NCERT Class 8 Mathematics Ganita Prakash Part 2 Chapter 01 Fractions in Disguise PDF Download

Chapter 1: Fractions in Disguise

1.1 Fractions as Percentages

You might have heard statements like, "Mega Sale - up to 50% off!" or "Hiya scored 83% in her board exams".

Do you know what the symbol '%' means?

This symbol is read as per cent.

The word 'per cent' is derived from the Latin phrase per centum, meaning 'by the hundred' or 'out of hundred'.

So, 25 per cent (25%) means 25 out of every 100 - like 25 people out of 100, 25 rupees out of 100 rupees, or 25 marks out of 100 marks.

If we say 50% of some quantity s, it means

\[ 50\% = 50 \times \frac{1}{100} \times s \text{ (50 times the unit fraction of s)} \]

\[ = \frac{50}{100} \times s = \frac{1}{2}s \]

Thus, percentages are simply fractions where the denominator is 100. Examples:

\[ 20\% = \frac{20}{100} = \frac{2}{10} = \frac{1}{5} \]

\[ 33\% = \frac{33}{100} \]

We saw that percentages are just fractions. Given any fraction, can we express it as a percentage? Yes, let us see how.

Expressing Fractions as Percentages

Example 1: Surya wants to use a deep orange colour to capture the sunset. He mixes some red paint and yellow paint to make this colour. The red paint makes up \( \frac{3}{4} \) of this mixture. What percentage of the colour is made with red?

\( \frac{3}{4} \) is 3 out of every 4.

That is, 6 out of every 8 (equivalent fraction).

That is, 30 out of every 40.

That is, 75 out of every 100.

This means 75%.

\[ \frac{3}{4} = \frac{6}{8} = \frac{30}{40} = \frac{75}{100} \]

Explaining this in a different way: To express \( \frac{3}{4} \) as a percentage, we need to find its equivalent fraction with 100 as the denominator. Two ways of going ahead are shown below.

So, \( \frac{3}{4} \) can be expressed as 75%.

[Figure: Bar model diagram showing the equivalence between 3/4 and 75%, See in your textbook]

Can you tell what percentage of the colour was made using yellow?

Teacher's Note

When converting a fraction to a percentage, you multiply by 100. Remember: \( \frac{3}{4} \times 100 = 75 \), which means the fraction becomes 75%. The key is that the percentage symbol '%' already means "out of 100", so you're essentially finding how many parts out of 100 your fraction represents.

Example 2: Surya won some prize money in a contest. He wants to save \( \frac{2}{5} \) of the money to purchase a new canvas. Express this quantity as a percentage.

Try to understand the different methods for solving this problem, as shown below.

Try completing Method 3 by filling the boxes.

Method 3

[Figure: Bar model diagram showing fifths mapped to percentage scale from 0% to 100%, with savings for canvas highlighted, See in your textbook]

Isn't it the same as finding \( \frac{2}{5} \)th of 100?

Several problems in mathematics can be approached and solved in different ways. While the method you came up with may be dear to you, it can be amusing and enriching to know how others thought about it.

A fraction is of a unit, while a percentage is per 100. Therefore, to express a fraction as a percentage, we can just multiply the fraction by 100.

Example 3: Given a percentage, can you express it as a fraction? For example, express 24% as a fraction.

Since a percentage is a fraction, 24% is the same as \( \frac{24}{100} \).

We can find other equivalent forms of \( \frac{24}{100} = \frac{12}{50} = \frac{6}{25} = \frac{48}{200} \).

In general, we can say that a percentage, z%, can be expressed by any of the fractions that are equivalent to \( \frac{z}{100} \).

Teacher's Note

Different fractions can represent the same percentage. For instance, 24% equals \( \frac{24}{100} \), \( \frac{12}{50} \), and \( \frac{6}{25} \) all at once. On a test, always reduce the fraction to its simplest form unless the question asks otherwise. So write 24% as \( \frac{6}{25} \), not \( \frac{24}{100} \).

Figure it Out

1. Express the following fractions as percentages.
   1. \( \frac{3}{5} \)
   2. \( \frac{7}{14} \)
   3. \( \frac{9}{20} \)
   4. \( \frac{72}{150} \)
   5. \( \frac{1}{3} \)
   6. \( \frac{5}{11} \)
2. Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white?
   1. 10%
   2. 15%
   3. 25%
   4. 60%
   5. 40%
   6. None of these

Key Points

• Percent means "out of 100". A percentage is a fraction with denominator 100.
• To convert a fraction to a percentage, multiply the fraction by 100.
• To convert a percentage to a fraction, write it as a fraction with denominator 100, then simplify.
• Any fraction can be expressed as a percentage using equivalent fractions or by the formula: \( \frac{a}{b} \times 100 = \text{percentage} \).

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