ICSE Class 8 Maths Number Systems Chapter 07 Fractions and Decimals

Fractions And Decimals

Common Fractions

Operations Involving Common Fractions

Decimal Fractions

Operations Involving Decimal Fractions

Introduction

Let us recall what we have learnt about common fractions and decimal fractions in previous classes.

Common Fractions

1. A common fraction is a part of a natural number.

2. Common fraction \(\frac{3}{4}\) represents 3 parts out of 4 equal parts, where 3 above the division line is known as the numerator and 4 below the division line is known as the denominator.

3. The numerator is less than the denominator in proper fractions like \(\frac{1}{2}, \frac{3}{5}, \frac{5}{8}\).

4. The numerator is greater than the denominator in improper fractions like \(\frac{3}{2}, \frac{7}{5}, \frac{9}{8}\).

5. A mixed fraction, like \(1\frac{3}{4}\), is the sum of a natural number 1 and a proper fraction \(\frac{3}{4}\).

6. The denominators of like fractions, like \(\frac{3}{7}, \frac{4}{7}, \frac{1}{7}\), are the same.

7. The denominators of unlike fractions, like \(\frac{2}{3}, \frac{1}{5}, \frac{2}{9}\), are different.

8. The value of equivalent common fractions is the same. Equivalent common fractions are obtained by multiplying or dividing the numerator as well as the denominator of the fraction by the same number. For e.g., \(\frac{3}{4} = \frac{6}{8} = \frac{9}{12}\)

9. A common fraction may be converted into a decimal fraction. \(\frac{1}{2} = 0.5, \frac{5}{16} = 0.3125\)

Decimal Fractions

1. A decimal fraction is a part of a power of 10.

2. The decimal fraction 0.04 represents 4 parts out of 100 equal parts. The number of digits after the decimal point is the number of decimal places.

3. The value of the decimal equivalents of proper fractions, like 0.25, 0.4, 0.625, is less than 1.

4. The value of the decimal equivalents of improper fractions, like 1.5, 1.4, 1.125, is greater than 1.

5. The decimal fraction 3.4 is the sum of its integral part 3 and its fractional part 0.4.

6. Like decimal fractions, like 0.728, 10.001, have the same number of decimal places.

7. Unlike decimal fractions, like 1.5, 0.05, do not have the same number of decimal places.

8. The value of equivalent decimal fractions is the same. Equivalent decimals are obtained by adding zeroes to the extreme right of the decimal part. For e.g., 3.6 = 3.60 = 3.600

9. A decimal fraction may be converted into a common fraction. \(1.25 = \frac{5}{4}, 0.875 = \frac{7}{8}\)

Common Fractions

Reduction To Simplest Form

A common fraction, also known as vulgar fraction, is reduced to its simplest form or lowest terms, by dividing the numerator and denominator by their HCF.

Example 1: Reduce \(\frac{385}{539}\) to its simplest form.

\(385 = 5 \times 7 \times 11\)

\(539 = 7 \times 7 \times 11\)

HCF of 385 and 539 = 77

Thus \(\frac{385 \div 77}{539 \div 77} = \frac{5}{7}\)

Try this!

Reduce \(\frac{582}{1024}\) to its simplest form.

Comparing Common Fractions

1. Convert the given fractions into equivalent fractions with the same denominator. The fraction with the greater numerator is greater. Or

2. Convert the given fractions into equivalent fractions with the same numerator. The fraction with the smaller denominator is greater. Or

3. Divide the numerators by their denominators and then compare the decimal values.

Example 2: Write the following fractions in ascending order.

\(\frac{5}{6}, \frac{3}{4}, \frac{2}{1}, \frac{11}{12}, \frac{2}{3}\)

The LCM of the denominators is 12. Converting the given fractions into equivalent fractions with 12 as denominator, we have

\(\frac{5 \times 2}{6 \times 2} = \frac{10}{12}, \frac{3 \times 3}{4 \times 3} = \frac{9}{12}, \frac{2 \times 12}{1 \times 12} = \frac{24}{12}, \frac{11 \times 1}{12 \times 1} = \frac{11}{12}, \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

Beginning with the least numerator, the fractions in ascending order are;

\(\frac{8}{12}, \frac{9}{12}, \frac{10}{12}, \frac{11}{12}, \frac{24}{12}\) or \(\frac{2}{3}, \frac{3}{4}, \frac{5}{6}, \frac{11}{12}, \frac{2}{1}\)

Example 3: Write the following fractions in descending order.

\(1\frac{2}{3}, \frac{15}{16}, 1\frac{3}{13}, 1\frac{3}{7}, \frac{6}{9}\)

Convert the mixed fractions into common fractions:

\(\frac{5}{3}, \frac{15}{16}, \frac{16}{13}, \frac{10}{7}, \frac{6}{9}\)

Here we find it simpler to find the LCM of the numerators. LCM of numerators = 30. Converting given fractions into equivalent fractions with 30 as numerator, we have

\(\frac{5 \times 6}{3 \times 6} = \frac{30}{18}, \frac{15 \times 2}{16 \times 2} = \frac{30}{32}, \frac{16 \times 10}{13 \times 10} = \frac{30}{130},\) \(\frac{10 \times 3}{7 \times 3} = \frac{30}{21}, \frac{6 \times 5}{9 \times 5} = \frac{30}{45}\)

Beginning with the smallest denominator, the fractions in descending order are:

\(\frac{30}{18}, \frac{30}{21}, \frac{30}{32}, \frac{30}{45}, \frac{30}{130}\) or \(\frac{2}{3}, \frac{3}{7}, \frac{15}{16}, \frac{6}{9}, \frac{1}{13}\)

Example 4: Write the following fractions in ascending order.

\(\frac{16}{33}, \frac{7}{15}, \frac{9}{19}, \frac{2}{5}, \frac{10}{21}\)

As finding the LCM of numerators or denominators is not convenient, find the decimal values of the given fractions up to at least 3 decimal places.

\(\frac{16}{33} = 0.485, \frac{7}{15} = 0.467, \frac{9}{19} = 0.474,\)

\(\frac{2}{5} = 0.4, \frac{10}{21} = 0.476\)

As 0.4 - 0.467 - 0.474 - 0.476 - 0.485, the fractions in ascending order are:

\(\frac{2}{5}, \frac{7}{15}, \frac{9}{19}, \frac{10}{21}, \frac{16}{33}\)

Try this!

Write the following fractions:

1. In descending order: \(\frac{2}{9}, \frac{5}{12}, \frac{7}{15}, \frac{10}{14}\)

2. In ascending order: \(\frac{15}{8}, \frac{5}{6}, \frac{10}{12}, \frac{5}{15}\)

Operations Involving Common Fractions

Addition And Subtraction

Example 5: Evaluate \(\frac{5}{7} + \frac{1}{5} - \frac{3}{14}\)

Adding and subtracting equivalent fractions with LCM of denominators as the common denominator, we get

\(\frac{5}{7} + \frac{1}{5} - \frac{3}{14} = \frac{(5 \times 10) + (1 \times 14) - (3 \times 5)}{70}\)

\(= \frac{50 + 14 - 15}{70}\)

\(= \frac{49}{70} = \frac{7}{10}\)

Example 6: Evaluate \(1\frac{7}{8} + 2\frac{1}{2} - 2\frac{11}{12}\)

Converting mixed fractions into common fractions, we get

\(1\frac{7}{8} + 2\frac{1}{2} - 2\frac{11}{12} = \frac{15}{8} + \frac{5}{2} - \frac{35}{12}\)

\(= \frac{45 + 60 - 70}{24}\)

\(= \frac{35}{24} = 1\frac{11}{24}\)

Multiplication And Division

Example 7: Evaluate \(\frac{6}{7} \times 2\frac{1}{3}\).

\(\frac{6}{7} \times 2\frac{1}{3} = \frac{6}{7} \times \frac{7}{3} = \frac{6 \times 7}{7 \times 3}\)

\(= \frac{42}{21} = 2\)

Example 8: Evaluate \(3\frac{5}{7} \div \frac{13}{14}\)

\(3\frac{5}{7} \div \frac{13}{14} = \frac{26}{7} \div \frac{13}{14}\)

\(= \frac{26}{7} \times \frac{14}{13} = 4\)

Try this!

Evaluate

1. \(\frac{3}{12} - \frac{7}{8} + \frac{6}{10}\)

2. \(\frac{2}{5} \times \frac{10}{18}\)

3. \(\frac{5}{8} \div \frac{10}{16}\)

Finding HCF And LCM Of Common Fractions

HCF of common fractions \(= \frac{\text{HCF of the numerators}}{\text{LCM of the denominators}}\)

LCM of common fractions \(= \frac{\text{LCM of the numerators}}{\text{HCF of the denominators}}\)

Example 9: (i) Which is the greatest fraction that will divide \(\frac{9}{14}, \frac{3}{7},\) and \(\frac{6}{21}\) exactly without leaving any remainders?

(ii) Which is the smallest fraction that can be divided by the above fractions without leaving any remainders?

(i) HCF of the given fractions

\(= \frac{\text{HCF of 9, 3, and 6}}{\text{LCM of 14, 7, and 21}} = \frac{3}{42}\)

CHECK: \(\frac{9}{14} \div \frac{3}{42} = \frac{9}{14} \times \frac{42}{3} = 9\);

\(\frac{3}{7} \div \frac{3}{42} = \frac{3}{7} \times \frac{42}{3} = 6\); \(\frac{6}{21} \div \frac{3}{42} = \frac{6}{21} \times \frac{42}{3} = 4\)

(ii) LCM of the given fractions

\(= \frac{\text{LCM of 9, 3, and 6}}{\text{HCF of 14, 7, and 21}} = \frac{18}{7}\)

CHECK: \(\frac{18}{7} \div \frac{9}{14} = \frac{18}{7} \times \frac{14}{9} = 4\);

\(\frac{18}{7} \div \frac{3}{7} = \frac{18}{7} \times \frac{7}{3} = 6\); \(\frac{18}{7} \div \frac{6}{21} = \frac{18}{7} \times \frac{21}{6} = 9\)

Teacher's Note

Understanding fractions helps students split bills at restaurants or divide recipe ingredients when cooking with family, making math practical and relatable.

Simplification And Word Problems

The order of operations follows the rule of BODMAS as detailed in the chapter on directed numbers.

Example 10: Simplify

\(2\frac{5}{6} - \left[3\frac{2}{3} - \frac{2}{5}\left\{\frac{2}{3} - \left(\frac{1}{2} - 2\frac{1}{7} - \frac{5}{14}\right)\right\}\right]\)

\(= \frac{17}{6} - \left[\frac{11}{3} - \frac{7}{5}\left\{\frac{3}{2} - \left(\frac{15}{7} - \frac{19}{14}\right)\right\}\right]\)

\(= \frac{17}{6} - \left[\frac{11}{3} - \frac{7}{5}\left\{\frac{3}{2} - \frac{30 - 19}{14}\right\}\right]\)

\(= \frac{17}{6} - \left[\frac{11}{3} - \frac{7}{5}\left\{\frac{35 - 15}{21}\right\}\right]\)

\(= \frac{17}{6} - \left[\frac{11}{3} - \frac{7}{5} \times \frac{20}{21}\right]\)

\(= \frac{17}{6} - \left[\frac{11}{3} - \frac{4}{3}\right]\)

\(= \frac{17}{6} - \frac{7}{3}\)

\(= \frac{17 - 14}{6}\)

\(= \frac{3}{6} = \frac{1}{2}\)

Example 11: \(\frac{2}{3}\) of the girls in a class are over 5 feet tall, \(\frac{1}{6}\) of the girls in that class are over 5 feet 3 inches tall. If 27 girls are between 5 feet and 5 feet 3 inches, how many girls are there in that class?

Let there be x number of girls in that class.

Given \(\frac{2}{3}x - \frac{1}{6}x = 27\)

\(\Rightarrow \frac{4x - x}{6} = 27\)

\(\Rightarrow \frac{3x}{6} = 27\)

\(\Rightarrow x = 27 \times 2 = 54\)

Thus, there are 54 girls in that class.

CHECK: Girls - 5': \(\frac{2}{3} \times 54 = 36\), Girls - 5'3": \(\frac{1}{6} \times 54 = 9\), Girls - 5', - 5'3": \(36 - 9 = 27\)

Teacher's Note

Word problems about groups of students help learners see how fractions apply to real classroom scenarios, making abstract math concepts concrete and memorable.

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