SHSAT: Percent change: know your base
Chain discounts, taxes and increases with multipliers, work backward to an earlier amount, and find percent change from the right base.
When a problem stacks several percent changes, how do you know which amount each percent is taken of?
Short answer
Each percent is taken of the amount just before it. Turn each change into a multiplier, multiply in order, and divide by the multiplier to work backward.
Know first: Converting percents to decimals, Multiplying decimals
Step 1 of 5
Every percent has a base
A percent is always a percent of something, its base. In a chain of changes, each new percent uses the amount right before it as its base. That's why you can't combine a discount and a tax by adding or subtracting their percents: the tax is taken of the smaller, discounted price.
The fastest way to handle a chain is with multipliers. Turn each change into a decimal, then multiply them in order.
| Change | Multiplier | Why |
|---|---|---|
| 15% off | × 0.85 | 100% − 15% = 85% remains |
| 8% tax | × 1.08 | 100% + 8% |
| 40% increase | × 1.4 | 100% + 40% |
| undo a 30% increase | ÷ 1.3 | reverse the multiplier |
| percent change | (new − old) ÷ old | the base is the starting amount |
| percent of two groups combined | add the counts, then divide by the combined total | groups of different sizes can't be averaged directly |
A chain of changes multiplies: 20% off and then 10% off is × 0.8 × 0.9 = × 0.72, a 28% discount in all.
Step 2 of 5
Apply each change to the new amount
Worked example: A discount, then tax
A jacket costs $80. It is discounted 15%, and then a 5% tax is added to the sale price. What is the final cost?
- A$12.00
- B$68.00
- C$71.40
- D$72.00
- 1
Discount
80 × 0.85 = $68.
- 2
Tax on the sale price
68 × 1.05 = 68 + 3.40 = $71.40.
- 3
One line
80 × 0.85 × 1.05 = $71.40.
- 4
See why the other choices are there
$12.00 is the discount. $68.00 forgets the tax. $72.00 either combines the changes into a single 10% discount or taxes the $80 price instead of the sale price.
Answer
$71.40
Wrong: After a 20% increase, a price is $90. Before: 90 − 20% of 90 = $72.
Right: After a 20% increase, a price is $90. Before: 90 ÷ 1.2 = $75. Check: 75 × 1.2 = 90.
The 20% was taken of the earlier price, not of $90. Undo a multiplier by dividing.
Step 3 of 5
Track the base at every step
Check yourself · Question 1
A price increases by 20% and then decreases by 20%. The final price is what percent of the starting price?
Answer: B
Multiply: 1.2 × 0.8 = 0.96, so the final price is 96% of the starting price. For example, $100 becomes $120, then $120 − $24 = $96.
Trap answer: 100%
Cancelling equal percents
- Why it looks right
- Up 20% and down 20% sound like opposite moves of the same size.
- Why it's wrong
- The increase is 20% of the starting price; the decrease is 20% of the larger, raised price. The decrease takes away more.
- The right answer
- 96%, from 1.2 × 0.8 = 0.96.
Check yourself · Question 2
A club's membership grows from 125 to 150. What is the percent increase?
Answer: B
Change: 150 − 125 = 25. Divide by the starting amount: 25 ÷ 125 = 0.2 = 20%.
Check yourself · Question 3
An account has $800 and earns 5% interest per year, compounded once a year. No money is added or taken out. How much interest is earned during the second year alone?
Answer: B
Year 1: 800 × 0.05 = $40, so the balance becomes $840. Year 2: 840 × 0.05 = $42.
Common mistake
For percent change, I divide the change by the new amount.
- Why it's tempting
- The new amount is the number you're thinking about at the end of the problem.
- Do this instead
- Percent change always uses the starting amount as its base: (new − old) ÷ old.
Step 4 of 5
What to remember
Remember
- Turn each change into a multiplier and multiply in order; each percent uses the amount right before it.
- To undo a change, divide by its multiplier.
- Percent change is (new − old) ÷ old, and percentage points are a plain difference of two percents.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what amount was the tax taken of, and which wrong choice would you get by taxing the starting price?