Decimals

Write each of the following decimal numbers in words:

(i) 30.5

(ii) 0.03

(iii) 108.56

(iv) 47.20

(v) 5.008

(vi) 26.039

Answer

(i) 30.5 is read as thirty point five.

(ii) 0.03 is read as zero point zero three.

(iii) 108.56 is read as one hundred eight point five six.

(iv) 47.20 is read as forty seven point two zero.

(v) 5.008 is read as five point zero zero eight.

(vi) 26.039 is read as twenty six point zero three nine.

Write each of the following decimal numbers in the place value table:

(i) 4.2

(ii) 0.3

(iii) 205.9

(iv) 0.29

(v) 2.08

(vi) 7200.812

(vii) 38.007

Answer

The given decimal numbers are written in the place value table as under :

Write the following decimal numbers in the expanded form:

(i) 123.7

(ii) 43.06

(iii) 509.306

Answer

(i) We have,

$123.7 = 1 \times 100 + 2 \times 10 + 3 \times 1 + 7 \times \dfrac{1}{10} \\[1em] = 100 + 20 + 3 + \dfrac{7}{10}$

Hence, 123.7 = 100 + 20 + 3 + $\dfrac{7}{10}$

(ii) We have,

$43.06 = 4 \times 10 + 3 \times 1 + 0 \times \dfrac{1}{10} + 6 \times \dfrac{1}{100} \\[1em] = 40 + 3 + \dfrac{6}{100}$

Hence, 43.06 = 40 + 3 + $\dfrac{6}{100}$

(iii) We have,

$509.306 = 5 \times 100 + 0 \times 10 + 9 \times 1 + 3 \times \dfrac{1}{10} + 0 \times \dfrac{1}{100} + 6 \times \dfrac{1}{1000} \\[1em] = 500 + 9 + \dfrac{3}{10} + \dfrac{6}{1000}$

Hence, 509.306 = 500 + 9 + $\dfrac{3}{10} + \dfrac{6}{1000}$

Write each of the following as a decimal number:

(i) $200 + 60 + 5 + \dfrac{3}{10}$

(ii) $50 + \dfrac{1}{10} + \dfrac{6}{100}$

(iii) $70 + 6 + \dfrac{7}{10} + \dfrac{9}{1000}$

(iv) $600 + 7 + \dfrac{3}{100} + \dfrac{6}{1000}$

Answer

(i) We have,

$200 + 60 + 5 + \dfrac{3}{10} \\[1em] = 2 \times 100 + 6 \times 10 + 5 \times 1 + 3 \times \dfrac{1}{10} \\[1em] = 265.3$

Hence, 265.3 is the decimal number

(ii) We have,

$50 + \dfrac{1}{10} + \dfrac{6}{100} \\[1em] = 5 \times 10 + 0 \times 1 + 1 \times \dfrac{1}{10} + 6 \times \dfrac{1}{100} \\[1em] = 50.16$

Hence, 50.16 is the decimal number

(iii) We have,

$70 + 6 + \dfrac{7}{10} + \dfrac{9}{1000} \\[1em] = 7 \times 10 + 6 \times 1 + 7 \times \dfrac{1}{10} + 0 \times \dfrac{1}{100} + 9 \times \dfrac{1}{1000} \\[1em] = 76.709$

Hence, 76.709 is the decimal number

(iv) We have,

$600 + 7 + \dfrac{3}{100} + \dfrac{6}{1000} \\[1em] = 6 \times 100 + 0 \times 10 + 7 \times 1 + 0 \times \dfrac{1}{10} + 3 \times \dfrac{1}{100} + 6 \times \dfrac{1}{1000} \\[1em] = 607.036$

Hence, 607.036 is the decimal number

Write each of the following as decimals:

(i) Two ones and five tenths

(ii) Two tens and nine tenths

(iii) Six hundred point eight

(iv) Two hundred five and five hundredths

(v) Seven and fifteen thousandths

Answer

(i) Two ones and five tenths

$= 2 + \dfrac{5}{10} = 2 + 0.5 = 2.5$

Hence, the decimal form is 2.5.

(ii) Two tens and nine tenths

$= 20 + \dfrac{9}{10} = 20 + 0.9 = 20.9$

Hence, the decimal form is 20.9.

(iii) Six hundred point eight

$= 600 + \dfrac{8}{10} = 600 + 0.8 = 600.8$

Hence, the decimal form is 600.8.

(iv) Two hundred five and five hundredths

$= 205 + \dfrac{5}{100} = 205 + 0.05 = 205.05$

Hence, the decimal form is 205.05.

(v) Seven and fifteen thousandths

$= 7 + \dfrac{15}{1000} = 7 + 0.015 = 7.015$

Hence, the decimal form is 7.015.

Write the number given in the following place value table in decimal form:

Answer

Reading each row of the place value table and writing the digits in the correct decimal places, we get:

(i) 7102.306

(ii) 211.902

(iii) 3053.015

(iv) 70.03

(v) 5.40

(vi) 719.028

Show the following decimal numbers on the number line:

(i) 0.4

(ii) 1.9

(iii) 1.1

(iv) 2.5

Answer

The unit lengths between successive whole numbers on the number line are divided into 10 equal parts. Where each part represents 0.1

In the given figure,

A = 0.4, B = 1.1, C = 1.9 and D = 2.5

Write the decimal numbers represented by the points A, B, C and D on the given number line:

Answer

The unit lengths between successive whole numbers on the number line are divided into 10 equal parts. Reading the position of each point, we get:

A = 0.8,   B = 1.3,   C = 2.2,   D = 2.9

Between which two numbers in tenths place on the number line does each of the given number lie?

(i) 0.06

(ii) 0.45

(iii) 0.66

(iv) 0.92

Answer

(i) The number 0.06 lies between 0.0 and 0.1 on the number line.

(ii) The number 0.45 lies between 0.4 and 0.5 on the number line.

(iii) The number 0.66 lies between 0.6 and 0.7 on the number line.

(iv) The number 0.92 lies between 0.9 and 1.0 on the number line.

Write the following decimal fractions as decimal numbers:

(i) $\dfrac{531}{10}$

(ii) $\dfrac{422}{100}$

(iii) $\dfrac{58301}{1000}$

(iv) $\dfrac{7}{10}$

(v) $\dfrac{3}{100}$

(vi) $\dfrac{37}{1000}$

Answer

(i) $\dfrac{531}{10} = 53.1$

Hence, the decimal form is 53.1.

(ii) $\dfrac{422}{100} = 4.22$

Hence, the decimal form is 4.22.

(iii) $\dfrac{58301}{1000} = 58.301$

Hence, the decimal form is 58.301.

(iv) $\dfrac{7}{10} = 0.7$

Hence, the decimal form is 0.7.

(v) $\dfrac{3}{100} = 0.03$

Hence, the decimal form is 0.03.

(vi) $\dfrac{37}{1000} = 0.037$

Hence, the decimal form is 0.037.

Write the following decimal numbers as decimal fractions:

(i) 54.01

(ii) 318.105

(iii) 0.37

(iv) 0.047

(v) 0.03

(vi) 34.5

Answer

(i) 54.01 has 2 decimal places, so the denominator is 100.

$54.01 = \dfrac{5401}{100}$

Hence, the decimal fraction is $\dfrac{5401}{100}$.

(ii) 318.105 has 3 decimal places, so the denominator is 1000.

$318.105 = \dfrac{318105}{1000}$

Hence, the decimal fraction is $\dfrac{318105}{1000}$.

(iii) 0.37 has 2 decimal places, so the denominator is 100.

$0.37 = \dfrac{37}{100}$

Hence, the decimal fraction is $\dfrac{37}{100}$.

(iv) 0.047 has 3 decimal places, so the denominator is 1000.

$0.047 = \dfrac{47}{1000}$

Hence, the decimal fraction is $\dfrac{47}{1000}$.

(v) 0.03 has 2 decimal places, so the denominator is 100.

$0.03 = \dfrac{3}{100}$

Hence, the decimal fraction is $\dfrac{3}{100}$.

(vi) 34.5 has 1 decimal place, so the denominator is 10.

$34.5 = \dfrac{345}{10}$

Hence, the decimal fraction is $\dfrac{345}{10}$.

Write the following decimal numbers as fractions in lowest terms:

(i) 0.8

(ii) 0.04

(iii) 0.125

(iv) 0.225

(v) 0.066

(vi) 0.092

Answer

(i) Writing the decimal 0.8 as a fraction,

$0.8 = \dfrac{8}{10} = \dfrac{4}{5}$

Therefore, the required fraction is $\dfrac{4}{5}$.

(ii) Writing the decimal 0.04 as a fraction,

$0.04 = \dfrac{4}{100} = \dfrac{1}{25}$

Therefore, the required fraction is $\dfrac{1}{25}$.

(iii) Writing the decimal 0.125 as a fraction,

$0.125 = \dfrac{125}{1000} = \dfrac{1}{8}$

Therefore, the required fraction is $\dfrac{1}{8}$.

(iv) Writing the decimal 0.225 as a fraction,

$0.225 = \dfrac{225}{1000} = \dfrac{9}{40}$

Therefore, the required fraction is $\dfrac{9}{40}$.

(v) Writing the decimal 0.066 as a fraction,

$0.066 = \dfrac{66}{1000} = \dfrac{33}{500}$

Therefore, the required fraction is $\dfrac{33}{500}$.

(vi) Writing the decimal 0.092 as a fraction,

$0.092 = \dfrac{92}{1000} = \dfrac{23}{250}$

Therefore, the required fraction is $\dfrac{23}{250}$.

Convert the following decimal numbers into mixed fractions:

(i) 31.6

(ii) 3.25

(iii) 7.025

(iv) 95.95

Answer

(i) Writing the decimal 31.6 as a fraction,

$31.6 = \dfrac{316}{10} = \dfrac{158}{5} = 31\dfrac{3}{5}$

Therefore, the required mixed fraction is $31\dfrac{3}{5}$.

(ii) Writing the decimal 3.25 as a fraction,

$3.25 = \dfrac{325}{100} = \dfrac{13}{4} = 3\dfrac{1}{4}$

Therefore, the required mixed fraction is $3\dfrac{1}{4}$.

(iii) Writing the decimal 7.025 as a fraction,

$7.025 = \dfrac{7025}{1000} = \dfrac{281}{40} = 7\dfrac{1}{40}$

Therefore, the required mixed fraction is $7\dfrac{1}{40}$.

(iv) Writing the decimal 95.95 as a fraction,

$95.95 = \dfrac{9595}{100} = \dfrac{1919}{20} = 95\dfrac{19}{20}$

Therefore, the required mixed fraction is $95\dfrac{19}{20}$.

Convert the following fractions into decimal numbers:

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

(ii) $\dfrac{6}{25}$

(iii) $\dfrac{112}{125}$

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

(v) $\dfrac{3}{8}$

(vi) $7\dfrac{3}{40}$

Answer

(i) Converting the fraction $\dfrac{4}{5}$ into a decimal,

$\dfrac{4}{5} = \dfrac{4 \times 2}{5 \times 2} = \dfrac{8}{10} = 0.8$

Therefore, the decimal form is 0.8.

(ii) Converting the fraction $\dfrac{6}{25}$ into a decimal,

$\dfrac{6}{25} = \dfrac{6 \times 4}{25 \times 4} = \dfrac{24}{100} = 0.24$

Therefore, the decimal form is 0.24.

(iii) Converting the fraction $\dfrac{112}{125}$ into a decimal,

$\dfrac{112}{125} = \dfrac{112 \times 8}{125 \times 8} = \dfrac{896}{1000} = 0.896$

Therefore, the decimal form is 0.896.

(iv) Converting the fraction $\dfrac{3}{4}$ into a decimal,

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

Therefore, the decimal form is 0.75.

(v) Converting the fraction $\dfrac{3}{8}$ into a decimal,

$\dfrac{3}{8} = \dfrac{3 \times 125}{8 \times 125} = \dfrac{375}{1000} = 0.375$

Therefore, the decimal form is 0.375.

(vi) Converting the fraction $7\dfrac{3}{40}$ into a decimal,

$7\dfrac{3}{40} = 7 + \dfrac{3}{40} = 7 + \dfrac{3 \times 25}{40 \times 25} = 7 + \dfrac{75}{1000} = 7 + 0.075 = 7.075$

Therefore, the decimal form is 7.075.

Convert the following unlike decimal numbers to like decimal numbers:

(i) 17.5, 3.912

(ii) 5.04, 13.1902

(iii) 2.451, 3.7, 28.34

(iv) 3.1, 2.678, 27.0103

Answer

(i) The maximum number of decimal places in the given numbers is 3. So, we convert each of the decimals into one having 3 decimal places by annexing zeros.

17.5 = 17.500

3.912 = 3.912

As 17.500 and 3.912 have equal number of decimal places, they are like decimals.

(ii) The maximum number of decimal places in the given numbers is 4. So, we convert each of the decimals into one having 4 decimal places by annexing zeros.

5.04 = 5.0400

13.1902 = 13.1902

As 5.0400 and 13.1902 have equal number of decimal places, they are like decimals.

(iii) The maximum number of decimal places in the given numbers is 3. So, we convert each of the decimals into one having 3 decimal places by annexing zeros.

2.451 = 2.451

3.7 = 3.700

28.34 = 28.340

As 2.451, 3.700 and 28.340 have equal number of decimal places, they are like decimals.

(iv) The maximum number of decimal places in the given numbers is 4. So, we convert each of the decimals into one having 4 decimal places by annexing zeros.

3.1 = 3.1000

2.678 = 2.6780

27.0103 = 27.0103

As 3.1000, 2.6780 and 27.0103 have equal number of decimal places, they are like decimals.

In each of the following pairs of decimal numbers, state which number is greater?

(i) 0.3, 0.4

(ii) 1, 0.99

(iii) 1.09, 1.093

(iv) 0.5, 0.05

Answer

(i) The given decimals are 0.3 and 0.4. Their whole number parts are equal. Comparing their tenths digits, 3 < 4.

Therefore, 0.4 is greater.

(ii) The given decimals are 1 and 0.99. Comparing their whole number parts, 1 > 0.

Therefore, 1 is greater.

(iii) The given decimals are 1.09 and 1.093.

Writing them as like decimals: 1.090 and 1.093.

Comparing the decimal parts, 090 < 093.

Therefore, 1.093 is greater.

(iv) The given decimals are 0.5 and 0.05.

Writing them as like decimals: 0.50 and 0.05.

Comparing the decimal parts, 50 > 05.

Therefore, 0.5 is greater.

In each of the following pairs of decimal numbers, state which number is smaller:

(i) 45.78, 345.8

(ii) 37.701, 37.71

(iii) 5.907, 5.903

Answer

(i) The given decimals are 45.78 and 345.8. Comparing their whole number parts, 45 < 345.

Therefore, 45.78 is smaller.

(ii) The given decimals are 37.701 and 37.71.

Writing them as like decimals: 37.701 and 37.710.

Comparing the decimal parts, 701 < 710.

Therefore, 37.701 is smaller.

(iii) The given decimals are 5.907 and 5.903. Their whole number parts are equal. Comparing the decimal parts, 907 > 903.

Therefore, 5.903 is smaller.

Arrange the following decimal numbers in ascending order:

(i) 27.35, 27.305, 2.7, 2.543

(ii) 4.53, 4.07, 29.1, 0.9, 0.709

Answer

(i) The maximum number of decimal places in the given decimals is 3. So, we convert each of the given decimals into like decimals having three decimal places by annexing zeros.

The decimals are: 27.350, 27.305, 2.700, 2.543

Clearly, 2.543 < 2.700 < 27.305 < 27.350.

Hence, the given decimals in ascending order are:

2.543 < 2.7 < 27.305 < 27.35

(ii) The maximum number of decimal places in the given decimals is 3. So, we convert each of the given decimals into like decimals having three decimal places by annexing zeros.

The decimals are: 4.530, 4.070, 29.100, 0.900, 0.709

Clearly, 0.709 < 0.900 < 4.070 < 4.530 < 29.100.

Hence, the given decimals in ascending order are:

0.709 < 0.9 < 4.07 < 4.53 < 29.1

Arrange the following decimal numbers in descending order:

(i) 3.303, 33.03, 3.3, 30.33

(ii) 72.5, 2.75, 27.505, 0.275, 2.507

Answer

(i) The maximum number of decimal places in the given decimals is 3. So, we convert each of the given decimals into like decimals having three decimal places by annexing zeros.

The decimals are: 3.303, 33.030, 3.300, 30.330

Clearly, 33.030 > 30.330 > 3.303 > 3.300.

Hence, the given decimals in descending order are:

33.03 > 30.33 > 3.303 > 3.3

(ii) The maximum number of decimal places in the given decimals is 3. So, we convert each of the given decimals into like decimals having three decimal places by annexing zeros.

The decimals are: 72.500, 2.750, 27.505, 0.275, 2.507

Clearly, 72.500 > 27.505 > 2.750 > 2.507 > 0.275.

Hence, the given decimals in descending order are:

72.5 > 27.505 > 2.75 > 2.507 > 0.275

Add:

(i) 17.5, 8.8

(ii) 9.999, 0.03

(iii) 5.87, 1.03, 0.1

(iv) 23.71, 9.9, 4.023

(v) 4.5, 16.024, 7.99

(vi) 8.79, 23.001, 5.41, 0.875

Answer

(i) Writing the decimals in column form and adding, we get:

$\begin{array}{r} \phantom{+}17.4500 \\ -\space 7.9702 \\ \hline 9.4798 \end{array}$

Therefore, the sum is 26.3.

(ii) Converting the given decimals into like decimals, each having 3 places of decimal, we get:

9.999, 0.030

Writing these decimals in column form and adding, we get:

$\begin{array}{r} \phantom{+}9.999 \\ +\space 0.030 \\ \hline 10.029 \end{array}$

Therefore, the sum is 10.029.

(iii) Converting the given decimals into like decimals, each having 2 places of decimal, we get:

5.87, 1.03, 0.10

Writing these decimals in column form and adding, we get:

$\begin{array}{r} \phantom{+}5.87 \\ 1.03 \\ +\space 0.10 \\ \hline 7.00 \end{array}$

Therefore, the sum is 7.

(iv) Converting the given decimals into like decimals, each having 3 places of decimal, we get:

23.710, 9.900, 4.023

Writing these decimals in column form and adding, we get:

$\begin{array}{r} \phantom{+}23.710 \\ 9.900 \\ +\space 4.023 \\ \hline 37.633 \end{array}$

Therefore, the sum is 37.633.

(v) Converting the given decimals into like decimals, each having 3 places of decimal, we get:

4.500, 16.024, 7.990

Writing these decimals in column form and adding, we get:

$\begin{array}{r} \phantom{+}4.500 \\ 16.024 \\ +\space 7.990 \\ \hline 28.514 \end{array}$

Therefore, the sum is 28.514.

(vi) Converting the given decimals into like decimals, each having 3 places of decimal, we get:

8.790, 23.001, 5.410, 0.875

Writing these decimals in column form and adding, we get:

$\begin{array}{r} \phantom{+}8.790 \\ 23.001 \\ 5.410 \\ +\space 0.875 \\ \hline 38.076 \end{array}$

Therefore, the sum is 38.076.

Calculate:

(i) 5.82 - 2.65

(ii) 19.01 - 12.234

(iii) 15.4 + 3.015 - 14.237

(iv) 7.4 - 2.19 - 0.456 - 3.5

(v) 19.27 - 3.6 - 8.812 + 0.84

(vi) 6.4 - 2.351 - 1.45 - 0.999

Answer

(i) Writing the decimals in column form and subtracting, we get:

$\begin{array}{r} \phantom{+}5.82 \\ -\space 2.65 \\ \hline 3.17 \end{array}$

Therefore, 5.82 - 2.65 = 3.17.

(ii) Converting the given decimals into like decimals, each having 3 places of decimal, we get 19.010 and 12.234.

Writing in column form and subtracting, we get:

$\begin{array}{r} \phantom{+}19.010 \\ -\space 12.234 \\ \hline 6.776 \end{array}$

Therefore, 19.01 - 12.234 = 6.776.

(iii) Writing each of the given numbers with three decimal places, we get:

Given expression = 15.400 + 3.015 - 14.237

Now,

$\begin{array}{r} \phantom{+}15.400 \\ +\space 3.015 \\ \hline 18.415 \end{array} \qquad \text{and} \qquad \begin{array}{r} \phantom{+}18.415 \\ -\space 14.237 \\ \hline 4.178 \end{array}$

Therefore, 15.4 + 3.015 - 14.237 = 4.178.

(iv) Writing each of the given numbers with three decimal places, we get:

Given expression = 7.400 - 2.190 - 0.456 - 3.500

$\phantom{Given expression}$ = 7.400 - (2.190 + 0.456 + 3.500)

Now,

$\begin{array}{r} \phantom{+}2.190 \\ 0.456 \\ +\space 3.500 \\ \hline 6.146 \end{array}$

So,

$\begin{array}{r} \phantom{+}7.400 \\ -\space 6.146 \\ \hline 1.254 \end{array}$

Therefore, 7.4 - 2.19 - 0.456 - 3.5 = 1.254.

(v) Writing each of the given numbers with three decimal places, we get:

Given expression = 19.270 - 3.600 - 8.812 + 0.840

$\phantom{Given expression}$ = (19.270 + 0.840) - (3.600 + 8.812)

Now,

$\begin{array}{r} \phantom{+}19.270 \\ +\space 0.840 \\ \hline 20.110 \end{array} \qquad \text{and} \qquad \begin{array}{r} \phantom{+}3.600 \\ +\space 8.812 \\ \hline 12.412 \end{array}$

So,

$\begin{array}{r} \phantom{+}20.110 \\ -\space 12.412 \\ \hline 7.698 \end{array}$

Therefore, 19.27 - 3.6 - 8.812 + 0.84 = 7.698.

(vi) Writing each of the given numbers with three decimal places, we get:

Given expression = 6.400 - 2.351 - 1.450 - 0.999 = 6.400 - (2.351 + 1.450 + 0.999)

Now,

$\begin{array}{r} \phantom{+}2.351 \\ 1.450 \\ +\space 0.999 \\ \hline 4.800 \end{array}$

So,

$\begin{array}{r} \phantom{+}6.400 \\ -\space 4.800 \\ \hline 1.600 \end{array}$

Therefore, 6.4 - 2.351 - 1.45 - 0.999 = 1.6.

What number added to 0.756 gives 1?

Answer

Let the required number be x. Then,

0.756 + x = 1

Solving for x,

x = 1 - 0.756

Rewriting the numbers with three decimal places,

x = 1.000 - 0.756

Writing in column form and subtracting, we get:

$\begin{array}{r} \phantom{+}1.000 \\ -\space 0.756 \\ \hline 0.244 \end{array}$

Therefore, the required number is 0.244.

By how much should 17.45 be decreased to get 7.9702?

Answer

Let the required decrease be x. Then,

17.45 - x = 7.9702

Solving for x,

x = 17.45 - 7.9702

Rewriting the numbers with four decimal places,

x = 17.4500 - 7.9702

Writing in column form and subtracting, we get:

$\begin{array}{r} \phantom{+}17.4500 \\ -\space 7.9702 \\ \hline 9.4798 \end{array}$

Therefore, 17.45 should be decreased by 9.4798.

Evaluate the following:

(i) 3.7 × 4.5

(ii) 12.08 × 9.3

(iii) 238.06 × 7.5

(iv) 0.79 × 32.4

(v) 3.6 × 1.4 × 0.7

(vi) 9.01 × 2.5 × 1.6

Answer

(i) Ignoring the decimal points, we multiply 37 and 45.

37 × 45 = 1665

The two given numbers together have 1 + 1 = 2 decimal places.

So, placing the decimal point such that the product has 2 decimal places,

3.7 × 4.5 = 16.65

Therefore, 3.7 × 4.5 = 16.65.

(ii) Ignoring the decimal points, we multiply 1208 and 93.

1208 × 93 = 112344

The two given numbers together have 2 + 1 = 3 decimal places.

So, placing the decimal point such that the product has 3 decimal places,

12.08 × 9.3 = 112.344

Therefore, 12.08 × 9.3 = 112.344.

(iii) Ignoring the decimal points, we multiply 23806 and 75.

23806 × 75 = 1785450

The two given numbers together have 2 + 1 = 3 decimal places.

So, placing the decimal point such that the product has 3 decimal places,

238.06 × 7.5 = 1785.450 = 1785.45

Therefore, 238.06 × 7.5 = 1785.45.

(iv) Ignoring the decimal points, we multiply 79 and 324.

79 × 324 = 25596

The two given numbers together have 2 + 1 = 3 decimal places.

So, placing the decimal point such that the product has 3 decimal places,

0.79 × 32.4 = 25.596

Therefore, 0.79 × 32.4 = 25.596.

(v) Ignoring the decimal points, we multiply 36, 14 and 7.

36 × 14 × 7 = 504 × 7 = 3528

The three given numbers together have 1 + 1 + 1 = 3 decimal places.

So, placing the decimal point such that the product has 3 decimal places,

3.6 × 1.4 × 0.7 = 3.528

Therefore, 3.6 × 1.4 × 0.7 = 3.528.

(vi) Ignoring the decimal points, we multiply 901, 25 and 16.

901 × 25 × 16 = 22525 × 16 = 360400

The three given numbers together have 2 + 1 + 1 = 4 decimal places.

So, placing the decimal point such that the product has 4 decimal places,

9.01 × 2.5 × 1.6 = 36.0400 = 36.04

Therefore, 9.01 × 2.5 × 1.6 = 36.04.

Calculate the following:

(i) 70.756 ÷ 4

(ii) 2.46 ÷ 6

(iii) 3.016 ÷ 8

(iv) 8.64 ÷ 3.6

(v) 72.8 ÷ 0.04

(vi) 0.144 ÷ 0.02

Answer

(i) By actual division, we have:

$\begin{array}{l} \phantom{4}{\space \phantom{} 17.689} \\ 4\overline{\smash{\big)}\space 70.756} \\ \phantom{4}\phantom{}\underline{-4} \\ \phantom{4)\space)}30 \\ \phantom{4\space}\underline{-28} \\ \phantom{4)\space 28}27 \\ \phantom{4)\space)}\underline{-24} \\ \phantom{4)\space 28))}35 \\ \phantom{4)\space 28}\underline{-32} \\ \phantom{4)\space 28))))}36 \\ \phantom{4)\space 28))}\underline{-36} \\ \phantom{4)\space 28))))}\times \end{array}$

Therefore, 70.756 ÷ 4 = 17.689.

(ii) By actual division, we have:

$\begin{array}{l} \phantom{6)\space }0.41 \\ 6\overline{\smash{\big)}\space 2.46} \\ \phantom{(}\underline{-0\phantom{.}} \\ \phantom{6(\space) )}24 \\ \phantom{\space)))}\underline{-24} \\ \phantom{6())\space 2.}6 \\ \phantom{6\space 2.)}\underline{-6} \\ \phantom{6)\space 2.4}\times \end{array}$

Therefore, 2.46 ÷ 6 = 0.41.

(iii) By actual division, we have:

$\begin{array}{l} \phantom{8)\space }0.377 \\ 8\overline{\smash{\big)}\space 3.016} \\ \phantom{8}\underline{-0\phantom{.}} \\ \phantom{8)\space }30 \\ \phantom{\space }\underline{-24} \\ \phantom{8)\space 1}61 \\ \phantom{\space 1}\underline{-56} \\ \phantom{8)\space 1\phantom{1}}56 \\ \phantom{\space ()\phantom{1}}\underline{-56} \\ \phantom{8)\space 3.1}\times \end{array}$

Therefore, 3.016 ÷ 8 = 0.377.

(iv) To divide 8.64 by 3.6, multiply both the dividend and divisor by 10 so that the divisor becomes a whole number:

$8.64 \div 3.6 = \dfrac{8.64 \times 10}{3.6 \times 10} = \dfrac{86.4}{36}$

By actual division, we have:

$\begin{array}{l} \phantom{36)\space }2.4 \\ 36\overline{\smash{\big)}\space 86.4} \\ \phantom{36}\underline{-72} \\ \phantom{36(\space)}144 \\ \phantom{+\space)}\underline{-144} \\ \phantom{36\space 86.}\times \end{array}$

Therefore, 8.64 ÷ 3.6 = 2.4.

(v) To divide 72.8 by 0.04, multiply both the dividend and divisor by 100 so that the divisor becomes a whole number:

$72.8 \div 0.04 = \dfrac{72.8 \times 100}{0.04 \times 100} = \dfrac{7280}{4} = 1820$

Therefore, 72.8 ÷ 0.04 = 1820.

(vi) To divide 0.144 by 0.02, multiply both the dividend and divisor by 100 so that the divisor becomes a whole number:

$0.144 \div 0.02 = \dfrac{0.144 \times 100}{0.02 \times 100} = \dfrac{14.4}{2} = 7.2$

Therefore, 0.144 ÷ 0.02 = 7.2.

Multiply each of the following numbers by 10, 100 and 1000 (orally):

(i) 4.7

(ii) 3.45

(iii) 0.234

Answer

When we multiply a decimal number by 10, 100 or 1000, the decimal point shifts one, two or three places to the right respectively.

(i) For 4.7:

4.7 × 10 = 47

4.7 × 100 = 470

4.7 × 1000 = 4700

(ii) For 3.45:

3.45 × 10 = 34.5

3.45 × 100 = 345

3.45 × 1000 = 3450

(iii) For 0.234:

0.234 × 10 = 2.34

0.234 × 100 = 23.4

0.234 × 1000 = 234

Divide each of the following numbers by 10, 100 and 1000 (orally):

(i) 4.7

(ii) 3.45

(iii) 23.01

Answer

When we divide a decimal number by 10, 100 or 1000, the decimal point shifts one, two or three places to the left respectively.

(i) For 4.7:

4.7 ÷ 10 = 0.47

4.7 ÷ 100 = 0.047

4.7 ÷ 1000 = 0.0047

(ii) For 3.45:

3.45 ÷ 10 = 0.345

3.45 ÷ 100 = 0.0345

3.45 ÷ 1000 = 0.00345

(iii) For 23.01:

23.01 ÷ 10 = 2.301

23.01 ÷ 100 = 0.2301

23.01 ÷ 1000 = 0.02301

Find the value of the following:

(i) (3.5)²

(ii) (0.4)³

Answer

(i) We have,

(3.5)² = 3.5 × 3.5

Ignoring the decimal points, 35 × 35 = 1225.

The two factors together have 1 + 1 = 2 decimal places, so:

(3.5)² = 12.25

Therefore, (3.5)² = 12.25.

(ii) We have,

(0.4)³ = 0.4 × 0.4 × 0.4

Ignoring the decimal points, 4 × 4 × 4 = 64.

The three factors together have 1 + 1 + 1 = 3 decimal places, so:

(0.4)³ = 0.064

Therefore, (0.4)³ = 0.064.

Express as rupees using decimals:

(i) 75 paise

(ii) 1025 paise

(iii) 63 rupees 9 paise

Answer

We know that 1 paisa = $\dfrac{1}{100}$ rupee = ₹0.01.

(i) 75 paise = ₹$\dfrac{75}{100}$ = ₹0.75.

(ii) 1025 paise = ₹$\dfrac{1025}{100}$ = ₹10.25.

(iii) 63 rupees 9 paise = ₹63 + ₹$\dfrac{9}{100}$ = ₹63 + ₹0.09 = ₹63.09.

Express as cm using decimals:

(i) 8 mm

(ii) 263 mm

(iii) 13 cm 3 mm

Answer

We know that 1 mm = $\dfrac{1}{10}$ cm = 0.1 cm.

(i) 8 mm = $\dfrac{8}{10}$ cm = 0.8 cm.

(ii) 263 mm = $\dfrac{263}{10}$ cm = 26.3 cm.

(iii) 13 cm 3 mm = 13 cm + $\dfrac{3}{10}$ cm = 13 cm + 0.3 cm = 13.3 cm.

Express as metres using decimals:

(i) 6 cm

(ii) 528 cm

(iii) 7 m 55 cm

Answer

We know that 1 cm = $\dfrac{1}{100}$ m = 0.01 m.

(i) 6 cm = $\dfrac{6}{100}$ m = 0.06 m.

(ii) 528 cm = $\dfrac{528}{100}$ m = 5.28 m.

(iii) 7 m 55 cm = 7 m + $\dfrac{55}{100}$ m = 7 m + 0.55 m = 7.55 m.

Express as km using decimals:

(i) 5 m

(ii) 888 m

(iii) 15 km 88 m

Answer

We know that 1 m = $\dfrac{1}{1000}$ km = 0.001 km.

(i) 5 m = $\dfrac{5}{1000}$ km = 0.005 km.

(ii) 888 m = $\dfrac{888}{1000}$ km = 0.888 km.

(iii) 15 km 88 m = 15 km + $\dfrac{88}{1000}$ km = 15 km + 0.088 km = 15.088 km.

Express as kg using decimals:

(i) 37 g

(ii) 100 g

(iii) 5 kg 8 g

Answer

We know that 1 g = $\dfrac{1}{1000}$ kg = 0.001 kg.

(i) 37 g = $\dfrac{37}{1000}$ kg = 0.037 kg.

(ii) 100 g = $\dfrac{100}{1000}$ kg = 0.1 kg.

(iii) 5 kg 8 g = 5 kg + $\dfrac{8}{1000}$ kg = 5 kg + 0.008 kg = 5.008 kg.

Express as kL using decimals:

(i) 6 L

(ii) 555 L

(iii) 3 kL 95 L

Answer

We know that 1 L = $\dfrac{1}{1000}$ kL = 0.001 kL.

(i) 6 L = $\dfrac{6}{1000}$ kL = 0.006 kL.

(ii) 555 L = $\dfrac{555}{1000}$ kL = 0.555 kL.

(iii) 3 kL 95 L = 3 kL + $\dfrac{95}{1000}$ kL = 3 kL + 0.095 kL = 3.095 kL.

Anita bought 2 m 70 cm cloth for her shirt and 2 m 85 cm cloth for her trouser. Find the total length of the cloth bought by her.

Answer

Length of cloth for shirt = 2 m 70 cm

= 2 m + $\dfrac{70}{100}$ m = 2 m + 0.70 m = 2.70 m

Length of cloth for trouser = 2 m 85 cm

= 2 m + $\dfrac{85}{100}$ m = 2 m + 0.85 m = 2.85 m

Total length of the cloth = 2.70 + 2.85

Writing in column form and adding:

$\begin{array}{r} \phantom{+}2.70 \\ +\space 2.85 \\ \hline 5.55 \end{array}$

Hence, the total length of the cloth bought by Anita is 5.55 m or 5 m 55 cm.

Sunita travelled 15 km 268 m by bus, 7 km 7 m by car and 500 m on foot in order to reach her school. How far is her school from her residence?

Answer

Distance travelled by bus = 15 km 268 m

= 15 km + $\dfrac{268}{1000}$ km = 15 km + 0.268 km = 15.268 km

Distance travelled by car = 7 km 7 m

= 7 km + $\dfrac{7}{1000}$ km = 7 km + 0.007 km = 7.007 km

Distance travelled on foot = 500 m

= $\dfrac{500}{1000}$ km = 0.500 km

Total distance = 15.268 + 7.007 + 0.500

Writing in column form and adding:

$\begin{array}{r} \phantom{+}15.268 \\ 7.007 \\ +\space 0.500 \\ \hline 22.775 \end{array}$

Hence, Sunita's school is 22.775 km or 22 km 775 m from her residence.

Rahul bought 4 kg 90 g apples, 2 kg 60 g grapes and 5 kg 300 g mangoes. Find the total weight of all the fruits he bought.

Answer

Weight of apples = 4 kg 90 g

= 4 kg + $\dfrac{90}{1000}$ kg = 4 kg + 0.090 kg = 4.090 kg

Weight of grapes = 2 kg 60 g

= 2 kg + $\dfrac{60}{1000}$ kg = 2 kg + 0.060 kg = 2.060 kg

Weight of mangoes = 5 kg 300 g

= 5 kg + $\dfrac{300}{1000}$ kg = 5 kg + 0.300 kg = 5.300 kg

Total weight = 4.090 + 2.060 + 5.300

Writing in column form and adding:

$\begin{array}{r} \phantom{+}4.090 \\ 2.060 \\ +\space 5.300 \\ \hline 11.450 \end{array}$

Hence, the total weight of the fruits bought is 11.450 kg or 11 kg 450 g.

Rani has ₹18.50. She bought one ice cream for ₹11.75. How much money does she have now?

Answer

Money Rani had = ₹18.50

Cost of ice cream = ₹11.75

Money left with her = 18.50 - 11.75

Writing in column form and subtracting:

$\begin{array}{r} \phantom{+}18.50 \\ -\space 11.75 \\ \hline 6.75 \end{array}$

Hence, Rani now has ₹6.75.

Tina had 20 m 5 cm long cloth. She cuts 4 m 50 cm length of cloth from this for making a curtain. How much length of cloth is left with her?

Answer

Total length of cloth = 20 m 5 cm

= 20 m + $\dfrac{5}{100}$ m = 20 m + 0.05 m = 20.05 m

Length of cloth cut for curtain = 4 m 50 cm

= 4 m + $\dfrac{50}{100}$ m = 4 m + 0.50 m = 4.50 m

Length of cloth left = 20.05 - 4.50

Writing in column form and subtracting:

$\begin{array}{r} \phantom{+}20.05 \\ -\space 4.50 \\ \hline 15.55 \end{array}$

Hence, the length of cloth left with Tina is 15.55 m or 15 m 55 cm.

Ruby bought a watermelon weighing 5 kg 300 g. Out of which she gave 2 kg 680 g to her neighbour. What is the weight of the watermelon left with Ruby?

Answer

Weight of watermelon = 5 kg 300 g

= 5 kg + $\dfrac{300}{1000}$ kg = 5 kg + 0.300 kg = 5.300 kg

Weight given to neighbour = 2 kg 680 g

= 2 kg + $\dfrac{680}{1000}$ kg = 2 kg + 0.680 kg = 2.680 kg

Weight left with Ruby = 5.300 - 2.680

Writing in column form and subtracting:

$\begin{array}{r} \phantom{+}5.300 \\ -\space 2.680 \\ \hline 2.620 \end{array}$

Hence, the weight of the watermelon left with Ruby is 2.620 kg or 2 kg 620 g.

The cost of 1 metre of cloth is ₹35.80. What will be cost of 9.8 metres of cloth?

Answer

Cost of 1 metre of cloth = ₹35.80

Cost of 9.8 metres of cloth = ₹35.80 × 9.8

Ignoring the decimal points, 3580 × 98 = 350840.

The two factors together have 2 + 1 = 3 decimal places, so:

35.80 × 9.8 = 350.840 = 350.84

Hence, the cost of 9.8 metres of cloth is ₹350.84.

Farida bought some bags of cement, each weighing 49.8 kg. If the total weight of all the bags is 1792.8 kg, how many bags did she buy?

Answer

Weight of each bag = 49.8 kg

Total weight of all the bags = 1792.8 kg

Number of bags = $\dfrac{1792.8}{49.8}$

Multiplying both the numerator and denominator by 10, we get:

Number of bags = $\dfrac{1792.8 \times 10}{49.8 \times 10} = \dfrac{17928}{498}$

By actual division, we have:

$\begin{array}{l} \phantom{498)\space 179}36 \\ 498\overline{\smash{\big)}\space 17928} \\ \phantom{498}\underline{-1494} \\ \phantom{498)\space )1}2988 \\ \phantom{498)\space)}\underline{-2988} \\ \phantom{498)\space 1792}\times \end{array}$

Hence, Farida bought 36 bags of cement.

Fill in the following blanks:

(i) The decimal point in a decimal number is placed between ones digit and .... digit.

(ii) The place value of the digit 3 in the decimal number 15.437 is .....

(iii) The decimal number 27.025 has ..... decimal places.

(iv) The decimal number 5.06 is read as five point .....

(v) If an object is divided into 1000 equal parts, then its 27 parts are represented by .....

(vi) Two decimal numbers having different number of ..... are called unlike decimal numbers.

(vii) 4 tens, 3 ones, 2 tenths, 0 hundredths and 5 thousandths in decimal form is written as .....

(viii) The smallest decimal number upto three decimal places is ......

(ix) The largest four digit decimal number less than 1 using the digits 1, 5, 3 and 8 once is ...

Answer

(i) The decimal point in a decimal number is placed between ones digit and tenths digit.

(ii) The place value of the digit 3 in the decimal number 15.437 is $\dfrac{3}{100}$.

(iii) The decimal number 27.025 has 3 decimal places.

(iv) The decimal number 5.06 is read as five point zero six.

(v) If an object is divided into 1000 equal parts, then its 27 parts are represented by 0.027.

(vi) Two decimal numbers having different number of decimal places are called unlike decimal numbers.

(vii) 4 tens, 3 ones, 2 tenths, 0 hundredths and 5 thousandths in decimal form is written as 43.205.

(viii) The smallest decimal number upto three decimal places is 0.001.

(ix) The largest four digit decimal number less than 1 using the digits 1, 5, 3 and 8 once is 0.8531.

State whether the following statements are true (T) or false (F):

(i) Every decimal number can be represented by a point on a number line.

(ii) Fractions with denominator 10, 100, 1000, ..... are called decimal fractions.

(iii) A decimal number having 3 decimal places can be written as a fraction with denominator 1000.

(iv) The value of a decimal number remains the same if any number of extra zeros are written at the end of a decimal number.

(v) If a decimal number is multiplied by 10, then the decimal point moves by one place to the left.

Answer

(i) True. Every decimal number corresponds to a unique point on a number line.

(ii) True. By definition, fractions with denominator 10, 100, 1000, ... are called decimal fractions.

(iii) True. For a decimal number with 3 decimal places, the denominator of its decimal fraction is 1000.

(iv) True. For example, 3.7 = 3.70 = 3.700.

(v) False. When a decimal number is multiplied by 10, the decimal point moves by one place to the right, not to the left.

Five and seven hundredths is equal to

1. 5.7

2. 5.07

3. 5.70

4. 0.57

Answer

Five and seven hundredths = $5 + \dfrac{7}{100} = 5 + 0.07 = 5.07$

Hence, option 2 is the correct option.

Sixty three thousandths is equal to

1. 0.63

2. 0.603

3. 0.063

4. 0.630

Answer

Sixty three thousandths = $\dfrac{63}{1000} = 0.063$

Hence, option 3 is the correct option.

$3\dfrac{7}{100}$ is equal to

1. 3.07

2. 3.7

3. 3.70

4. 3.007

Answer

$3\dfrac{7}{100} = 3 + \dfrac{7}{100} = 3 + 0.07 = 3.07$

Hence, option 1 is the correct option.

$5\dfrac{3}{1000}$ is equal to

1. 5.03

2. 5.3

3. 5.003

4. 5.0003

Answer

$5\dfrac{3}{1000} = 5 + \dfrac{3}{1000} = 5 + 0.003 = 5.003$

Hence, option 3 is the correct option.

The place value of the digit 7 in the decimal number 5.0378 is

1. 7

2. $\dfrac{7}{10}$

3. $\dfrac{7}{100}$

4. $\dfrac{7}{1000}$

Answer

In the number 5.0378, the digit 7 is at the thousandths place.

So, place value of 7 = $\dfrac{7}{1000}$.

Hence, option 4 is the correct option.

The place value of the digit 0 in the decimal number 13.405 is

1. 0

2. $\dfrac{1}{10}$

3. $\dfrac{1}{100}$

4. none of these

Answer

In the number 13.405, the digit 0 is at the hundredths place. The place value of 0 at any position is 0.

Hence, option 1 is the correct option.

The value of $5 + \dfrac{7}{10} + \dfrac{3}{1000}$ is

1. 5.73

2. 5.703

3. 5.073

4. 0.753

Answer

$5 + \dfrac{7}{10} + \dfrac{3}{1000} = 5 + 0.7 + 0.003 = 5.703$

Hence, option 2 is the correct option.

The value of $\dfrac{3}{25}$ is

1. 1.2

2. 0.012

3. 0.12

4. none of these

Answer

$\dfrac{3}{25} = \dfrac{3 \times 4}{25 \times 4} = \dfrac{12}{100} = 0.12$

Hence, option 3 is the correct option.

The value of $5\dfrac{1}{25}$ is

1. 5.4

2. 5.25

3. 5.04

4. 5.004

Answer

$5\dfrac{1}{25} = 5 + \dfrac{1}{25} = 5 + \dfrac{1 \times 4}{25 \times 4} = 5 + \dfrac{4}{100} = 5 + 0.04 = 5.04$

Hence, option 3 is the correct option.

The decimal number not equivalent to 5.7 is

1. 5.70

2. 5.07

3. 5.700

4. 5.7000

Answer

The value of a decimal number remains the same if extra zeros are written at the end. So, 5.7 = 5.70 = 5.700 = 5.7000. But 5.07 is different from 5.7.

Hence, option 2 is the correct option.

When 0.04 is written as a fraction in its simplest form, then the sum of numerator and denominator is

1. 7

2. 21

3. 26

4. 104

Answer

$0.04 = \dfrac{4}{100} = \dfrac{1}{25}$

Sum of numerator and denominator = 1 + 25 = 26.

Hence, option 3 is the correct option.

13.572 correct to the tenth place is

1. 10

2. 13.57

3. 14.5

4. 13.6

Answer

To round 13.572 to the tenth place, look at the digit in the hundredths place, which is 7. Since 7 ≥ 5, we round up the tenths digit.

So, 13.572 ≈ 13.6.

Hence, option 4 is the correct option.

1 g is equal to

1. 0.1 kg

2. 0.01 kg

3. 0.001 kg

4. 0.0001 kg

Answer

$1 \text{ g} = \dfrac{1}{1000} \text{ kg} = 0.001 \text{ kg}$

Hence, option 3 is the correct option.

2 km 7 m is equal to

1. 2.7 km

2. 2.07 km

3. 2.007 km

4. 2.0007 km

Answer

$2 \text{ km } 7 \text{ m} = 2 \text{ km} + \dfrac{7}{1000} \text{ km} = 2 \text{ km} + 0.007 \text{ km} = 2.007 \text{ km}$

Hence, option 3 is the correct option.

Among 2.34, 2.43, 2.344 and 2.4, the greatest number is

1. 2.34

2. 2.43

3. 2.344

4. 2.4

Answer

Converting the given decimals into like decimals (3 decimal places):

2.340, 2.430, 2.344, 2.400

Clearly, 2.430 is the greatest. So, the greatest number is 2.43.

Hence, option 2 is the correct option.

5.2 - 3.6 is equal to

1. 0.16

2. 2.6

3. 0.26

4. 1.6

Answer

$\begin{array}{r} \phantom{+}5.2 \\ -\space 3.6 \\ \hline 1.6 \end{array}$

Hence, option 4 is the correct option.

A decimal number lying between 2.2 and 2.22 is

1. 2.12

2. 2.23

3. 2.219

4. 2.3

Answer

Writing 2.2 and 2.22 as like decimals, we get 2.200 and 2.220.

Among the options, 2.219 lies between 2.200 and 2.220.

Hence, option 3 is the correct option.

0.023 lies between

1. 0.2 and 0.3

2. 0.02 and 0.03

3. 0.029 and 0.03

4. 0.026 and 0.024

Answer

Writing 0.023 with 3 decimal places, we see that 0.020 < 0.023 < 0.030. So, 0.023 lies between 0.02 and 0.03.

Hence, option 2 is the correct option.

0.7499 lies between

1. 0.7 and 0.74

2. 0.759 and 0.799

3. 0.749 and 0.75

4. 0.74992 and 0.75

Answer

Writing all as like decimals, 0.7490 < 0.7499 < 0.7500. So, 0.7499 lies between 0.749 and 0.75.

Hence, option 3 is the correct option.

Which of the following decimal numbers is the greatest?

1. 0.182

2. 0.038

3. 0.219

4. 0.291

Answer

Comparing all four decimal numbers, clearly 0.291 > 0.219 > 0.182 > 0.038. So, 0.291 is the greatest.

Hence, option 4 is the correct option.

Which of the following decimal numbers is the smallest?

1. 0.108

2. 1.08

3. 0.801

4. 0.81

Answer

Converting to like decimals (3 decimal places): 0.108, 1.080, 0.801, 0.810.

Clearly, 0.108 < 0.801 < 0.810 < 1.080. So, 0.108 is the smallest.

Hence, option 1 is the correct option.

0.003 × 0.2 is equal to

1. 0.6

2. 0.06

3. 0.006

4. 0.0006

Answer

Ignoring the decimal points, 3 × 2 = 6.

The two factors together have 3 + 1 = 4 decimal places.

So, 0.003 × 0.2 = 0.0006.

Hence, option 4 is the correct option.

0.45 ÷ 0.9 is equal to

1. 50

2. 5

3. 0.5

4. 0.05

Answer

$0.45 \div 0.9 = \dfrac{0.45 \times 10}{0.9 \times 10} = \dfrac{4.5}{9} = 0.5$

Hence, option 3 is the correct option.

Statement I: In the number 34.17, the whole number part is 34 and the decimal part is 0.17.

Statement II: $34.17 = 3 \times 10 + 4 \times 1 + 1 \times \dfrac{1}{10} + 7 \times \dfrac{1}{100}$

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: In the number 34.17, the part to the left of the decimal point is 34 (whole number part) and the part to the right of the decimal point is 0.17 (decimal part). So, Statement I is true.

Statement II: Writing 34.17 in expanded form,

$34.17 = 3 \times 10 + 4 \times 1 + 1 \times \dfrac{1}{10} + 7 \times \dfrac{1}{100}$

So, Statement II is true.

Hence, option 3 is the correct option.

Statement I: $\dfrac{1}{20}$ is a terminating fraction.

Statement II: The denominator of $\dfrac{1}{20}$ is the 20th multiple of its numerator.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: We have $\dfrac{1}{20} = \dfrac{1 \times 5}{20 \times 5} = \dfrac{5}{100} = 0.05$, which is a terminating decimal. So, Statement I is true.

Statement II: The denominator of $\dfrac{1}{20}$ is 20, and $20 = 20 \times 1$, which is the 20th multiple of the numerator 1. So, Statement II is true.

Hence, option 3 is the correct option.

Statement I: According to a report from 2021, about 24.62% of our country's total land area is covered by forests. When we round this percentage to the nearest whole number, it represents $\dfrac{1}{4}$ th of the total land area.

Statement II: 24.62 rounded off to the nearest whole number is 25. Also, 25% is $\dfrac{1}{4}$.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement II: To round 24.62 to the nearest whole number, look at the tenths digit, which is 6 ≥ 5, so we round up. So 24.62 ≈ 25.

Also, $25$. So Statement II is true.

Statement I: 24.62% ≈ 25% = $\dfrac{1}{4}$. So Statement I is also true.

Hence, option 3 is the correct option.

Statement I: 42.500 and 42.123 two unlike decimal numbers.

Statement II: 42.500 = 42.5

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: Both 42.500 and 42.123 have 3 decimal places, so they are like decimals, not unlike. So, Statement I is false.

Statement II: The value of a decimal number does not change if extra zeros are written at the end of its decimal part. So $42.500 = 42.5$. So, Statement II is true.

Hence, option 2 is the correct option.

Statement I: The cost of 68.45 m of cloth at the rate of ₹400 per metre is ₹273.80 × 100

Statement II: 40.0 kL = 40,000.00 L

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: Cost of 68.45 m of cloth = ₹$(68.45 \times 400)$ = ₹27380.

Also, ₹273.80 × 100 = ₹27380. So, Statement I is true.

Statement II: $1 \text{ kL} = 1000 \text{ L}$, so $40.0 \text{ kL} = 40 \times 1000 = 40000 \text{ L} = 40,000.00 \text{ L}$. So, Statement II is true.

Hence, option 3 is the correct option.

Convert the following decimal numbers into fractions (in lowest terms):

(i) 6.015

(ii) 0.876

(iii) 23.375

Answer

(i) Writing the decimal 6.015 as a fraction,

$6.015 = \dfrac{6015}{1000} = \dfrac{1203}{200} = 6\dfrac{3}{200}$

Therefore, the required fraction is $6\dfrac{3}{200}$.

(ii) Writing the decimal 0.876 as a fraction,

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

Therefore, the required fraction is $\dfrac{219}{250}$.

(iii) Writing the decimal 23.375 as a fraction,

$23.375 = \dfrac{23375}{1000} = \dfrac{187}{8}= 23\dfrac{3}{8}$

Therefore, the required fraction is $23\dfrac{3}{8}$.

Write the following fractions as decimal numbers:

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

(ii) $2\dfrac{31}{125}$

(iii) $13\dfrac{7}{40}$

Answer

(i) Converting the fraction $\dfrac{5}{8}$ into a decimal,

$\dfrac{5}{8} = \dfrac{5 \times 125}{8 \times 125} = \dfrac{625}{1000} = 0.625$

Therefore, the decimal form is 0.625.

(ii) Converting the fraction $2\dfrac{31}{125}$ into a decimal,

$2\dfrac{31}{125} = 2 + \dfrac{31}{125} = 2 + \dfrac{31 \times 8}{125 \times 8} = 2 + \dfrac{248}{1000} = 2 + 0.248 = 2.248$

Therefore, the decimal form is 2.248.

(iii) Converting the fraction $13\dfrac{7}{40}$ into a decimal,

$13\dfrac{7}{40} = 13 + \dfrac{7}{40} = 13 + \dfrac{7 \times 25}{40 \times 25} = 13 + \dfrac{175}{1000} = 13 + 0.175 = 13.175$

Therefore, the decimal form is 13.175.

Arrange the following decimal numbers in ascending order:

(i) 123.8, 74.205, 74.209, 7.4209

(ii) 85.01, 85.1, 85.001, 85.103

Answer

(i) The maximum number of decimal places in the given decimals is 4. So, we convert each of the given decimals into like decimals having four decimal places by annexing zeros.

The decimals are: 123.8000, 74.2050, 74.2090, 7.4209

Clearly, 7.4209 < 74.2050 < 74.2090 < 123.8000.

Hence, the given decimals in ascending order are:

7.4209 < 74.205 < 74.209 < 123.8

(ii) The maximum number of decimal places in the given decimals is 3. So, we convert each of the given decimals into like decimals having three decimal places by annexing zeros.

The decimals are: 85.010, 85.100, 85.001, 85.103

Clearly, 85.001 < 85.010 < 85.100 < 85.103.

Hence, the given decimals in ascending order are:

85.001 < 85.01 < 85.1 < 85.103

Arrange the following decimal numbers in descending order:

(i) 6.45, 4.65, 6.405, 64.5

(ii) 73.5, 35.7, 7.35, 7.035

Answer

(i) The maximum number of decimal places in the given decimals is 3. So, we convert each of the given decimals into like decimals having three decimal places by annexing zeros.

The decimals are: 6.450, 4.650, 6.405, 64.500

Clearly, 64.500 > 6.450 > 6.405 > 4.650.

Hence, the given decimals in descending order are:

64.5 > 6.45 > 6.405 > 4.65

(ii) The maximum number of decimal places in the given decimals is 3. So, we convert each of the given decimals into like decimals having three decimal places by annexing zeros.

The decimals are: 73.500, 35.700, 7.350, 7.035

Clearly, 73.500 > 35.700 > 7.350 > 7.035.

Hence, the given decimals in descending order are:

73.5 > 35.7 > 7.35 > 7.035

If the school bags of Garima and Nakul weigh 5.2 kg and 4.832 kg respectively, find

(i) the total weight

(ii) the difference in weight of the bags

Answer

Converting the given decimals into like decimals having three decimal places, we get 5.200 kg and 4.832 kg.

(i) The total weight = 5.200 + 4.832

Writing in column form and adding:

$\begin{array}{r} \phantom{+}5.200 \\ +\space 4.832 \\ \hline 10.032 \end{array}$

Hence, the total weight of the bags is 10.032 kg.

(ii) The difference in weight = 5.200 - 4.832

Writing in column form and subtracting:

$\begin{array}{r} \phantom{+}5.200 \\ -\space 4.832 \\ \hline 0.368 \end{array}$

Hence, the difference in weight of the bags is 0.368 kg.

Evaluate the following:

(i) 31.42 - 17.853 - 6.43

(ii) 13.01 - 5.428 - 3.703 + 2.99

Answer

(i) Writing each of the given numbers with three decimal places, we get:

Given expression = 31.420 - 17.853 - 6.430

$\phantom{Given expression}$ = 31.420 - (17.853 + 6.430)

Now,

$\begin{array}{r} \phantom{+}17.853 \\ +\space 6.430 \\ \hline 24.283 \end{array}$

So,

$\begin{array}{r} \phantom{+}31.420 \\ -\space 24.283 \\ \hline 7.137 \end{array}$

Therefore, 31.42 - 17.853 - 6.43 = 7.137.

(ii) Writing each of the given numbers with three decimal places, we get:

Given expression = 13.010 - 5.428 - 3.703 + 2.990

$\phantom{Given expression}$ = (13.010 + 2.990) - (5.428 + 3.703)

Now,

$\begin{array}{r} \phantom{+}13.010 \\ +\space 2.990 \\ \hline 16.000 \end{array} \qquad \text{and} \qquad \begin{array}{r} \phantom{+}5.428 \\ +\space 3.703 \\ \hline 9.131 \end{array}$

So,

$\begin{array}{r} \phantom{+}16.000 \\ -\space 9.131 \\ \hline 6.869 \end{array}$

Therefore, 13.01 - 5.428 - 3.703 + 2.99 = 6.869.

By how much does the sum of 15.453 and 31.647 exceed the sum of 18.47 and 19.506?

Answer

First sum = 15.453 + 31.647

$\begin{array}{r} \phantom{+}15.453 \\ +\space 31.647 \\ \hline 47.100 \end{array}$

Second sum = 18.47 + 19.506

Converting into like decimals, we get 18.470 and 19.506.

$\begin{array}{r} \phantom{+}18.470 \\ +\space 19.506 \\ \hline 37.976 \end{array}$

Required difference = 47.100 - 37.976

$\begin{array}{r} \phantom{+}47.100 \\ -\space 37.976 \\ \hline 9.124 \end{array}$

Hence, the sum of 15.453 and 31.647 exceeds the sum of 18.47 and 19.506 by 9.124.

Convert 2435 m to km and express the result as mixed fraction.

Answer

We know that 1 m = $\dfrac{1}{1000}$ km.

So, 2435 m = $\dfrac{2435}{1000} = 2.435$ km

Reducing to lowest terms,

$\dfrac{2435}{1000} = \dfrac{487}{200}$

Converting to a mixed fraction,

$\dfrac{487}{200} = 2\dfrac{87}{200}$

Hence, 2435 m = 2.435 km and 2435 m = $2\dfrac{87}{200}$ km.

Namita travels 20 km 50 m every day. Out of this she travels 10 km 200 m by bus and the rest by auto. How much distance does she travel by auto?

Answer

Total distance travelled = 20 km 50 m

= 20 km + $\dfrac{50}{1000}$ km = 20 km + 0.050 km = 20.050 km

Distance travelled by bus = 10 km 200 m

= 10 km + $\dfrac{200}{1000}$ km = 10 km + 0.200 km = 10.200 km

Distance travelled by auto = 20.050 - 10.200

Writing in column form and subtracting:

$\begin{array}{r} \phantom{+}20.050 \\ -\space 10.200 \\ \hline 9.850 \end{array}$

Hence, Namita travels 9.850 km or 9 km 850 m by auto.

Asha purchased 5 kg 400 g rice, 2 kg 20 g sugar and 10 kg 850 g flour. Find the total weight of his purchases.

Answer

Weight of rice = 5 kg 400 g

= 5 kg + $\dfrac{400}{1000}$ kg = 5 kg + 0.400 kg = 5.400 kg

Weight of sugar = 2 kg 20 g

= 2 kg + $\dfrac{20}{1000}$ kg = 2 kg + 0.020 kg = 2.020 kg

Weight of flour = 10 kg 850 g

= 10 kg + $\dfrac{850}{1000}$ kg = 10 kg + 0.850 kg = 10.850 kg

Total weight = 5.400 + 2.020 + 10.850

Writing in column form and adding:

$\begin{array}{r} \phantom{+}5.400 \\ 2.020 \\ +\space 10.850 \\ \hline 18.270 \end{array}$

Hence, the total weight of Asha's purchases is 18.270 kg or 18 kg 270 g.

1 kg of pure milk contains 0.263 kg of fat. How much fat is there 15.5 kg of milk?

Answer

Quantity of fat in 1 kg of milk = 0.263 kg

Quantity of fat in 15.5 kg of milk = 0.263 × 15.5

Ignoring the decimal points, 263 × 155 = 40765.

The two factors together have 3 + 1 = 4 decimal places, so:

0.263 × 15.5 = 4.0765

Hence, there is 4.0765 kg of fat in 15.5 kg of milk.

The product of two numbers is 15.275. If one number is 4.7, find the other.

Answer

Product of the two numbers = 15.275

One number = 4.7

The other number = $\dfrac{15.275}{4.7}$

Multiplying both the numerator and denominator by 10, we get:

The other number = $\dfrac{15.275 \times 10}{4.7 \times 10} = \dfrac{152.75}{47}$

By actual division, we have:

$\begin{array}{l} \phantom{47)\space 1}3.25 \\ 47\overline{\smash{\big)}\space 152.75} \\ \phantom{47}\underline{-141} \\ \phantom{47)))\space }117 \\ \phantom{47)))\space}\underline{-94\phantom{1}} \\ \phantom{47)))\space 1}235 \\ \phantom{47)))\space }\underline{-235} \\ \phantom{47)\space 152.7}\times \end{array}$

Hence, the other number is 3.25.

On her birthday, Ayushi is taking her 5 friends to a movie and treats them with cold drink. The cost of a ticket is ₹150 and a cold drink costs ₹28.50. How much Ayushi has to spend?

Answer

Total number of persons = Ayushi + 5 friends = 6

Cost of one ticket = ₹150

Cost of 6 tickets = ₹150 × 6 = ₹900

Cost of one cold drink = ₹28.50

Cost of 6 cold drinks = ₹28.50 × 6 = ₹171.00

Total amount Ayushi has to spend = ₹900 + ₹171

Writing in column form and adding:

$\begin{array}{r} \phantom{+}900.00 \\ +\space 171.00 \\ \hline 1071.00 \end{array}$

Hence, Ayushi has to spend ₹1,071.

Write digits in the boxes of the number:

$3\boxed{\phantom{6}}6\boxed{\phantom{6}}.8\boxed{\phantom{6}} \boxed{\phantom{6}}$ to obtain (i) greatest possible number (ii) smallest possible number. Repetition of digits in a number is not allowed.

Answer

The number is of the form 3⬜6⬜.8 ⬜⬜ where the digits 3, 6 and 8 are already used. The four boxes have to be filled with four distinct digits chosen from {0, 1, 2, 4, 5, 7, 9}.

(i) Greatest possible number:

To get the greatest number, we fill the boxes with the four largest available digits in decreasing order from the leftmost box: 9, 7, 5, 4.

So the boxes are filled as: 3967.854

Greatest possible number = 3967.854

(ii) Smallest possible number:

To get the smallest number, we fill the boxes with the four smallest available digits in increasing order from the leftmost box: 0, 1, 2, 4.

So the boxes are filled as: 3061.824

Smallest possible number = 3061.824

Arrange the following numbers in descending order:

$5\dfrac{3}{4}$, 5.721, $5\dfrac{7}{8}$, $5\dfrac{17}{25}$, 5.693

Answer

First, convert each mixed fraction into a decimal number.

$5\dfrac{3}{4} = 5 + \dfrac{3}{4} = 5 + \dfrac{3 \times 25}{4 \times 25} = 5 + \dfrac{75}{100} = 5 + 0.75 = 5.75$

$5\dfrac{7}{8} = 5 + \dfrac{7}{8} = 5 + \dfrac{7 \times 125}{8 \times 125} = 5 + \dfrac{875}{1000} = 5 + 0.875 = 5.875$

$5\dfrac{17}{25} = 5 + \dfrac{17}{25} = 5 + \dfrac{17 \times 4}{25 \times 4} = 5 + \dfrac{68}{100} = 5 + 0.68 = 5.68$

So the given numbers in decimal form are:

5.75, 5.721, 5.875, 5.68, 5.693

Converting into like decimals, each having three decimal places:

5.750, 5.721, 5.875, 5.680, 5.693

Clearly, 5.875 > 5.750 > 5.721 > 5.693 > 5.680.

Hence, the given numbers in descending order are:

$5\dfrac{7}{8}$ > $5\dfrac{3}{4}$ > 5.721 > 5.693 > $5\dfrac{17}{25}$