ICSE Class 8 Maths Chapter 07 Percentage

Chapter 7 - Percentage

Percent

The word percent means 'per hundred' or 'out of hundred'. Percent is denoted by the symbol %.

Thus, 76% means 76 out of hundred = \(\frac{76}{100}\).

39.4% means 39.4 out of hundred = \(\frac{39.4}{100}\).

So, the symbol % stands for one-hundredth = \(\frac{1}{100}\).

Percentage

Percentage is the numerator of a fraction with denominator 100.

In the fraction \(\frac{r}{100}\), percentage = r. It is written as r%.

Remarks

If the denominator of a fraction is not 100, then convert it into an equivalent fraction with denominator 100.

For example, consider the fraction \(\frac{27}{40}\). It can be written as

\[\frac{27}{40} = \frac{27}{40} \times \frac{100}{100} = \frac{27}{2} \times \frac{5}{100} = \frac{135}{2} = \frac{67.5}{100}\]

which is a fraction with denominator 100, therefore, percentage = 67.5.

In practice, the word percent and percentage both are used synonymously.

Key Points In Solving Problems On Percentages

To convert a percentage into a fraction, replace the % sign with \(\frac{1}{100}\) and reduce the fraction to simplest form.

To convert a fraction into a percentage, multiply the fraction by 100 and put the % sign.

To convert a percentage into a ratio, first convert the given percentage into a fraction and then write it as a ratio.

To convert a ratio into a percentage, first write the ratio as a fraction and then convert it into a percentage.

To convert a percentage into a decimal, remove the % sign and move the decimal point two places to the left.

To convert a decimal into a percentage, move the decimal point two places to the right (adding zeros if necessary) and put the % sign.

To find certain percentage of a given quantity:

\[x\% \text{ of a given quantity} = \frac{x}{100} \times \text{given quantity}\]

Expressing one quantity as a percentage of another quantity:

\[\text{To express } x \text{ as a percentage of } y, \text{ percentage} = \left(\frac{x}{y} \times 100\right)\%\]

Both quantities must be of same kind (in same units).

If certain percent of a quantity is given, then to find the quantity:

Let x% of a quantity be y, then

\[\frac{x}{100} \text{ of the quantity} = y \Rightarrow \text{required quantity} = \frac{y}{x} \times 100\]

\[\text{Thus, if } x\% \text{ of a quantity is } y, \text{ then quantity} = \frac{y}{x} \times 100\]

Percentage increase/decrease in a quantity:

\[\text{Percentage increase} = \left(\frac{\text{increase in quantity}}{\text{original quantity}} \times 100\right)\%\]

\[\text{Percentage decrease} = \left(\frac{\text{decrease in quantity}}{\text{original quantity}} \times 100\right)\%\]

If a quantity increases by x%, then

new quantity = original quantity + increase in the quantity

= original quantity + x% of original quantity

= original quantity + \(\frac{x}{100}\) of original quantity

= \(\left(1 + \frac{x}{100}\right)\) of original quantity

\[\text{Thus, if a quantity increases by } x\%, \text{ then } \text{new quantity} = \left(1 + \frac{x}{100}\right) \text{ of original quantity}\]

If a quantity decreases by x%, then

new quantity = original quantity - decrease in the quantity

= original quantity - x% of original quantity

= original quantity - \(\frac{x}{100}\) of original quantity

= \(\left(1 - \frac{x}{100}\right)\) of original quantity

\[\text{Thus, if a quantity decreases by } x\%, \text{ then } \text{new quantity} = \left(1 - \frac{x}{100}\right) \text{ of original quantity}\]

Teacher's Note

Percentages are used daily in calculating discounts at stores, understanding test scores, and determining tax amounts. Understanding these concepts helps make smart financial decisions.

Example 1

(i) Convert \(2\frac{1}{12}\)% into fraction.

(ii) Convert \(1\frac{11}{16}\) into percentage.

(iii) Convert 21 : 80 into percentage.

(iv) Express \(2\frac{7}{8}\)% as a decimal.

Solution

(i) \(2\frac{1}{12}\)% = \(\frac{25}{12}\)% = \(\frac{25}{12} \times \frac{1}{100}\) = \(\frac{1}{48}\).

(ii) \(1\frac{11}{16}\) = \(\frac{27}{16}\) = \(\left(\frac{27}{16} \times 100\right)\)% = \(\frac{27 \times 25}{4}\)% = \(\frac{675}{4}\)% = 168.75%.

(iii) 21 : 80 = \(\frac{21}{80}\) = \(\left(\frac{21}{80} \times 100\right)\)% = \(\frac{105}{4}\)% = 26.25%.

(iv) \(2\frac{7}{8}\)% = 2.875% = 0.02875.

Teacher's Note

Converting between fractions, percentages, and decimals is essential for interpreting nutrition labels, understanding weather forecasts, and reading financial reports.

Example 2

Convert \(\frac{5}{24}\) into a percentage correct to four significant figures.

Solution

\(\frac{5}{24}\) = \(\left(\frac{5}{24} \times 100\right)\)% = \(\frac{125}{6}\)% = 20.833%

= 20.83%, correct to four significant figures.

Teacher's Note

Rounding percentages to significant figures is important in scientific reporting and when precision affects decisions like medication dosages.

Example 3

(i) Find \(3\frac{1}{8}\)% of 75 kg.

(ii) What percent is 15 paise of 2 rupees 70 paise?

Solution

(i) \(3\frac{1}{8}\)% of 75 kg = \(\frac{25}{8}\)% of 75 kg = \(\frac{25}{100}\) of 75 kg = \(\left(\frac{25}{800} \times 75\right)\) kg

= \(\frac{75}{32}\) kg = \(2\frac{11}{32}\) kg.

(ii) 2 rupees 70 paise = (2 x 100 + 70) paise = 270 paise

Required percentage = \(\left(\frac{15}{270} \times 100\right)\)% = \(\frac{50}{9}\)% = \(5\frac{5}{9}\)%.

Teacher's Note

These calculations appear in real situations like calculating ingredient amounts in recipes and determining what percentage of your budget goes to different expenses.

Example 4

(i) If 9.5% of a number is 76, find the number.

(ii) Increase the number 240 by 15%.

(iii) Decrease the number 275 by 8%.

Solution

(i) Let the required number by x.

According to the given condition, 9.5% of x = 76

\(\Rightarrow \frac{9.5}{100} \times x = 76 \Rightarrow \frac{95}{1000} \times x = 76 \Rightarrow x = \frac{76 \times 1000}{95}\) = 800

Hence, the required number is 800.

(ii) New number = \(\left(1 + \frac{15}{100}\right) \times 240\)

= \(\frac{115}{100} \times 240\) = \(\frac{23}{20} \times 240\) = 276.

(iii) New number = \(\left(1 - \frac{8}{100}\right) \times 275\)

= \(\frac{92}{100} \times 275\) = \(\frac{23}{25} \times 275\) = 253.

Teacher's Note

These techniques are used when calculating salary increases, price changes after discounts, and weight loss or gain percentages in health contexts.

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