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Fractions in Disguise - Using Percentages

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A percentage is a fraction with a denominator of 100100. The term 'percent' comes from the Latin 'per centum', meaning 'per hundred'.

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To convert a fraction ab\frac{a}{b} into a percentage, multiply the fraction by 100100 and add the %\% sign: ab×100%\frac{a}{b} \times 100\%.

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To convert a percentage x%x\% into a fraction, remove the %\% sign and divide by 100100: x100\frac{x}{100}. Then, simplify the fraction to its lowest terms.

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Decimals can also be viewed as fractions in disguise. To convert a decimal to a percentage, multiply by 100100 (move the decimal point two places to the right).

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Percentage change (increase or decrease) is calculated by finding the absolute change and dividing it by the original value, then multiplying by 100100.

📐Formulae

Fraction to Percentage=(NumeratorDenominator×100)%\text{Fraction to Percentage} = \left( \frac{\text{Numerator}}{\text{Denominator}} \times 100 \right)\%

Percentage of a Quantity=Rate100×Total Quantity\text{Percentage of a Quantity} = \frac{\text{Rate}}{100} \times \text{Total Quantity}

Percentage Change=Final Value−Initial ValueInitial Value×100%\text{Percentage Change} = \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \times 100\%

Value after Increase=Original Value×(1+Percentage Increase100)\text{Value after Increase} = \text{Original Value} \times \left( 1 + \frac{\text{Percentage Increase}}{100} \right)

💡Examples

Problem 1:

Convert the fraction 38\frac{3}{8} into a percentage.

Solution:

38×100%=3008%=37.5%\frac{3}{8} \times 100\% = \frac{300}{8}\% = 37.5\%

Explanation:

To convert any fraction to a percentage, we multiply it by 100100 and simplify.

Problem 2:

A basket contains 4040 fruits. If 15%15\% of them are oranges, how many oranges are there?

Solution:

15% of 40=15100×40=320×40=3×2=615\% \text{ of } 40 = \frac{15}{100} \times 40 = \frac{3}{20} \times 40 = 3 \times 2 = 6

Explanation:

We express the percentage as a fraction 15100\frac{15}{100} and multiply it by the total quantity.

Problem 3:

The price of a book decreased from Rs 250250 to Rs 200200. Find the percentage decrease.

Solution:

Decrease in price: 250−200=50250 - 200 = 50. Percentage decrease: 50250×100%=15×100%=20%\frac{50}{250} \times 100\% = \frac{1}{5} \times 100\% = 20\%

Explanation:

First find the absolute decrease, then divide by the original price (250250) and multiply by 100100.

Problem 4:

Calculate the difference between 80,000,00080,000,000 and 34,567,89234,567,892 and express the result as a percentage of 80,000,00080,000,000.

Solution:

80000000−3456789245432108\begin{array}{r} 80000000 \\ -34567892 \\ \hline 45432108 \end{array} Percentage: 4543210880000000×100%=56.790135%\frac{45432108}{80000000} \times 100\% = 56.790135\%

Explanation:

The subtraction is performed first using vertical arithmetic, and then the result is converted to a percentage by comparing it to the original value.