How to Solve: Percentage Increase and Decrease

How to Solve: Percentage Increase and Decrease

GCSE & A-Level Maths

Introduction

"Mastering percentage increase and decrease lets you tackle real-life money problems—salary raises, discounts, inflation—and secures 5-10 marks in your GCSE/A-Level exam. One wrong step here could cost you a grade, but follow this guide, and you’ll solve them in under 60 seconds."

What You Need To Know First

1. Basic percentages – How to find 10%, 50%, etc., of a number.
2. Decimal multiplication – Converting percentages to decimals (e.g., 20% = 0.2).
3. Reverse percentages – Finding the original value after a change (covered later).

Key Vocabulary

Formulas To Know

1. Percentage Increase

Formula: New Value = Original Value × (1 + Percentage Increase as a Decimal) Variables: - Original Value = Starting amount. - Percentage Increase = % added (e.g., 15% = 0.15). - New Value = Result after increase.

MEMORISE THIS – Not given on exam sheets.

2. Percentage Decrease

Formula: New Value = Original Value × (1 – Percentage Decrease as a Decimal) Variables: - Original Value = Starting amount. - Percentage Decrease = % removed (e.g., 10% = 0.10). - New Value = Result after decrease.

MEMORISE THIS – Not given on exam sheets.

3. Finding the Percentage Change

Formula: Percentage Change = [(New Value – Original Value) / Original Value] × 100 Variables: - Original Value = Starting amount. - New Value = Amount after change. - Percentage Change = % increase or decrease.

GIVEN ON EXAM SHEET (but memorise for speed).

4. Reverse Percentage (Finding Original Value)

Formula: Original Value = New Value / (1 ± Percentage Change as a Decimal) - Use + if it was an increase. - Use – if it was a decrease.

MEMORISE THIS – Not given on exam sheets.

Step-by-Step Method

For Percentage Increase/Decrease Problems:

1. Identify if it’s an increase or decrease.
2. Convert the percentage to a decimal (e.g., 25% → 0.25).
3. Adjust the decimal:
4. Increase: Add to 1 (e.g., 1 + 0.25 = 1.25).
5. Decrease: Subtract from 1 (e.g., 1 – 0.25 = 0.75).
6. Multiply the original value by the adjusted decimal to get the new value.
7. Check your answer makes sense (e.g., a decrease should be smaller).

For Reverse Percentage Problems:

1. Identify if the change was an increase or decrease.
2. Convert the percentage to a decimal.
3. Adjust the decimal:
4. Increase: 1 + decimal (e.g., 1.20 for 20% increase).
5. Decrease: 1 – decimal (e.g., 0.80 for 20% decrease).
6. Divide the new value by the adjusted decimal to find the original.
7. Verify by recalculating the change.

Worked Examples

Example 1 – Basic (Percentage Increase)

Question: A laptop costs £600. Its price increases by 15%. What is the new price?

Solution: 1. Increase → Use 1 + 0.15 = 1.15. 2. Multiply: £600 × 1.15 = £690.

Answer: £690.

What we did and why: - We converted 15% to 0.15 and added it to 1 to get the multiplier (1.15). - Multiplying by 1.15 gives the new value after a 15% increase.

Example 2 – Medium (Percentage Decrease)

Question: A jacket is reduced by 30% in a sale. It now costs £56. What was the original price?

Solution: 1. Decrease → Use 1 – 0.30 = 0.70. 2. Reverse percentage: Original = £56 ÷ 0.70 = £80.

Answer: £80.

What we did and why: - We converted 30% to 0.30 and subtracted from 1 to get the multiplier (0.70). - Since £56 is after the decrease, we divided by 0.70 to find the original price.

Example 3 – Exam-Style (Disguised Problem)

Question: A phone’s value decreases by 20% in the first year and then by 10% in the second year. If it’s worth £360 after 2 years, what was its original price?

Solution: 1. First decrease (20%): Multiplier = 1 – 0.20 = 0.80. 2. Second decrease (10%): Multiplier = 1 – 0.10 = 0.90. 3. Combined multiplier: 0.80 × 0.90 = 0.72. 4. Reverse percentage: Original = £360 ÷ 0.72 = £500.

Answer: £500.

What we did and why: - We multiplied the multipliers (0.80 × 0.90) to find the total change over 2 years. - Then, we divided the final value by the combined multiplier to find the original price.

Common Mistakes

Exam Traps

1-Minute Recap

"Here’s the fastest way to solve percentage increase/decrease problems: 1. For increases: Multiply by (1 + decimal) (e.g., 15% = 1.15). 2. For decreases: Multiply by (1 – decimal) (e.g., 10% = 0.90). 3. For reverse %: Divide by the multiplier (e.g., new price ÷ 0.80 for a 20% decrease). 4. For multi-step %: Multiply the multipliers (e.g., 20% then 10% = 0.80 × 0.90). 5. Always check: Does the answer make sense? A decrease should be smaller, an increase larger.

Now go crush those exam questions!