Percentage Change Problems

One formula, many problems

Most percentage problems beyond simple increases and decreases are really the same calculation wearing different costumes — profit, loss, error, growth. The single tool you need is the percentage change formula:

percentage change = (change ÷ original) × 100

The golden rule: the number you divide by is always the original (starting) value. Get that right and the rest follows.

Percentage profit and loss

In money problems the "original" is the cost price (what you paid). The change is profit or loss.

• Profit = selling price − cost price
• Loss = cost price − selling price
• percentage profit/loss = (profit or loss ÷ cost price) × 100

Worked example 1: profit You buy a bike for $40 and sell it for $50.

1. Profit = 50 − 40 = $10.
2. (10 ÷ 40) × 100 = 25% profit.

Worked example 2: loss A phone bought for $300 is sold for $240.

1. Loss = 300 − 240 = $60.
2. (60 ÷ 300) × 100 = 20% loss.

Dividing by the cost price (not the selling price) is the step people most often get wrong.

Percentage error

When you measure or estimate, the percentage error says how big the mistake is compared with the true value.

percentage error = (size of error ÷ true value) × 100

Worked example 3: a wall is truly 50 cm long but is measured as 51 cm.

1. Error = 51 − 50 = 1 cm.
2. (1 ÷ 50) × 100 = 2% error.

A 1 cm error on a 50 cm wall (2%) is far worse than a 1 cm error on a 500 cm wall (0.2%) — percentage error reveals this, while the raw error does not.

Comparing two percentage changes

Raw differences can mislead; percentages let you compare fairly.

Change	Difference	Percentage increase
20 → 25	+5	(5 ÷ 20) × 100 = 25%
50 → 60	+10	(10 ÷ 50) × 100 = 20%

The second jumped by more in raw terms (+10 vs +5), but as a percentage the first is bigger (25% vs 20%) because it started from a smaller base.

The "rise then fall" trap

Suppose a price rises 10%, then falls 10%. Does it return to the start? No.

Using multipliers from percentage increase and decrease:

• After the rise: × 1.10
• After the fall: × 0.90
• Combined: 1.10 × 0.90 = 0.99

That is a 1% overall decrease. The fall is calculated on the larger amount, so it removes more than the rise added. A $100 item becomes $110, then $99 — not $100.

Growth over repeated changes

Repeated percentage changes multiply, they do not add. A population growing 5% per year for 3 years is multiplied by 1.05 three times:

1.05 × 1.05 × 1.05 = 1.05³ ≈ 1.158, an overall increase of about 15.8%, not 15%. This is the seed of compound interest.

Try it yourself

1. A shop buys mugs at $6 and sells them at $9. Find the percentage profit.
2. A car worth $12,000 loses 15% of its value, then loses another 15% the next year. What is its value after two years, and what overall percentage has it lost?
3. A recipe says 200 g but you weigh out 210 g. Find the percentage error.

(Answers: 50%; $8,670, about 27.75% lost; 5%.)

Why this matters

Percentage change problems run through business, science, statistics and finance — profit margins, measurement accuracy, depreciation and growth. They build directly on percentage increase and decrease and lead into the compound-growth ideas in money, budgeting and interest.