UK K12 GCSE/A-Level: Year 7 KS3 Mathematics - Percentage, Finding, Increasing, Decreasing

Learning objectives

By the end of this topic, students will be able to: - Find percentages of quantities using percentages as decimals and fractions - Increase or decrease quantities by given percentages - Apply percentage calculations to real-world contexts - Use percentages to compare and order quantities - Solve problems involving percentage changes, including percentage increases and decreases

Core concepts

To find a percentage of a quantity, we can use the formula:

Percentage = (Part/Whole) × 100

For example, if we want to find 25% of £100, we can use the formula:

25% = (£100 / £100) × 100 = 25

This means that 25% of £100 is £25.

To increase a quantity by a given percentage, we can multiply the quantity by the percentage as a decimal. For example, if we want to increase £100 by 25%, we can multiply £100 by 0.25 (which is 25% expressed as a decimal):

£100 × 0.25 = £25

So, £100 increased by 25% is £125 (£100 + £25).

To decrease a quantity by a given percentage, we can multiply the quantity by the percentage as a decimal. For example, if we want to decrease £100 by 25%, we can multiply £100 by 0.25 (which is 25% expressed as a decimal):

£100 × 0.25 = £25

So, £100 decreased by 25% is £75 (£100 - £25).

Worked examples

Example 1: Finding a percentage of a quantity

A book costs £80. If we pay a 10% deposit, how much will we pay?

To find the deposit, we can use the formula:

Percentage = (Part/Whole) × 100

In this case, the part is the deposit, and the whole is the cost of the book (£80). We want to find 10% of £80:

10% = (£80 / £80) × 100 = 10

So, the deposit is 10% of £80, which is £8.

Example 2: Increasing a quantity by a given percentage

A shop sells a t-shirt for £25. If the price increases by 15%, how much will the t-shirt cost?

To increase the price by 15%, we can multiply the original price by 0.15 (which is 15% expressed as a decimal):

£25 × 0.15 = £3.75

So, the new price of the t-shirt is £25 + £3.75 = £28.75.

Example 3: Decreasing a quantity by a given percentage

A company has £1000 in the bank. If they lose 20% of their money, how much will they have left?

To decrease the amount by 20%, we can multiply the original amount by 0.20 (which is 20% expressed as a decimal):

£1000 × 0.20 = £200

So, the company will have £1000 - £200 = £800 left.

Common misconceptions

• Students may confuse the formula for finding a percentage of a quantity with the formula for increasing or decreasing a quantity.
• Students may not realize that increasing or decreasing a quantity by a percentage is the same as multiplying the quantity by the percentage as a decimal.
• Students may not understand that percentage changes can be negative (e.g., a decrease).

Exam tips

• Make sure to read the question carefully and understand what is being asked.
• Use the formula for finding a percentage of a quantity, increasing or decreasing a quantity, or comparing quantities.
• Check your units and make sure they are consistent.
• Use real-world contexts to help you understand the problem.

MCQs with explanations

Question 1: Finding a percentage of a quantity [F]

A box of chocolates costs £20. If we pay a 15% sales tax, how much will we pay?

A) £20 B) £23 C) £24 D) £25

Correct answer: B) £23

Why the distractors fail: - A) £20 is the original price, not the price with tax. - C) £24 is 20% more than the original price, not 15%. - D) £25 is the original price plus 25%, not 15%.

Question 2: Increasing a quantity by a given percentage [H]

A company increases the price of a product by 12%. If the original price is £50, what is the new price?

A) £56 B) £55 C) £54 D) £52

Correct answer: A) £56

Why the distractors fail: - B) £55 is the original price plus 10%, not 12%. - C) £54 is the original price minus 4%, not plus 12%. - D) £52 is the original price minus 4%, not plus 12%.

Question 3: Decreasing a quantity by a given percentage [F]

A shop sells a t-shirt for £25. If the price decreases by 10%, how much will the t-shirt cost?

A) £22.50 B) £23 C) £24 D) £25

Correct answer: A) £22.50

Why the distractors fail: - B) £23 is the original price minus 7%, not 10%. - C) £24 is the original price plus 4%, not minus 10%. - D) £25 is the original price, not the price with the decrease.

Question 4: Comparing quantities [H]

A company has £1000 in the bank. If they lose 20% of their money and then gain 15% of the remaining amount, how much will they have?

A) £800 B) £850 C) £900 D) £950

Correct answer: B) £850

Why the distractors fail: - A) £800 is the original amount minus 20%, not the final amount. - C) £900 is the original amount plus 10%, not the final amount. - D) £950 is the original amount plus 50%, not the final amount.

Question 5: Percentage changes [H]

A company increases the price of a product by 12% and then decreases the price by 8%. What is the overall percentage change?

A) -4% B) -2% C) 4% D) 2%

Correct answer: B) -2%

Why the distractors fail: - A) -4% is the decrease, not the overall change. - C) 4% is the increase, not the overall change. - D) 2% is the overall change, but it is a decrease, not an increase.

Short-answer questions

1. A shop sells a t-shirt for £25. If the price increases by 15%, how much will the t-shirt cost?

(Answer should be £28.75)

1. A company has £1000 in the bank. If they lose 20% of their money, how much will they have left?

(Answer should be £800)

1. A box of chocolates costs £20. If we pay a 15% sales tax, how much will we pay?

(Answer should be £23)

1. A company increases the price of a product by 12% and then decreases the price by 8%. What is the overall percentage change?

(Answer should be -2%)

1. A shop sells a t-shirt for £25. If the price decreases by 10%, how much will the t-shirt cost?

(Answer should be £22.50)