PSAT Percent Change: Discounts and Original Prices

For PSAT percent-change problems, turn each increase or decrease into a multiplier. Multiply to find the new amount, or divide by that multiplier to recover the original amount. For successive changes, apply each change to the result of the previous step.

College Board lists percentages within Problem-Solving and Data Analysis in its PSAT/NMSQT Math overview. The problems here are original practice examples, not official test items. All prices are hypothetical, with no taxes or fees unless stated.

Start by naming the reference amount

A percentage describes a fraction of a particular amount. Before calculating, finish this sentence: “This percent is taken from ___.” For a discount, that blank is the price immediately before that discount. For an overall percent change, compare the final amount with the original starting amount.

Let $r$ be the rate written as a decimal. For example, $15\%=0.15$. An increase keeps the original amount and adds $r$ times that amount; a decrease subtracts $r$ times it.

$$\text{new amount}=\text{original amount}\times(1+r)$$

$$\text{new amount}=\text{original amount}\times(1-r)$$

A $15\%$ increase uses $1.15$; a $15\%$ decrease uses $0.85$. OpenStax’s percent applications lesson explains percent increase, percent decrease, and the relationship between a discount and the original price.

Example: find the percent decrease

A drawing tablet’s price falls from $$96$ to $$72$. What is the percent decrease?

The decrease is $96-72=24$ dollars. Divide that change by the original price, $96$, to find its share of the starting amount.

$$\frac{96-72}{96}=\frac{24}{96}=0.25=25\%$$

Check by applying the discount: $96\times0.75=72$. Dividing by the new price would answer a different question: $24\div72$ measures the change relative to $72$, not the original $96$.

Example: two discounts use different starting prices

A desk lamp is listed at $$80$. A sale takes $15\%$ off, then a coupon takes $20\%$ off the sale price. What is the final price, and what single discount would produce it?

1. After the first discount: $80\times0.85=68$ dollars.
2. The coupon applies to $68$, so the final price is $68\times0.80=54.40$ dollars.
3. The combined multiplier is $0.85\times0.80=0.68$. You pay $68\%$ of the original price, so the equivalent discount is $32\%$.

$$80\times0.85\times0.80=54.40$$

Adding the rates to get $35\%$ off would give $$52$, which is too low. The second discount is $20\%$ of $$68$, or $$13.60$, rather than $20\%$ of $$80$. The actual savings are $12+13.60=25.60$ dollars. Check: $25.60\div80=0.32$.

Example: work backward to the original price

An art kit costs $$63$ after a $30\%$ discount. What was its original price?

The sale price is $70\%$ of the original price. If $P$ is the original price in dollars, the relationship is $0.70P=63$. Divide by the fraction that remains.

$$P=\frac{63}{0.70}=90$$

The original price was $$90$. Check: $90\times0.30=27$ dollars off, and $90-27=63$. Adding $30\%$ of the sale price instead gives $63\times1.30=81.90$, which uses the wrong reference amount.

For two successive discounts, divide by their combined multiplier. If a final price is $F$ after the lamp example’s two discounts, the original price is $F\div(0.85\times0.80)$. This reversal requires a nonzero multiplier; after a $100\%$ discount, a zero final price cannot identify the original price.

If setting up the equation is the difficult part, the linear-equation guide explains how to preserve equality while isolating an unknown.

Example: equal increase and decrease rates do not cancel

A tank holds $120$ liters of water. The volume rises by $25\%$, then falls by $25\%$. How much water remains?

The multipliers are $1.25$ and $0.75$, so $120\times1.25\times0.75=112.5$ liters remain. The increase adds $30$ liters, but the decrease removes $37.5$ liters because it applies to the larger intermediate amount of $150$ liters.

The combined multiplier is $0.9375$, which represents an overall $6.25\%$ decrease. Check: $(120-112.5)\div120=0.0625$. Equal increase and decrease rates use different reference amounts.

A short setup you can reuse

• Given: label the original, intermediate, and final amounts that the question provides.
• Asked: decide whether you need a price, a discount amount, or a percentage.
• Multiplier: write one factor per change, in the stated order.
• Check: apply the changes forward and compare with the information in the question.

Keep full precision through the calculation and round only as the question directs. If it specifies rounding at an intermediate step, follow that instruction.

Try five original percent-change questions

Use the stated percentages as exact. Write your multiplier or reference amount before calculating, and cover the answers until you finish.

1. A bookcase costs $$140$. Its price decreases by $18\%$. What is the new price?
2. A store takes $10\%$ off a $$75$ backpack, then takes another $12\%$ off the reduced price. Find the final price and the equivalent single discount.
3. After a $28\%$ discount, a board game costs $$46.80$. What was its original price?
4. A water tank holds $250$ liters. Its contents increase by $8\%$, then decrease by $10\%$. How many liters remain, and what is the overall percent change?
5. A camera bag costs $$76.50$ after successive discounts of $15\%$ and $10\%$. What was the original price?

Answers and checks

1. $$114.80$. Use $140\times0.82=114.80$. The discount is $25.20$ dollars, and $25.20\div140=0.18$.
2. $$59.40$, equivalent to $20.8\%$ off. The price calculation is $75\times0.90\times0.88=59.40$. The multiplier is $0.792$, so $1-0.792=0.208=20.8\%$. Check: $(75-59.40)\div75=0.208$.
3. $$65$. The remaining fraction is $1-0.28=0.72$, so $46.80\div0.72=65$. Check: $65\times0.72=46.80$.
4. $243$ liters remain, an overall $2.8\%$ decrease. Calculate $250\times1.08\times0.90=243$. Check the change against the starting volume: $(250-243)\div250=0.028$.
5. $$100$. The combined multiplier is $0.85\times0.90=0.765$, so $76.50\div0.765=100$. Check: $100\times0.85=85$, then $85\times0.90=76.50$.

Make your next attempt more specific

For a missed question, record which amount the percentage applied to and which multiplier you used. If those were right, check the arithmetic and requested units. The practice error-log template helps you turn that observation into a short correction you can reuse.

Try a free PSAT practice question with Kibo when you are ready to continue.

Prepared with AI assistance. Original examples were checked with separate calculations; no human review is claimed. Sources checked October 8, 2026. Kibo is not affiliated with or endorsed by College Board or National Merit Scholarship Corporation.