doing math

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Percentages in Business

level 2 course

Growth rates, percentage points, undoing a discount, and changes that stack.

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the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

Percent change, or the relative change of a rate.

  1. Find the change: new minus old. For a fall, old minus new.
  2. Divide by the starting value, never the new one.
  3. Write the decimal as a percent. For two rates, the subtraction alone is the change in points.
worked example

Example: Monthly sales rise from $24,000 to $30,000. What is the percentage increase?

  1. $30,000 − $24,000 = $6,000.
  2. $6,000 ÷ $24,000 = 0.25 = 25%.

Answer: 25%

Undo a discount by dividing

You know the sale price and the percent off.

  1. Say what share you still pay: 30% off leaves 70%.
  2. Divide the sale price by that share, written as a decimal.
worked example

Example: A tent is on sale for $84, which is 30% off its original price. What was the original price?

  1. $84 is 70% of the original.
  2. $84 ÷ 0.7 = $120.
  3. Check: 30% of $120 is $36, and $120 − $36 = $84.

Answer: $120

Multiply the changes

One percent change follows another.

  1. Turn each change into a factor: up 20% is × 1.2, and down 25% is × 0.75.
  2. Multiply the factors.
  3. Subtract 1 and write the result as a percent, with a minus sign for a decrease.
worked example

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.)

  1. The factors are 0.5 and 1.5.
  2. 0.5 × 1.5 = 0.75.
  3. 0.75 − 1 = −0.25, a change of −25%.

Answer: −25%

Tips by skill

  • TipPercent change: Divide the change by the starting value. If the question already says it fell, give the size of the fall with no minus sign.
  • TipPercentage points: Points are a plain subtraction of the two rates. Do not divide.
  • TipRelative change of a rate: Subtract the rates to get the points, then divide by the starting rate.
  • TipUndo a discount: The sale price is the share you still pay, like 80% after 20% off. Divide by that share as a decimal.
  • TipUp then down: Turn each change into a factor, like 1.2 for up 20% or 0.8 for down 20%. Multiply, subtract 1, and read it as a percent.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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