Number

Fractions, Decimals and Percentages

Year 10 · Year 11

✓By the end of this lesson students will be able to convert fluently between fractions, decimals, and percentages.
✓By the end of this lesson students will be able to calculate percentage increase and decrease, including profit and loss.
✓By the end of this lesson students will be able to solve problems involving reverse percentages to find an original amount.
✓By the end of this lesson students will be able to calculate compound interest and other repeated percentage changes over time.

Key concepts

Conversions between Fractions, Decimals and Percentages

Fractions, decimals, and percentages are different ways of representing parts of a whole. Being able to convert between them is a fundamental skill. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100. To convert a decimal to a fraction, write the decimal as a fraction over a power of 10 (e.g., 0.75 = 75/100) and then simplify. To convert a percentage to a fraction, write it over 100 and simplify.

Percentage Change (Increase and Decrease)

Percentage change describes the relative change in a quantity. It can be an increase or a decrease. To calculate percentage change, find the difference between the new amount and the original amount (the 'change'), then divide this change by the original amount, and finally multiply by 100 to express it as a percentage. A positive result indicates an increase, and a negative result indicates a decrease. The multiplier method is often used for efficiency: to increase an amount by x%, multiply by (1 + x/100); to decrease an amount by x%, multiply by (1 - x/100).

Percentage Change = (Change / Original Amount) × 100%

Reverse Percentages

Reverse percentages involve finding the original amount before a percentage increase or decrease was applied. If an amount has been increased by x%, the new amount represents (100 + x)% of the original. If it has been decreased by x%, the new amount represents (100 - x)% of the original. To find the original amount, you divide the new amount by the corresponding multiplier (1 + x/100 for increase, or 1 - x/100 for decrease).

Original Amount = New Amount / Multiplier

Compound Interest and Repeated Percentage Change

Compound interest is interest calculated on the initial principal and also on the accumulated interest from previous periods. It means that the interest earned in each period is added to the principal for the next period, leading to exponential growth. This concept also applies to other situations involving repeated percentage changes, such as depreciation or population growth. The formula calculates the total amount after a certain number of periods.

A = P(1 + r/100)^n

Key facts to remember

1To convert a decimal to a percentage, multiply by 100.
2To convert a percentage to a decimal, divide by 100.
3To convert a fraction to a decimal, divide the numerator by the denominator.
4Percentage change = (Change / Original Amount) × 100%.
5A multiplier of (1 + decimal percentage) is used for percentage increase.
6A multiplier of (1 - decimal percentage) is used for percentage decrease.
7For reverse percentage problems, divide the new amount by the relevant multiplier to find the original amount.
8The compound interest formula is A = P(1 + r/100)^n, where A is the final amount, P is the principal, r is the annual interest rate, and n is the number of years.

Worked examples

Example 1

a) Convert 5/8 to a decimal and a percentage. b) Convert 0.36 to a percentage and a simplified fraction. c) Convert 72% to a decimal and a simplified fraction.

Ia) To convert 5/8 to a decimal, divide 5 by 8: 5 ÷ 8 = 0.625.

II To convert 0.625 to a percentage, multiply by 100: 0.625 × 100 = 62.5%.

IIIb) To convert 0.36 to a percentage, multiply by 100: 0.36 × 100 = 36%.

IV To convert 0.36 to a fraction, write it as 36/100. Simplify by dividing numerator and denominator by their highest common factor, which is 4: 36 ÷ 4 = 9, 100 ÷ 4 = 25. So, 0.36 = 9/25.

Vc) To convert 72% to a decimal, divide by 100: 72 ÷ 100 = 0.72.

VI To convert 72% to a fraction, write it as 72/100. Simplify by dividing numerator and denominator by their highest common factor, which is 4: 72 ÷ 4 = 18, 100 ÷ 4 = 25. So, 72% = 18/25.

Answer

a) 0.625, 62.5% b) 36%, 9/25 c) 0.72, 18/25

Example 2

A shop buys a coat for £80 and sells it for £104. Calculate the percentage profit.

IFirst, calculate the profit (the change in price): Profit = Selling Price - Cost Price = £104 - £80 = £24.

IINext, use the percentage change formula: Percentage Profit = (Profit / Original Cost) × 100%.

IIISubstitute the values: Percentage Profit = (£24 / £80) × 100%.

IVCalculate the fraction: 24/80 = 3/10 = 0.3.

VMultiply by 100: 0.3 × 100 = 30%.

Answer

The percentage profit is 30%.

Remember that 'original amount' for profit/loss is always the cost price.

Example 3

After a 15% price increase, a television now costs £345. What was its original price?

IIdentify the percentage increase: 15%.

IICalculate the multiplier for a 15% increase: 1 + (15/100) = 1 + 0.15 = 1.15.

IIIThe new price (£345) represents 115% of the original price.

IVTo find the original price, divide the new price by the multiplier: Original Price = New Price / Multiplier.

VOriginal Price = £345 / 1.15.

VIPerform the division: £345 / 1.15 = £300.

Answer

The original price of the television was £300.

Always divide by the multiplier for reverse percentage problems.

Example 4

Emily invests £4000 in a savings account that pays 2.5% compound interest per annum. How much will her investment be worth after 3 years? Give your answer to the nearest penny.

IIdentify the principal (P): P = £4000.

IIIdentify the annual interest rate (r): r = 2.5%.

IIIIdentify the number of years (n): n = 3.

IVUse the compound interest formula: A = P(1 + r/100)^n.

VSubstitute the values: A = 4000(1 + 2.5/100)^3.

VICalculate the term in the bracket: 1 + 0.025 = 1.025.

VIICalculate the multiplier raised to the power: (1.025)^3 = 1.076890625.

VIIIMultiply by the principal: A = 4000 × 1.076890625 = 4307.5625.

9Round the answer to the nearest penny (two decimal places): £4307.56.

Answer

Emily's investment will be worth £4307.56 after 3 years.

Do not round intermediate steps; only round the final answer as specified.

Common mistakes

✗Confusing percentage change with reverse percentage problems. Always identify if you are finding the change or the original amount.
✗Incorrectly calculating the multiplier (e.g., adding/subtracting the percentage directly instead of its decimal equivalent).
✗Forgetting to divide by the original amount when calculating percentage change.
✗Using the simple interest formula instead of the compound interest formula for repeated percentage changes.
✗Rounding intermediate calculations too early, leading to inaccuracies in the final answer, especially in compound interest problems.

Exam tips

★Read the question carefully to determine whether you need to calculate a percentage change, a reverse percentage, or compound interest.
★The multiplier method is highly efficient for percentage increase/decrease and reverse percentage problems. Practise using it.
★Show all your working steps clearly, especially for multi-step problems like compound interest, to gain partial marks even if your final answer is incorrect.
★Always check if your answer is reasonable. For example, an original price should be higher than a discounted price, and an investment with compound interest should grow over time.
★Pay close attention to the required degree of accuracy for your final answer, e.g., 'to the nearest penny' means two decimal places.

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