Percentages in Business
Growth rates, percentage points, undoing a discount, and changes that stack.
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the idea
A percent is always a percent of something, its base. Percent change divides the change by the starting value: sales that rise from $40,000 to $50,000 grew by $10,000, which is 25% of where they started.
When a rate changes, there are two ways to measure it. A conversion rate that goes from 20% to 25% rose 5 percentage points (a plain subtraction), and 25% relative to where it started (5 ÷ 20).
Percent changes in a row multiply. Each one applies to the new amount, not to the start.
techniques
Divide by the starting value
- Find the change: new minus old. For a fall, old minus new.
- Divide by the starting value, never the new one.
- Write the decimal as a percent. For two rates, the subtraction alone is the change in points.
worked example
Monthly sales rise from $24,000 to $30,000. What is the percentage increase?
- $30,000 − $24,000 = $6,000.
- $6,000 ÷ $24,000 = 0.25 = 25%.
Answer: 25%
Undo a discount by dividing
- Say what share you still pay: 30% off leaves 70%.
- Divide the sale price by that share, written as a decimal.
worked example
A tent is on sale for $84, which is 30% off its original price. What was the original price?
- $84 is 70% of the original.
- $84 ÷ 0.7 = $120.
- Check: 30% of $120 is $36, and $120 − $36 = $84.
Answer: $120
Multiply the changes
- Turn each change into a factor: up 20% is × 1.2, and down 25% is × 0.75.
- Multiply the factors.
- Subtract 1 and write the result as a percent, with a minus sign for a decrease.
worked example
Monthly sales fall 50% in March, then rise 50% in April. What is the overall percent change from the start? (Use a minus sign for a decrease.)
- The factors are 0.5 and 1.5.
- 0.5 × 1.5 = 0.75.
- 0.75 − 1 = −0.25, a change of −25%.
Answer: −25%
watch out for
- Dividing the change by the new value. Percent change always divides by the starting value.
- Calling a rise from 20% to 25% a 5% increase. It is 5 percentage points, and a 25% relative rise.
- Adding the discount back on. Adding 20% to a $96 sale price gives $115.20, not the original $120.
- Adding changes in a row. Up 20% then down 20% ends 4% below the start, not back where it began.
practice
Percent change
worked example
Monthly sales rise from $13,000 to $13,650. What is the percentage increase?
Answer: 5%
- Percent change = (new − old) ÷ old = ($13,650 − $13,000) ÷ $13,000 = $650 ÷ $13,000 = 5%.
- Always divide by the starting value.
Percentage points
worked example
A delivery firm's on-time rate falls from 46% to 40%. By how many percentage points did it fall? Give the number of points.
Answer: 6
- Percentage points are a straight subtraction of the two rates: 46 − 40 = 6 points.
Relative change of a rate
worked example
A repeat-customer rate moves from 20% to 22%. Relative to where it started, by what percent did it rise?
Answer: 10%
- Relative change = point change ÷ starting rate = (22 − 20) ÷ 20 = 2 ÷ 20 = 10%.
Undo a discount
worked example
A desk chair is on sale for $207, which is 10% off its original price. What was the original price?
Answer: $230.00
- After 10% off you pay 90% of the original, so original = $207 ÷ 0.9 = $230.
- Check: 10% of $230 is $23, and $230 − $23 = $207.
Up then down
worked example
A price rises 25% in March, then falls 50% in April. What is the overall percent change from the start? (Use a minus sign for a decrease.)
Answer: -37.5%
- Multiply the factors: 1.25 × 0.5 = 0.625, so the overall change is −37.5%.
- The April change is taken from the new March level, not from the start.