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Number - Fractions, decimals, and percentages

Grade 11IGCSE

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

🔑Concepts

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Converting between fractions, decimals, and percentages (FDP).

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Arithmetic operations with fractions: Addition and subtraction require a common denominator; multiplication involves multiplying numerators and denominators; division uses the reciprocal ('keep, change, flip').

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Terminating vs. Recurring decimals: Recognizing that fractions with denominators whose prime factors are only 2 and 5 terminate.

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Percentage change: Calculating the increase or decrease relative to the original value.

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Reverse Percentages: Finding the original value after a percentage change has occurred (working backwards).

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Compound Interest: Calculating the total amount after a repeated percentage increase over time.

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Converting recurring decimals into fractions using algebraic methods.

📐Formulae

Percentage Change=DifferenceOriginal Value×100\text{Percentage Change} = \frac{\text{Difference}}{\text{Original Value}} \times 100

New Value=Original Value×Multiplier\text{New Value} = \text{Original Value} \times \text{Multiplier}

Original Value=New ValueMultiplier\text{Original Value} = \frac{\text{New Value}}{\text{Multiplier}}

Compound Interest Amount=P(1+r100)n\text{Compound Interest Amount} = P(1 + \frac{r}{100})^n

Multiplier for x% increase=1+x100\text{Multiplier for } x\% \text{ increase} = 1 + \frac{x}{100}

Multiplier for x% decrease=1−x100\text{Multiplier for } x\% \text{ decrease} = 1 - \frac{x}{100}

💡Examples

Problem 1:

Calculate 213−1342\frac{1}{3} - 1\frac{3}{4}, giving your answer as a simplified fraction.

Solution:

73−74=2812−2112=712\frac{7}{3} - \frac{7}{4} = \frac{28}{12} - \frac{21}{12} = \frac{7}{12}

Explanation:

Convert mixed numbers to improper fractions. Find a common denominator (12). Subtract the numerators and keep the denominator.

Problem 2:

A laptop is sold for $595 after a 15% discount. Calculate the original price of the laptop.

Solution:

Multiplier=1−0.15=0.85\text{Multiplier} = 1 - 0.15 = 0.85. Original Price=595÷0.85=700\text{Original Price} = 595 \div 0.85 = 700.

Explanation:

This is a reverse percentage problem. A 15% discount means the sale price is 85% of the original. Divide the sale price by the decimal multiplier (0.85) to find the 100% value.

Problem 3:

Write the recurring decimal 0.27˙0.2\dot{7} as a fraction in its simplest form.

Solution:

Let x=0.2777...x = 0.2777..., 10x=2.777...10x = 2.777..., 100x=27.777...100x = 27.777.... 100x−10x=27.777...−2.777...⇒90x=25⇒x=2590=518100x - 10x = 27.777... - 2.777... \Rightarrow 90x = 25 \Rightarrow x = \frac{25}{90} = \frac{5}{18}.

Explanation:

Use algebra to eliminate the recurring part. Multiply xx by powers of 10 to create two equations where the decimals after the point match, then subtract the equations and solve for xx.

Problem 4:

Invest $4000 at a rate of 3% per year compound interest. Calculate the total interest earned after 5 years.

Solution:

Total Amount=4000×(1.03)5≈4637.10\text{Total Amount} = 4000 \times (1.03)^5 \approx 4637.10. Interest=4637.10−4000=637.10\text{Interest} = 4637.10 - 4000 = 637.10.

Explanation:

Use the compound interest formula P(1+r)nP(1+r)^n. To find only the interest, subtract the original principal from the final total amount.