ICSE Class 7 Maths Chapter 05 Decimal Fractions

Chapter 5: Decimal Fractions (Decimals)

5.1 Definition

A decimal fraction or a decimal number is a fraction whose denominator can be expressed as 10 or some higher power of 10.

(i) \(\frac{7}{10}, \frac{13}{100}, \frac{851}{1000}, \frac{79}{10^4}, \frac{2547}{10^7}\) etc., are all decimal fractions.

(ii) Since, \(\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}\), i.e., \(\frac{3}{5}\) can be expressed as a fraction with denominator 10, therefore \(\frac{3}{5}\) is a decimal fraction.

(iii) \(\frac{7}{8} = \frac{7 \times 125}{8 \times 125} = \frac{875}{1000}\), i.e., \(\frac{7}{8}\) can be expressed as a fraction with denominator 1000 (i.e. \(10^3\), which is higher power of 10), therefore \(\frac{7}{8}\) is also a decimal fraction.

For the same reason, each of \(\frac{7}{20}, \frac{16}{25}, \frac{33}{50}, \frac{64}{125}\), etc., is also a decimal fraction.

In order to express a given decimal fraction in shorter form, the denominator is not written, but its absence is shown by a dot called a decimal point, inserted in a proper place.

e.g., \(\frac{3}{10} = 0.3; \frac{213}{100} = 2.13; \frac{7}{100} = 0.07; \frac{59}{10^4} = \frac{59}{10000} = 0.0059\), etc.

When there is no number to the left of the decimal point, generally, a zero is written.

(i) .72 is written as 0.72

(ii) .004 is written as 0.004 and so on.

2.4 means 2 + 0.4. Here, 2 is the integral part and 0-4 is the decimal part of the number 2.4.

Any extra zero or zeroes written after the decimal part of a number does not change its value.

e.g., value of 3.5 is the same as 3.50 or 3.500 or 3.5000 and so on.

5.2 Reading Decimal Numbers

The integral part is read according to its value and decimal part is read by naming each digit, in order, separately.

e.g., (i) 21.45 will be read as: Twenty one point four-five.

(ii) 152.639 will be read as: One fifty two point six-three-nine.

(iii) 0.08 will be read as: Point zero-eight or zero-point zero-eight.

In decimal system, the first place on the right of the decimal point is called tenths' place, second place to the right of decimal is called hundredths' place and so on.

Similarly, the first place on the left of decimal is the units' place, the second place on the left of decimal is the tens' place, and so on.

e.g., in number 5.46; 5 is at units' place, 4 is at tenths' place and 6 is at hundredths' place.

The following table shows the place values of different digits in a decimal number:

5.3 Converting A Decimal Number Into A Vulgar Fraction

Remove decimal point from the given decimal number. And, in its denominator write as many zeroes, as the number of digits in the decimal part, to the right of 1. Then simplify, if possible, to get the fraction obtained to its lowest terms.

Thus, \(0.47 = \frac{47}{100}; 2.739 = \frac{2739}{1000}; 0.0244 = \frac{244}{10000} = \frac{61}{2500}\) and so on.

5.4 Converting A Given Fraction Into A Decimal Fraction

When the denominator of the given fraction is 10, 100, 1000, etc.:

Counting from extreme right to left, mark the decimal point after as many digits of the numerator as there are zeroes in the denominator.

Thus, (i) \(\frac{259}{10} = 25.9\), (ii) \(\frac{259}{100} = 2.59\), (iii) \(\frac{259}{1000} = 0.259\), (iv) \(\frac{259}{10000} = 0.0259\) and so on.

When the denominator of the given fraction is other than 10, 100, 1000, etc.:

Divide in an ordinary way and mark the decimal point in the quotient just after the division of unit digit is completed. After this, any number of zeroes (one by one) can be placed to complete the division.

Thus, (i) \(\frac{15}{4} = 3.75\)

(ii) \(\frac{27}{5} = 5.4\)

5.5 Decimal Places

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

Thus, 28.497 has 3 decimal places, 153.46 has 2 decimal places, 0.5497 has 4 decimal places and so on.

Example 1

Convert each of the following decimal fractions into vulgar fraction in lowest terms:

(i) 0.125 (ii) 5.08 (iii) 26.25

Solution

(i) \(0.125 = \frac{125}{1000} = \frac{1}{8}\)

(ii) \(5.08 = \frac{508}{100} = \frac{127}{25} = 5\frac{2}{25}\)

OR, \(5.08 = 5 + 0.08 = 5 + \frac{8}{100} = 5 + \frac{2}{25} = 5\frac{2}{25}\)

(iii) \(26.25 = \frac{2625}{100} = \frac{105}{4} = 26\frac{1}{4}\)

OR, \(26.25 = 26 + 0.25 = 26 + \frac{25}{100} = 26 + \frac{1}{4} = 26\frac{1}{4}\)

Example 2

Convert each of the following into decimal fraction:

(i) \(5\frac{3}{8}\) (ii) \(\frac{2}{25}\) (iii) \(2\frac{7}{100}\)

Solution

(i) \(5\frac{3}{8} = \frac{5 \times 8 + 3}{8} = \frac{43}{8} = 5.375\)

OR, \(5\frac{3}{8} = 5 + \frac{3}{8} = 5 + 0.375 = 5.375\)

(ii) \(\frac{2}{25} = 0.08\)

OR, \(\frac{2}{25} = \frac{2 \times 4}{25 \times 4} = \frac{8}{100} = 0.08\)

(iii) \(2\frac{7}{100} = \frac{2 \times 100 + 7}{100} = \frac{207}{100} = 2.07\)

OR, \(2\frac{7}{100} = 2 + \frac{7}{100} = 2 + 0.07 = 2.07\)

Exercise 5(A)

1. Convert the following into fractions in their lowest terms:

(i) 3.75 (ii) 0.5 (iii) 2.04 (iv) 0.65 (v) 2.405 (vi) 0.085 (vii) 8.025

2. Convert into decimal fractions:

(i) \(2\frac{4}{5}\) (ii) \(\frac{79}{100}\) (iii) \(\frac{37}{10,000}\) (iv) \(\frac{7543}{10^4}\) (v) \(\frac{3}{4}\) (vi) \(9\frac{3}{5}\) (vii) \(8\frac{5}{8}\) (viii) \(5\frac{7}{8}\)

3. Write the number of decimal places in:

(i) 0.4762 (ii) 7.00349 (iii) 8235.403 (iv) 35.4 (v) 2.608 (vi) 0.000879

4. Write the following decimals as word statements:

(i) 0.4, 0.9, 0.1 (ii) 1.9, 4.4, 7.5 (iii) 0.02, 0.56, 13.06 (iv) 0.005, 0.207, 111.519 (v) 0.8, 0.08, 0.008, 0.0008 (vi) 256.1, 10.22, 0.634

Teacher's Note

Understanding decimal fractions helps students comprehend how to read prices in a shop, measure ingredients in cooking, or track their athletic performance times in sports competitions.

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