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Chapter 3

Decimals - Exercise 3(A)

Class - 7 RS Aggarwal Mathematics Solutions

Exercise 3(A)

Question 1(i)

Convert the following decimals into like decimals :

8.1, 9.4, 6.05, 12

Answer

Given:

8.1, 9.4, 6.05, 12

To convert into like decimals, we make the number of decimal places equal.

The maximum number of decimal places is 2 (in 6.05).

So,

8.1 = 8.10
9.4 = 9.40
6.05 = 6.05
12 = 12.00

∴ Like decimals are 8.10, 9.40, 6.05 and 12.00

Question 1(ii)

Convert the following decimals into like decimals :

6.35, 11.8, 9.125, 32.04, 9

Answer

Given:

6.35, 11.8, 9.125, 32.04, 9

The maximum number of decimal places is 3 (in 9.125).

So,

6.35 = 6.350
11.8 = 11.800
9.125 = 9.125
32.04 = 32.040
9 = 9.000

∴ Like decimals are 6.350, 11.800, 9.125, 32.040 and 9.000

Question 1(iii)

Convert the following decimals into like decimals :

6.005, 2, 0.6, 1.78, 14.3

Answer

Given:

6.005, 2, 0.6, 1.78, 14.3

The maximum number of decimal places is 3 (in 6.005).

So,

6.005 = 6.005
2 = 2.000
0.6 = 0.600
1.78 = 1.780
14.3 = 14.300

∴ Like decimals are 6.005, 2.000, 0.600, 1.780 and 14.300

Question 1(iv)

Convert the following decimals into like decimals :

0.72, 53.76, 16, 8.304, 2.3754

Answer

Given:

0.72, 53.76, 16, 8.304, 2.3754

Maximum decimal places = 4

So,

0.72 = 0.7200
53.76 = 53.7600
16 = 16.0000
8.304 = 8.3040
2.3754 = 2.3754

∴ Like decimals are 0.7200, 53.7600, 16.0000, 8.3040 and 2.3754

Question 2(i)

Write each of the following decimals into expanded form :

16.74

Answer

We have:

16.74 = 10 + 6 + 0.7 + 0.04 \hspace{2cm}[Separating according to place values]

=(1×10)+(6×1)+(7×110)+(4×1100)=10+6+710+4100= (1 \times 10) + (6 \times 1) + \Big(7 \times \dfrac{1}{10}\Big) + \Big(4 \times\dfrac{1}{100}\Big) \\[1em] = 10 + 6 + \dfrac{7}{10} + \dfrac{4}{100}

∴ The expanded form of 16.74 is (10+6+710+4100)\Big(10 + 6 + \dfrac{7}{10} + \dfrac{4}{100}\Big)

Question 2(ii)

Write each of the following decimals into expanded form :

243.689

Answer

We have:

243.689 = 200 + 40 + 3 + 0.6 + 0.08 + 0.009 [Separating according to place values]

=(2×100)+(4×10)+(3×1)+(6×110)+(8×1100)+(9×11000)=200+40+3+610+8100+91000= (2 \times 100) + (4 \times 10) + (3 \times 1) + \Big(6 \times \dfrac{1}{10}\Big) + \Big(8 \times \dfrac{1}{100}\Big) + \Big(9 \times \dfrac{1}{1000}\Big) \\[1em] = 200 + 40 + 3 + \dfrac{6}{10} + \dfrac{8}{100} + \dfrac{9}{1000}

∴ The expanded form of 243.689 is (200+40+3+610+8100+91000)\Big(200 + 40 + 3 + \dfrac{6}{10} + \dfrac{8}{100} + \dfrac{9}{1000}\Big)

Question 2(iii)

Write each of the following decimals into expanded form :

9.3578

Answer

We have:

9.3578 = 9 + 0.3 + 0.05 + 0.007 + 0.0008 [Separating according to place values]

=(9×1)+(3×110)+(5×1100)+(7×11000)+(8×110000)=9+310+5100+71000+810000= (9 \times 1) + \Big(3 \times \dfrac{1}{10}\Big) + \Big(5 \times \dfrac{1}{100}\Big) + \Big(7 \times \dfrac{1}{1000}\Big) + \Big(8 \times \dfrac{1}{10000}\Big) \\[1em] = 9 + \dfrac{3}{10} + \dfrac{5}{100} + \dfrac{7}{1000} + \dfrac{8}{10000}

∴ The expanded form of 9.3578 is (9+310+5100+71000+810000)\Big(9 + \dfrac{3}{10} + \dfrac{5}{100} + \dfrac{7}{1000} + \dfrac{8}{10000}\Big)

Question 2(iv)

Write each of the following decimals into expanded form :

2314.57

Answer

We have:

2314.57 = 2000 + 300 + 10 + 4 + 0.5 + 0.07 [Separating according to place values]

=(2×1000)+(3×100)+(1×10)+(4×1)+(5×110)+(7×1100)=2000+300+10+4+510+7100= (2 \times 1000) + (3 \times 100) + (1 \times 10) + (4 \times 1) + \Big(5 \times \dfrac{1}{10}\Big) + \Big(7 \times \dfrac{1}{100}\Big) \\[1em] = 2000 + 300 + 10 + 4 + \dfrac{5}{10} + \dfrac{7}{100}

∴ The expanded form of 2314.57 is (2000+300+10+4+510+7100)\Big(2000 + 300 + 10 + 4 + \dfrac{5}{10} + \dfrac{7}{100}\Big)

Question 3(i)

Express the following as a fraction in simplest form :

0.8

Answer

We have:

0.8

There is 1 decimal place, so we remove the decimal point and divide by 10.

=0.8=810=45[Dividing both by 2]\phantom{=} 0.8 = \dfrac{8}{10} \\[1em] = \dfrac{4}{5} \quad \text{[Dividing both by 2]}

∴ The answer is 45\dfrac{4}{5}

Question 3(ii)

Express the following as a fraction in simplest form :

0.45

Answer

We have:

0.45

There are 2 decimal places, so we remove the decimal point and divide by 100.

=0.45=45100=920[Dividing both by 5]\phantom{=} 0.45 = \dfrac{45}{100} \\[1em] = \dfrac{9}{20} \quad \text{[Dividing both by 5]}

∴ The answer is 920\dfrac{9}{20}

Question 3(iii)

Express the following as a fraction in simplest form :

1.6

Answer

We have:

1.6

There is 1 decimal place, so we remove the decimal point and divide by 10.

=1.6=1610=85[Dividing both by 2]=135\phantom{=} 1.6 = \dfrac{16}{10} \\[1em] = \dfrac{8}{5} \quad \text{[Dividing both by 2]} \\[1em] = 1\dfrac{3}{5}

∴ The answer is 1351\dfrac{3}{5}

Question 3(iv)

Express the following as a fraction in simplest form :

4.75

Answer

We have:

4.75

There are 2 decimal places, so we remove the decimal point and divide by 100.

=4.75=475100=194[Dividing both by 25]=434\phantom{=} 4.75 = \dfrac{475}{100} \\[1em] = \dfrac{19}{4} \quad \text{[Dividing both by 25]} \\[1em] = 4\dfrac{3}{4}

∴ The answer is 4344\dfrac{3}{4}

Question 3(v)

Express the following as a fraction in simplest form :

0.345

Answer

We have:

0.345

There are 3 decimal places, so we remove the decimal point and divide by 1000.

=0.345=3451000=69200[Dividing both by 5]\phantom{=} 0.345 = \dfrac{345}{1000} \\[1em] = \dfrac{69}{200} \quad \text{[Dividing both by 5]}

∴ The answer is 69200\dfrac{69}{200}

Question 3(vi)

Express the following as a fraction in simplest form :

64.015

Answer

We have:

64.015

There are 3 decimal places, so we remove the decimal point and divide by 1000.

=64.015=640151000=12803200[Dividing both by 5]=643200\phantom{=} 64.015 = \dfrac{64015}{1000} \\[1em] = \dfrac{12803}{200} \quad \text{[Dividing both by 5]} \\[1em] = 64\dfrac{3}{200}

∴ The answer is 64320064\dfrac{3}{200}

Question 3(vii)

Express the following as a fraction in simplest form :

12.725

Answer

We have:

12.725

There are 3 decimal places, so we remove the decimal point and divide by 1000.

=12.725=127251000=50940[Dividing both by 25]=122940\phantom{=} 12.725 = \dfrac{12725}{1000} \\[1em] = \dfrac{509}{40} \quad \text{[Dividing both by 25]} \\[1em] = 12\dfrac{29}{40}

∴ The answer is 12294012\dfrac{29}{40}

Question 3(viii)

Express the following as a fraction in simplest form :

0.0024

Answer

We have:

0.0024

There are 4 decimal places, so we remove the decimal point and divide by 10000.

=0.0024=2410000=31250[Dividing both by 8]\phantom{=} 0.0024 = \dfrac{24}{10000} \\[1em] = \dfrac{3}{1250} \quad \text{[Dividing both by 8]}

∴ The answer is 31250\dfrac{3}{1250}

Question 4(i)

Convert the following fraction into a decimal :

1910\dfrac{19}{10}

Answer

We have:

=1910=1910[improper to mixed]=1+910=1+0.9=1.9\phantom{=} \dfrac{19}{10} = 1\dfrac{9}{10} \quad \text{[improper to mixed]} \\[1em] = 1 + \dfrac{9}{10} \\[1em] = 1 + 0.9 \\[1em] = 1.9

∴ The answer is 1.9

Question 4(ii)

Convert the following fraction into a decimal :

243100\dfrac{243}{100}

Answer

We have:

=243100=243100[improper to mixed]=2+43100=2+0.43=2.43\phantom{=} \dfrac{243}{100} = 2\dfrac{43}{100} \quad \text{[improper to mixed]} \\[1em] = 2 + \dfrac{43}{100} \\[1em] = 2 + 0.43 \\[1em] = 2.43

∴ The answer is 2.43

Question 4(iii)

Convert the following fraction into a decimal :

51100\dfrac{51}{100}

Answer

We have:

51100=0.51\dfrac{51}{100} = 0.51

∴ The answer is 0.51

Question 4(iv)

Convert the following fraction into a decimal :

9100\dfrac{9}{100}

Answer

We have:

9100=0.09\dfrac{9}{100} = 0.09

∴ The answer is 0.09

Question 4(v)

Convert the following fraction into a decimal :

721000\dfrac{72}{1000}

Answer

We have:

721000=0.072\dfrac{72}{1000} = 0.072

∴ The answer is 0.072

Question 4(vi)

Convert the following fraction into a decimal :

35\dfrac{3}{5}

Answer

We have:

=35=3×25×2[Multiplying by 2 to get denominator 10]=610=0.6\phantom{=} \dfrac{3}{5} = \dfrac{3 \times 2}{5 \times 2} \quad \text{[Multiplying by 2 to get denominator 10]} \\[1em] = \dfrac{6}{10} \\[1em] = 0.6

∴ The answer is 0.6

Question 4(vii)

Convert the following fraction into a decimal :

78\dfrac{7}{8}

Answer

We have:

=78=7×1258×125[Multiplying by 125 to get denominator 1000]=8751000=0.875\phantom{=} \dfrac{7}{8} = \dfrac{7 \times 125}{8 \times 125} \quad \text{[Multiplying by 125 to get denominator 1000]} \\[1em] = \dfrac{875}{1000} \\[1em] = 0.875

∴ The answer is 0.875

Question 4(viii)

Convert the following fraction into a decimal :

3740\dfrac{37}{40}

Answer

We have:

=3740=37×2540×25[Multiplying by 25 to get denominator 1000]=9251000=0.925\phantom{=} \dfrac{37}{40} = \dfrac{37 \times 25}{40 \times 25} \quad \text{[Multiplying by 25 to get denominator 1000]} \\[1em] = \dfrac{925}{1000} \\[1em] = 0.925

∴ The answer is 0.925

Question 4(ix)

Convert the following fraction into a decimal :

4384\dfrac{3}{8}

Answer

First, we convert the mixed fraction into an improper fraction:

=438=4×8+38=32+38=358\phantom{=} 4\dfrac{3}{8} = \dfrac{4 \times 8 + 3}{8} = \dfrac{32 + 3}{8} = \dfrac{35}{8}

Now, we divide 35 by 8. Since 35 is not exactly divisible by 8, we add a decimal point and zeros to continue the division:

4.3758)35.000‾−32.000‾3000−2400‾600−560‾40−40‾0\begin{array}{r} 4.375 \\ 8 \overline{) 35.000} \\ \underline{-32\phantom{.000}} \\ 30\phantom{00} \\ \underline{-24\phantom{00}} \\ 60\phantom{0} \\ \underline{-56\phantom{0}} \\ 40 \\ \underline{-40} \\ 0 \end{array}

358\dfrac{35}{8} = 4.375

∴ 4384\dfrac{3}{8} = 4.375

Question 4(x)

Convert the following fraction into a decimal :

91259\dfrac{1}{25}

Answer

First, we convert the mixed fraction into an improper fraction:

=9125=9×25+125=225+125=22625\phantom{=} 9\dfrac{1}{25} = \dfrac{9 \times 25 + 1}{25} = \dfrac{225 + 1}{25} = \dfrac{226}{25}

Now, we divide 226 by 25. Since 226 is not exactly divisible by 25, we add a decimal point and zeros to continue the division:

9.0425)226.00‾−22500‾100−100‾0\begin{array}{r} 9.04 \\ 25 \overline{) 226.00} \\ \underline{-225\phantom{00}} \\ 100 \\ \underline{-100} \\ 0 \end{array}

22625\dfrac{226}{25} = 9.04

∴ 91259\dfrac{1}{25} = 9.04

Question 4(xi)

Convert the following fractions into a decimal :

29162\dfrac{9}{16}

Answer

First, we convert the mixed fraction into an improper fraction:

=2916=2×16+916=32+916=4116\phantom{=} 2\dfrac{9}{16} = \dfrac{2 \times 16 + 9}{16} = \dfrac{32 + 9}{16} = \dfrac{41}{16}

Now, we divide 41 by 16. We add a decimal point and zeros to continue the division:

2.562516)41.0000‾−32.0000‾90000−80000‾10000−09600‾400−320‾80−80‾0\begin{array}{r} 2.5625 \\ 16 \overline{) 41.0000} \\ \underline{-32\phantom{.0000}} \\ 90\phantom{000} \\ \underline{-80\phantom{000}} \\ 100\phantom{00} \\ \underline{-096\phantom{00}} \\ 40\phantom{0} \\ \underline{-32\phantom{0}} \\ 80 \\ \underline{-80} \\ 0 \end{array}

4116\dfrac{41}{16} = 2.5625

∴ 29162\dfrac{9}{16} = 2.5625

Question 4(xii)

Convert the following fractions into a decimal :

4111254\dfrac{11}{125}

Answer

First, we convert the mixed fraction into an improper fraction:

411125=4×125+11125=500+11125=5111254\dfrac{11}{125} = \dfrac{4 \times 125 + 11}{125} = \dfrac{500 + 11}{125} = \dfrac{511}{125}

Now, we divide 511 by 125. We add a decimal point and zeros to continue the division:

4.088125)511.000‾−500.000‾11000−10000‾1000−1000‾0\begin{array}{r} 4.088 \\ 125 \overline{) 511.000} \\ \underline{-500\phantom{.000}} \\ 1100\phantom{0} \\ \underline{-1000\phantom{0}} \\ 1000 \\ \underline{-1000} \\ 0 \end{array}

511125\dfrac{511}{125} = 4.088

∴ 4111254\dfrac{11}{125} = 4.088

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