ICSE Class 8 Maths Arithmetical Problems Chapter 03 Percentage

Percentage

Let us briefly review what you have read about percentage in your previous classes. Per cent (or per centum or per centuri) means 'in a hundred'. Thus, when we say 15 per cent, we mean 15 out of 100. Symbolically, this is written as 15%.

Some Conversion Rules of Percentage

Rule 1 To convert a percentage into a fraction, write the number as the numerator and 100 as the denominator and then reduce this fraction to its simplest form. You can also express the fraction as a ratio.

Examples (i) \(12\% = \frac{12}{100} = \frac{3}{25}\) (or 3 : 25) (ii) \(39\% = \frac{39}{100}\) (or 39 : 100)

Rule 2 To convert a fraction into a percentage, multiply the fraction by 100 and add the % symbol. Since ratios are essentially fractions, you can also express a ratio as a percentage.

Examples (i) \(\frac{4}{5}\) (or 4 : 5) = \(\left(\frac{4}{5} \times 100\right)\% = 80\%\)

(ii) \(\frac{1}{8}\) (or 1 : 8) = \(\left(\frac{1}{8} \times 100\right)\% = 12.5\%\)

Rule 3 To convert a percentage into a decimal, remove the % sign and shift the decimal point in the number by two places to the left.

Examples (i) \(3\% = \left(\frac{3}{100}\right) = 0.03\) (ii) \(1.7\% = \left(\frac{1.7}{100}\right) = 0.017\)

Rule 4 To convert a decimal into a percentage, shift the decimal point in the number by two places to the right and add the % sign.

Examples (i) \(0.25 = \left(\frac{25}{100}\right) = 25\%\) (ii) \(1.35 = \left(\frac{135}{100}\right) = 135\%\)

(iii) \(0.7 = 0.70 = \left(\frac{70}{100}\right) = 70\%\)

Rule 5 To find a given percentage of a quantity, write the percentage as a fraction and multiply the quantity by this fraction.

\[x\% \text{ of } y = \frac{x}{100} \times y\]

Examples (i) \(36\% \text{ of } 400 = \frac{36}{100} \times 400 = 144\)

(ii) \(25\% \text{ of } 40 \text{ L} = \frac{25}{100} \times 40 \text{ L} = 10 \text{ L}\)

(iii) \(56\% \text{ of } 20 \text{ kg} = \frac{56}{100} \times 20 \text{ kg} = 11.2 \text{ kg}\)

Rule 6 To express a given quantity as a percentage of another quantity of the same kind, divide the given quantity by the other quantity, multiply the result by 100 and add the % sign.

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

Examples (i) 25 as percentage of 80 = \(\left(\frac{25}{80} \times 100\right)\% \text{ of } 80 = 31.25\% \text{ of } 80\)

(ii) 46 g as percentage of 230 g = \(\left(\frac{46}{230} \times 100\right)\% \text{ of } 230 \text{ g} = 20\% \text{ of } 230 \text{ g}\)

Rule 7 To find a quantity from a given percentage of the quantity, express the given percentage as a fraction and divide the given quantity by this fraction.

Examples (i) If 25% of a number is 136 then

the number \(= 136 \div \frac{25}{100} = 136 \times \frac{100}{25} = 136 \times 4 = 544.\)

(ii) If 30% of a quantity is 405 g then

the quantity \(= 405 \text{ g} \div \frac{30}{100} = 405 \text{ g} \times \frac{100}{30} = 1350 \text{ g} = 1.35 \text{ kg}.\)

Rule 8 To find the percentage change in a quantity, write the change in the quantity as the numerator and the original quantity as the denominator, and then multiply the fraction by 100.

\[\text{Percentage change} = \frac{\text{change in the quantity}}{\text{original quantity}} \times 100\]

Example If the number of tigers in India was 1600 in the year 2008 and 1400 in year 2009 then decrease in the number of tigers = 1600 - 1400 = 200.

Therefore, percentage change (or decrease) = \(\frac{200}{1600} \times 100 = 12.5\).

If a quantity increases by x%,

\[\text{new quantity} = \left(1 + \frac{x}{100}\right) \times \text{original quantity, original quantity} = \frac{\text{new quantity}}{1 + \frac{x}{100}}\]

If a quantity decreases by x%,

\[\text{new quantity} = \left(1 - \frac{x}{100}\right) \times \text{original quantity, original quantity} = \frac{\text{new quantity}}{1 - \frac{x}{100}}\]

Teacher's Note

When shopping, if an item is on sale for 30% off, we use percentage calculations to find the final price. This real-world application helps students understand why these rules matter in everyday transactions.

Worked Examples

Example (i) If 30 increases by 42%, what will be the new number?

(ii) If 56 decreases by 40%, what will be the new number?

Solution

(i) The new number \(= \left(1 + \frac{42}{100}\right) \times 30 = \frac{142}{100} \times 30 = \frac{426}{10} = 42.6.\)

(ii) The new number \(= \left(1 - \frac{40}{100}\right) \times 56 = \frac{60}{100} \times 56 = \frac{336}{10} = 33.6.\)

Alternatively

(i) The new number is 42% more than 30.

Therefore, the new number = (100 + 42)% of 30 = 142% of 30 = 1.42 × 30 = 42.6.

(ii) The new number is 40% less than 56.

Therefore, the new number = (100 - 40)% of 56 = 60% of 56 = 0.6 × 56 = 33.6.

Example The price of petrol increased by 8% to Rs 54 per litre. Find the old price.

Solution The old price = \(\frac{\text{Rs } 54}{1 + \frac{8}{100}} = \frac{\text{Rs } 54}{1 + \frac{2}{25}} = \frac{25}{27} \times \text{Rs } 54 = \text{Rs } 50\) (per litre).

Teacher's Note

When fuel prices change, understanding how to calculate the original price from the new price helps consumers and business owners make informed financial decisions about their budgets.

Solved Examples

Example 1 3224 students appeared in an examination and 75% passed. How many failed?

Solution The number of students who passed = \(75\% \text{ of } 3224 = \frac{75}{100} \times 3224 = 2418.\)

Hence, the number of students who failed = 3224 - 2418 = 806.

Alternative method

Since 75% passed, (100 - 75)%, i.e., 25% failed.

Therefore, the number of students who failed = \(25\% \text{ of } 3224 = \frac{25}{100} \times 3224 = 806.\)

Example 2 Rakesh gets a salary of Rs 40,000 per month. His salary increases by 8% every year. Find (i) the increase in his (monthly) salary after a year and (ii) his monthly salary after two years.

Solution

(i) The increase in his salary after a year = 8% of Rs 40000

\(= \frac{8}{100} \times \text{Rs } 40000 = \text{Rs } 3200.\)

(ii) His monthly salary after 1 year = Rs 40000 + Rs 3200 = Rs 43,200.

The next increase = 8% of Rs 43200 = \(\frac{8}{100} \times \text{Rs } 43200 = \text{Rs } 3456.\)

Hence, his monthly salary after 2 years = Rs 43200 + Rs 3456 = Rs 46,656.

Example 3 The price of a notebook was reduced by 10% in 2009. In 2010, it was increased by 10%. Find the percentage change in the price in two years.

Solution Let the price of the notebook = Rs 100.

Price after 10% reduction = Rs 100 - 10% of Rs 100 = \(\text{Rs } 100 - \frac{10}{100} \times \text{Rs } 100\)

= Rs 100 - Rs 10 = Rs 90.

Price after 10% increase = Rs 90 + 10% of Rs 90 = \(\text{Rs } 90 + \frac{10}{100} \times \text{Rs } 90 = \text{Rs } 99.\)

Thus, the price of the notebook decreased from Rs 100 to Rs 99, i.e., by Re 1 in two years.

Therefore, the percentage decrease in price after two years = \(\frac{1}{100} \times 100 = 1.\)

Teacher's Note

When a product's price changes multiple times, calculating the net percentage change shows that two equal percentage changes (one increase, one decrease) do not result in the original price due to different base values.

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