TACHS: Discount, then tax: chain the multipliers and undo them
Turn a discount and a tax into multipliers, combine them, and work backward to a starting price or a missing percent.
How do you find the price before a discount and a tax when you only know the final price?
Short answer
Turn each change into a multiplier and multiply them to get the combined effect. Divide the final price by that combined multiplier to get back to the start.
Know first: Writing a percent as a decimal, Multiplying and dividing decimals
Step 1 of 6
Every percent change is a multiplier
A 40% discount leaves 60% of the price, so it multiplies the price by 0.60. A 10% tax on top of a price makes it 110%, so it multiplies by 1.10.
When a tax is added after a discount, the tax is taken of the smaller, discounted price. That is why you cannot just add and subtract the percents. Instead, chain the multipliers: start × 0.60 × 1.10 = start × 0.66.
| Change | Multiplier | What it means |
|---|---|---|
| 10% off | × 0.90 | You pay 90% |
| 15% off | × 0.85 | You pay 85% |
| 40% off | × 0.60 | You pay 60% |
| 5% tax | × 1.05 | You pay 105% |
| 8% tax | × 1.08 | You pay 108% |
| 10% tax | × 1.10 | You pay 110% |
A discount multiplier is less than 1. A tax multiplier is more than 1.
Step 2 of 6
Undo both changes with one division
Worked example: A tablet case at 40% off plus tax
A tablet case is marked 40% off. Then 10% sales tax is added to the sale price. The customer pays $59.40. What was the price before the discount?
- A$65.34
- B$77.22
- C$90.00
- D$99.00
- 1
Write the multipliers
40% off is × 0.60, and 10% tax is × 1.10.
- 2
Combine them
0.60 × 1.10 = 0.66, so start × 0.66 = 59.40.
- 3
Divide
59.40 ÷ 0.66 = 90.
- 4
Check forward
90 × 0.60 = 54 after the discount, and 54 × 1.10 = 59.40 after tax.
- 5
See why the other choices are there
$65.34 adds 10% tax again instead of removing it. $77.22 treats the changes as a net 30% discount and adds 30% back. $99.00 undoes the discount but forgets the tax.
Answer
The price before the discount was $90.00.
Wrong: 40% off and then 10% tax is a 30% discount, so the price is multiplied by 0.70.
Right: 40% off and then 10% tax multiplies the price by 0.60 × 1.10 = 0.66, a 34% discount.
The 10% tax is taken of the discounted price, which is smaller than the starting price. So the tax adds back less than 10% of the start.
Step 3 of 6
Find a missing percent one step at a time
Some questions give the starting price and the final price and ask for the tax rate or the discount. Work through the chain in order, one change at a time.
A lamp costs $60 before a 20% discount and tax, and the receipt shows $50.40. After the discount it is 60 × 0.80 = $48. The tax multiplier is 50.40 ÷ 48 = 1.05, so the tax rate is 5%. The tax is measured against $48, the price it was charged on.
Step 4 of 6
Chain, divide, check
Check yourself · Question 1
An item gets a 20% discount, and then a 10% tax is added to the discounted price. The final price is what percent of the starting price?
Answer: B
Multiply the multipliers: 0.80 × 1.10 = 0.88, which is 88%. With a $100 item: $80 after the discount, then $88 after tax.
Trap answer: 90%
Adding percents taken of different amounts
- Why it looks right
- Taking 20 off and putting 10 back looks like a net change of 10.
- Why it's wrong
- The 20% is taken of the full price, but the 10% is taken of the discounted price. Those are different amounts, so the percents cannot be combined by adding.
- The right answer
- 88%, because 0.80 × 1.10 = 0.88.
Check yourself · Question 2
A backpack is marked 15% off. Then 8% tax is added to the sale price. The final price is $45.90. What was the price before the discount?
Answer: C
Combined multiplier: 0.85 × 1.08 = 0.918. Start: 45.90 ÷ 0.918 = 50. Check: 50 × 0.85 = 42.50, and 42.50 × 1.08 = 45.90.
Check yourself · Question 3
A $40 sweater is discounted. Then 5% tax is added to the sale price, and the final price is $35.70. What percent was the discount?
Answer: B
Undo the tax first: 35.70 ÷ 1.05 = 34, the sale price. Then 34 ÷ 40 = 0.85, so the buyer paid 85% and the discount was 15%. Check: 40 × 0.85 × 1.05 = 35.70.
Common mistake
"The sale price after 40% off is $27, so I add 40% back: $27 + $10.80 = $37.80."
- Why it's tempting
- Adding the same percent back feels like the exact reverse of taking it off.
- Do this instead
- The 40% was taken of the starting price, which is larger than $27. Divide by the multiplier instead: 27 ÷ 0.60 = $45. Check: 45 × 0.60 = 27.
Step 5 of 6
What to remember
Remember
- Write each change as a multiplier: a discount is less than 1, a tax is more than 1.
- Multiply the multipliers to combine changes. Never add or subtract the percents.
- Divide by the combined multiplier to work backward, then check forward in order.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what is the combined multiplier, and which wrong choice would you get by undoing only one of the two changes?