 × 100 = 37.25,

0.3725 × 1000 = 372.5.

Hence, 3.725, 37.25 and 372.5

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 8\overline{\smash{\big)}8.64\smash{\big(}} 1.08 \\ \phantom{}\underline{-8} \\ \phantom{+()}{64} \\ \phantom{+}\underline{-64} \\ \phantom{++}{0} \\ \end{array}$$

Hence, 8.64 ÷ 8 = 1.08

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 6\overline{\smash{\big)}0.0072\smash{\big(}} 0.0012 \\ \phantom{++}\underline{-6} \\ \phantom{+++}{12} \\ \phantom{++}\underline{-12} \\ \phantom{+++}{0} \\ \end{array}$$

Hence, 0.0072 ÷ 6 = 0.0012

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 16\overline{\smash{\big)}20.64\smash{\big(}} 1.29 \\ \phantom{+}\underline{-16} \\ \phantom{++}{46} \\ \phantom{+)}\underline{-32} \\ \phantom{++}{144} \\ \phantom{+(}\underline{-144} \\ \phantom{+++}{0} \\ \end{array}$$

Hence, 20.64 ÷ 16 = 1.29

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 15\overline{\smash{\big)}1.602\smash{\big(}} 0.1068 \\ \phantom{+}\underline{-15} \\ \phantom{++}{102} \\ \phantom{+()}\underline{-90} \\ \phantom{++(}{120} \\ \phantom{+()}\underline{-120} \\ \phantom{+++}{0} \\ \end{array}$$

Hence, 1.602 ÷ 15 = 0.1068

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 4\overline{\smash{\big)}13.08\smash{\big(}} 3.27 \\ \phantom{)}\underline{-12} \\ \phantom{+()}{10} \\ \phantom{+)}\underline{-8} \\ \phantom{++}{28} \\ \phantom{+)}\underline{-28} \\ \phantom{++)}{0} \\ \end{array}$$

Hence, 13.08 ÷ 4 = 3.27

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 9\overline{\smash{\big)}3.204\smash{\big(}} 0.356 \\ \phantom{)}\underline{-27} \\ \phantom{+()}{50} \\ \phantom{+}\underline{-45} \\ \phantom{++))}{54} \\ \phantom{+))}\underline{-54} \\ \phantom{+++}{0} \\ \end{array}$$

Hence, 3.204 ÷ 9 = 0.356

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 12\overline{\smash{\big)}3.024\smash{\big(}} 0.252 \\ \phantom{+}\underline{-24} \\ \phantom{++))}{62} \\ \phantom{+))}\underline{-60} \\ \phantom{++(()}{24} \\ \phantom{+(())}\underline{-24} \\ \phantom{++(())}{0} \\ \end{array}$$

Hence, 3.024 ÷ 12 = 0.252

To divide decimal numbers, place the decimal in the quotient just after the division of the integral part of the given decimal number.

Solving,

$$\begin{array}{l} 5\overline{\smash{\big)}5.15\smash{\big(}} 1.03 \\ \phantom{}\underline{-5} \\ \phantom{+)}{01} \\ \phantom{+}\underline{-0} \\ \phantom{+()}{15} \\ \phantom{+}\underline{-15} \\ \phantom{++}{0} \\ \end{array}$$

Hence, 5.15 ÷ 5 = 1.03

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

49.79 ÷ 10 = 4.979,

49.79 ÷ 100 = 0.4979,

49.79 ÷ 1000 = 0.04979.

Hence, 4.979, 0.4979 and 0.04979

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

0.923 ÷ 10 = 0.0923,

0.923 ÷ 100 = 0.00923,

0.923 ÷ 1000 = 0.000923.

Hence, 0.0923, 0.00923 and 0.000923

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

0.0704 ÷ 10 = 0.00704,

0.0704 ÷ 100 = 0.000704,

0.0704 ÷ 1000 = 0.0000704.

Hence, 0.00704, 0.000704 and 0.0000704

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

9.4 ÷ 0.47 = $\frac{9.4}{0.47} = \frac{9.4 \times 100}{0.47 \times 100} = \frac{940}{47}$

$$\begin{array}{l} 47\overline{\smash{\big)}940\smash{\big(}} 20 \\ \phantom{+}\underline{-94} \\ \phantom{++(}{00} \\ \end{array}$$

Hence, 9.4 ÷ 0.47 = 20

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

6.3 ÷ 0.09 = $\frac{6.3}{0.09} = \frac{6.3 \times 100}{0.09 \times 100} = \frac{630}{9}$

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

Hence, 6.3 ÷ 0.09 = 70

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

2.88 ÷ 1.2 = $\frac{2.88}{1.2} = \frac{2.88 \times 10}{1.2 \times 10} = \frac{28.8}{12}$

$$\begin{array}{l} 12\overline{\smash{\big)}28.8\smash{\big(}} 2.4 \\ \phantom{+}\underline{-24} \\ \phantom{++}{48} \\ \phantom{+(}\underline{-48} \\ \phantom{++))}{0} \\ \end{array}$$

Hence, 2.88 ÷ 1.2 = 2.4

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

8.64 ÷ 1.6 = $\frac{8.64}{1.6} = \frac{8.64 \times 10}{1.6 \times 10} = \frac{86.4}{16}$

$$\begin{array}{l} 16\overline{\smash{\big)}86.4\smash{\big(}} 5.4 \\ \phantom{+}\underline{-80} \\ \phantom{++}{64} \\ \phantom{+(}\underline{-64} \\ \phantom{++}{0} \\ \end{array}$$

Hence, 8.64 ÷ 1.6 = 5.4

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

37.188 ÷ 3.6 = $\frac{37.188}{3.6} = \frac{37.188 \times 10}{3.6 \times 10} = \frac{371.88}{36}$

$$\begin{array}{l} 36\overline{\smash{\big)}371.88\smash{\big(}} 10.33 \\ \phantom{+}\underline{-36} \\ \phantom{++}{118} \\ \phantom{+)}\underline{-108} \\ \phantom{++(}{108} \\ \phantom{+()}\underline{-108} \\ \phantom{+++}{0} \\ \end{array}$$

Hence, 37.188 ÷ 3.6 = 10.33

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

16.5 ÷ 0.15 = $\frac{16.5}{0.15} = \frac{16.5 \times 100}{0.15 \times 100} = \frac{1650}{15}$

$$\begin{array}{l} 15\overline{\smash{\big)}1650\smash{\big(}} 110 \\ \phantom{+}\underline{-15} \\ \phantom{++}{15} \\ \phantom{+(}\underline{-15} \\ \phantom{++}{00} \\ \end{array}$$

Hence, 16.5 ÷ 0.15 = 110

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

3.2 ÷ 0.005 = $\frac{3.2}{0.005} = \frac{3.2 \times 1000}{0.005 \times 1000} = \frac{3200}{5}$

$$\begin{array}{l} 5\overline{\smash{\big)}3200\smash{\big(}} 640 \\ \phantom{}\underline{-30} \\ \phantom{+()}{20} \\ \phantom{+}\underline{-20} \\ \phantom{+()}{00} \\ \end{array}$$

Hence, 3.2 ÷ 0.005 = 640

To divide by a decimal number, we first form a fraction with the dividend as the numerator and the divisor as the denominator. Then we multiply both the numerator and the denominator by 10, 100, 1000, etc. so that the divisor (denominator) becomes a whole number, and then we divide.

3.24 ÷ 0.0016 = $\frac{3.24}{0.0016} = \frac{3.24 \times 10000}{0.0016 \times 10000} = \frac{32400}{16}$

$$\begin{array}{l} 16\overline{\smash{\big)}32400\smash{\big(}} 2025 \\ \phantom{+}\underline{-32} \\ \phantom{++}{040} \\ \phantom{++}\underline{-32} \\ \phantom{+++()}{80} \\ \phantom{+++}\underline{-80} \\ \phantom{++++}{0} \\ \end{array}$$

Hence, 3.24 ÷ 0.0016 = 2025

(i) 7.85 × 10 = 78.5

(ii) 0.442 × 1000 = 442

(iii) 0.0924 × 100 = 9.24

(iv) 0.00187 × 10000 = 18.7

(v) 2.6 × 1000 = 2600

(vi) 0.08 × 1000 = 80

(vii) 96.7 ÷ 100 = 0.967

(viii) 5.2 ÷ 10 = 0.52

(ix) 33.15 ÷ 1000 = 0.03315

(x) 0.7 ÷ 100 = 0.007

(xi) 0.00672 × 10000 = 67.2

According to the order of operations, we first perform division (÷) and multiplication (×), and then addition (+) and subtraction (−).

9.32 - 28.54 ÷ 10 = 9.32 - 2.854 = 6.466

Hence, 9.32 - 28.54 ÷ 10 = 6.466

According to the order of operations, we first perform division (÷) and multiplication (×), and then addition (+) and subtraction (−).

0.234 × 10 + 62.8 = 2.34 + 62.8 = 65.14

Hence, 0.234 × 10 + 62.8 = 65.14

According to the order of operations, we first perform division (÷) and multiplication (×), and then addition (+) and subtraction (−).

3.06 × 100 - 889.4 ÷ 100 = 306 - 8.894 = 297.106

Hence, 3.06 × 100 - 889.4 ÷ 100 = 297.106

According to the order of operations, we first perform division (÷) and multiplication (×), and then addition (+) and subtraction (−).

2.86 × 7.5 + 45.4 ÷ 0.2 = 21.45 + 227 = 248.45

Hence, 2.86 × 7.5 + 45.4 ÷ 0.2 = 248.45

According to the order of operations, we first perform division (÷) and multiplication (×), and then addition (+) and subtraction (−).

97.82 × 0.03 - 0.54 ÷ 0.3 = 2.9346 - 1.8 = 1.1346

Hence, 97.82 × 0.03 - 0.54 ÷ 0.3 = 1.1346

Since ₹ 1 = 100 paise, we multiply the amount in rupees by 100.

(i) ₹ 8.40 = 8.40 × 100 paise = 840 p

Hence, ₹ 8.40 = 840 p

(ii) ₹ 0.97 = 0.97 × 100 paise = 97 p

Hence, ₹ 0.97 = 97 p

(iii) ₹ 0.09 = 0.09 × 100 paise = 9 p

Hence, ₹ 0.09 = 9 p

(iv) ₹ 62.35 = 62.35 × 100 paise = 6235 p

Hence, ₹ 62.35 = 6235 p

Since 100 paise = ₹ 1, we divide the amount in paise by 100.

(i) 55 p = ₹ $\frac{55}{100}$ = ₹ 0.55

Hence, 55 p = ₹ 0.55

(ii) 8 p = ₹ $\frac{8}{100}$ = ₹ 0.08

Hence, 8 p = ₹ 0.08

(iii) 695 p = ₹ $\frac{695}{100}$ = ₹ 6.95

Hence, 695 p = ₹ 6.95

(iv) 3279 p = ₹ $\frac{3279}{100}$ = ₹ 32.79

Hence, 3279 p = ₹ 32.79

Since 1 m = 100 cm, we multiply the length in metre by 100.

(i) 6 m = 6 × 100 cm = 600 cm

Hence, 6 m = 600 cm

(ii) 8.54 m = 8.54 × 100 cm = 854 cm

Hence, 8.54 m = 854 cm

(iii) 3.08 m = 3.08 × 100 cm = 308 cm

Hence, 3.08 m = 308 cm

(iv) 0.87 m = 0.87 × 100 cm = 87 cm

Hence, 0.87 m = 87 cm

(v) 0.03 m = 0.03 × 100 cm = 3 cm

Hence, 0.03 m = 3 cm

(vi) 25.04 m = 25.04 × 100 cm = 2504 cm

Hence, 25.04 m = 2504 cm

Since 100 cm = 1 m, we divide the length in cm by 100.

(i) 250 cm = $\frac{250}{100}$ m = 2.50 m

Hence, 250 cm = 2.50 m

(ii) 2328 cm = $\frac{2328}{100}$ m = 23.28 m

Hence, 2328 cm = 23.28 m

(iii) 86 cm = $\frac{86}{100}$ m = 0.86 m

Hence, 86 cm = 0.86 m

(iv) 4 cm = $\frac{4}{100}$ m = 0.04 m

Hence, 4 cm = 0.04 m

(v) 107 cm = $\frac{107}{100}$ m = 1.07 m

Hence, 107 cm = 1.07 m

Since 1 kg = 1000 g, we multiply the mass in kg by 1000.

(i) 6 kg = 6 × 1000 g = 6000 g

Hence, 6 kg = 6000 g

(ii) 5.543 kg = 5.543 × 1000 g = 5543 g

Hence, 5.543 kg = 5543 g

(iii) 0.078 kg = 0.078 × 1000 g = 78 g

Hence, 0.078 kg = 78 g

(iv) 3.62 kg = 3.62 × 1000 g = 3620 g

Hence, 3.62 kg = 3620 g

(v) 4.5 kg = 4.5 × 1000 g = 4500 g

Hence, 4.5 kg = 4500 g

Since 1000 g = 1 kg, we divide the mass in g by 1000.

(i) 7000 g = $\frac{7000}{1000}$ kg = 7 kg

Hence, 7000 g = 7 kg

(ii) 6839 g = $\frac{6839}{1000}$ kg = 6.839 kg

Hence, 6839 g = 6.839 kg

(iii) 445 g = $\frac{445}{1000}$ kg = 0.445 kg

Hence, 445 g = 0.445 kg

(iv) 93 g = $\frac{93}{1000}$ kg = 0.093 kg

Hence, 93 g = 0.093 kg

(v) 8 g = $\frac{8}{1000}$ kg = 0.008 kg

Hence, 8 g = 0.008 kg

(vi) 13545 g = $\frac{13545}{1000}$ kg = 13.545 kg

Hence, 13545 g = 13.545 kg

Since 100 paise = ₹ 1, we first convert each amount into rupees (dividing paise by 100) so that all amounts are in the same unit, and then add.

(i) ₹ 5.37 + ₹ 12 = ₹ 5.37 + ₹ 12.00 = ₹ 17.37

Hence, the sum = ₹ 17.37

(ii) 532 paise = ₹ 5.32

₹ 24.03 + ₹ 5.32 = ₹ 29.35

Hence, the sum = ₹ 29.35

(iii) 73 paise = ₹ 0.73

₹ 0.73 + ₹ 2.08 = ₹ 2.81

Hence, the sum = ₹ 2.81

(iv) 8 paise = ₹ 0.08

₹ 0.08 + ₹ 15.36 = ₹ 15.44

Hence, the sum = ₹ 15.44

Since 100 paise = ₹ 1, we first convert each amount into rupees (dividing paise by 100) so that both amounts are in the same unit, and then subtract.

(i) ₹ 63.22 - ₹ 35.74 = ₹ 27.48

Hence, the difference = ₹ 27.48

(ii) 286 paise = ₹ 2.86

₹ 7.02 - ₹ 2.86 = ₹ 4.16

Hence, the difference = ₹ 4.16

(iii) 121 paise = ₹ 1.21

₹ 1.21 - ₹ 0.55 = ₹ 0.66 = 66 paise

Hence, the difference = ₹ 0.66 = 66 paise

Since 100 cm = 1 m, we first convert each length into metre (dividing cm by 100) so that all lengths are in the same unit, and then add.

(i) 2.4 m + 1.78 m = 2.40 m + 1.78 m = 4.18 m

Hence, the sum = 4.18 m

(ii) 848 cm = 8.48 m

8.48 m + 2.90 m = 11.38 m

Hence, the sum = 11.38 m

(iii) 64 cm = 0.64 m

0.93 m + 0.64 m = 1.57 m

Hence, the sum = 1.57 m

Since 100 cm = 1 m, we first convert each length into metre (dividing cm by 100) so that both lengths are in the same unit, and then subtract.

(i) 19.6 m - 5.03 m = 19.60 m - 5.03 m = 14.57 m

Hence, the difference = 14.57 m

(ii) 428 cm = 4.28 m

1033 m - 4.28 m = 1028.72 m

Hence, the difference = 1028.72 m

(iii) 122 cm = 1.22 m

1.22 m - 0.84 m = 0.38 m

Hence, the difference = 0.38 m

Since 1000 g = 1 kg, we first convert each mass into kilogram (dividing g by 1000) so that all masses are in the same unit, and then add.

(i) 2.06 kg + 57.864 kg = 2.060 kg + 57.864 kg = 59.924 kg

Hence, the sum = 59.924 kg

(ii) 778 g = 0.778 kg

0.778 kg + 1.939 kg = 2.717 kg

Hence, the sum = 2.717 kg

(iii) 4023 g = 4.023 kg

0.065 kg + 4.023 kg = 4.088 kg

Hence, the sum = 4.088 kg

Since 1000 g = 1 kg, we first convert each mass into kilogram (dividing g by 1000) so that both masses are in the same unit, and then subtract.

(i) 15.6 kg - 9.462 kg = 15.600 kg - 9.462 kg = 6.138 kg

Hence, the difference = 6.138 kg

(ii) 4317 g = 4.317 kg

23 kg - 4.317 kg = 23.000 kg - 4.317 kg = 18.683 kg

Hence, the difference = 18.683 kg

(iii) 4169 g = 4.169 kg

4.169 kg - 0.798 kg = 3.371 kg

Hence, the difference = 3.371 kg

Cost of 1 fountain pen = ₹ 13.25

∴ Cost of 8 fountain pens = ₹ 13.25 × 8 = ₹ 106

Hence, the cost of 8 pens = ₹ 106.

Cost of 25 articles = ₹ 218.25

∴ Cost of 1 article = ₹ $\frac{218.25}{25}$ = ₹ 8.73

Hence, the cost of one article = ₹ 8.73

Length of the rod = 10.32 m

Since the rod is divided into 4 equal pieces,

∴ Length of each piece = $\frac{10.32}{4}$ = 2.58 m

Hence, the length of each piece = 2.58 m

Cloth needed for 1 shirt = 2.15 m

∴ Cloth needed for 5 shirts = 2.15 × 5 = 10.75 m

Hence, the total length of cloth required = 10.75 m

Distance walked in 1 hour = 2.150 km

Time = $1\frac{1}{2}$ hours = 1.5 hours

∴ Distance walked in 1.5 hours = 2.150 × 1.5 = 3.225 km

Hence, the distance walked = 3.225 km

Cost of 1 notebook = ₹ 15.25

∴ Total money for 83 notebooks = ₹ 15.25 × 83 = ₹ 1265.75

Hence, the total money obtained = ₹ 1,265.75

Length of 1 bedcover = 2.1 m

∴ Total length of 17 bedcovers = 2.1 × 17 = 35.7 m

Hence, the total length of 17 bedcovers = 35.7 m

Length of first rope = 10 m 67 cm = 10.67 m

Length of second rope = 16 m 32 cm = 16.32 m

Required difference = 16.32 m - 10.67 m = 5.65 m

Hence, the second rope is longer by 5.65 m = 5 m 65 cm

Weight of 12 cakes = 5 kg 604 g = 5.604 kg

(i) Weight of 1 cake = $\frac{5.604}{12}$ = 0.467 kg = 467 g

Hence, the weight of one cake = 0.467 kg = 467 g

(ii) Weight of 5 cakes = 0.467 × 5 = 2.335 kg

Hence, the weight of 5 cakes = 2.335 kg

The lengths of the three strings are:

50 m 75 cm = 50.75 m

68 m 58 cm = 68.58 m

121 m 3 cm = 121.03 m

Length of single string = 50.75 + 68.58 + 121.03 = 240.36 m

Now, this string is divided into 12 equal pieces.

∴ Length of each piece = $\frac{240.36}{12}$ = 20.03 m

Hence, the length of the single string = 240.36 m and the length of each piece = 20.03 m

To add decimal numbers, we write them as like decimals and then add. Here both numbers already have 2 decimal places.

$$\begin{array}{rr} & 2.73 \\ + & 27.30 \\ \hline & 30.03 \end{array}$$

2.73 + 27.30 = 30.03

Hence, option 1 is the correct option.

To subtract decimal numbers, we first make them like decimals. Writing 27.3 as 27.30, now both numbers have 2 decimal places, and then we subtract.

$$\begin{array}{rr} & 27.30 \\ - & 2.73 \\ \hline & 24.57 \end{array}$$

27.30 - 2.73 = 24.57

Hence, option 2 is the correct option.

On multiplying a decimal number by 10, 100, 1000, 10000, etc., the decimal point shifts to the right by as many places as there are zeroes in the multiplier. Since 10000 has 4 zeroes, the decimal point shifts 4 places to the right.

0.058 × 10000 = 580

Hence, option 3 is the correct option.

On dividing a decimal number by 10, 100, 1000, etc., the decimal point shifts to the left by as many places as there are zeroes in the divisor. Since 100 has 2 zeroes, the decimal point shifts 2 places to the left.

5.8 ÷ 100 = 0.058

Hence, option 4 is the correct option.

To multiply decimal numbers, we multiply them ignoring the decimal points and then mark the decimal point so that the number of decimal places in the product equals the sum of the decimal places of the numbers multiplied.

0.2 × 0.02 × 0.002 = 0.000008

Now, multiplying by 1000 shifts the decimal point 3 places to the right.

0.000008 × 1000 = 0.008

Hence, option 3 is the correct option.

Since 100 paise = ₹ 1, we first convert the paise into rupees by dividing by 100, and then add.

6 paise = ₹ 0.06

∴ ₹ 66 + ₹ 0.06 = ₹ 66.06

Hence, option 3 is the correct option.

Area of square = (side)² = 0.25 m²

∴ Side = $\sqrt{0.25}$ = 0.5 m

Perimeter = 4 × side = 4 × 0.5 = 2 m

Hence, option 2 is the correct option.

15 articles = 3 × 5 articles

∴ Cost of 15 articles = 3 × (cost of 5 articles) = ₹ 22.50 × 3

Hence, option 1 is the correct option.

We first change each fraction into a decimal fraction and then add them as like decimals.

$\frac{1}{100} = 0.01, \frac{1}{1000} = 0.001 \text { and } \frac{1}{10000}$ = 0.0001

∴ 0.0100 + 0.0010 + 0.0001 = 0.0111

Hence, option 3 is the correct option.

Since 1000 g = 1 kg, we convert grams into kilograms by dividing by 1000.

75 g = $\frac{75}{1000}$ kg = 0.075 kg

Hence, option 4 is the correct option.

On dividing 48.36 by 10, 100 and 1000, the decimal point shifts to the left by 1, 2 and 3 places respectively, giving 4.836, 0.4836 and 0.04836.

So, Assertion (A) is true.

Now, 48.36 ÷ 10 = $\frac{48.36}{10} = \frac{4836}{100 \times 10} = \frac{4836}{1000}$ = 4.836

and

48.36 ÷ 100 = $\frac{48.36}{100} = \frac{4836}{100 \times 100}$ = 0.4836.

So, Reason (R) is also correct.

Hence, option 3 is the correct option.

0.094 × 10 = 0.94, 0.094 × 100 = 9.40 and 0.094 × 1000 = 94, so Assertion (A) is true.

The Reason (R) correctly states the rule of shifting the decimal point to the right and is also true.

Hence, option 3 is the correct option.

0.4 × 0.04 × 100 = 0.4 × (0.04 × 100) = 0.4 × 4 = 1.6

So Statement 1 is true and Statement 2 correctly shows the working and is also true.

Hence, option 1 is the correct option.

The difference between two numbers being 17 does not necessarily mean m - n = 17, because we do not know which number is greater.

So Statement 1 is false.

Statement 2 correctly accounts for both cases (m - n = 17 when m > n and n - m = 17 when n > m), so it is true.

Hence, option 4 is the correct option.