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Percent Problems: Tax, Discounts, Markups, Interest, and Percent Change

Solve one- and two-step percent problems about money — sales tax, tips, discounts, markups, commission, simple interest, and percent change.

Practice this skill 25 min Percent problems: interest, tax, markup, percent change Leer en español
Key idea

What you'll learn

Percents are everywhere money changes hands: sales tax, tips, sale prices, store markups, commissions, raises, interest on loans and savings. The GED tests all of these, usually as two-step problems: find a percent of something, then add it or subtract it — or undo a percent change to find an original amount.

In this lesson you will learn to:

  • change between percents, decimals, and fractions;
  • find a part, a percent, or a whole with one simple equation;
  • compute tax, tips, discounts, markups, and commissions in one step using multipliers;
  • find percent increase and decrease;
  • handle successive percent changes and work backwards to an original price;
  • use the simple-interest formula I = Prt (it is on the formula sheet) with time in months or years.

Why it matters on the GED: Percent problems are among the most common questions on the Mathematical Reasoning test. Trap answers are built from predictable slips — stopping after the first step, dividing by the wrong number, or treating "20% then 20%" as "40%." Learn the traps and you pick up easy points.

Words to know

  • Percent (%) — "per hundred." 35\% = \tfrac{35}{100} = 0.35.
  • Base (whole) — the amount you are taking a percent of (the original price, the old value).
  • Part — the result of taking a percent of the base. 20% of $50 is the part $10.
  • Sales tax — a percent of the price added to the bill. 7% tax on $20 is $1.40.
  • Tip (gratuity) — a percent of the bill added for service.
  • Discount / markdown — a percent subtracted from a price. 25% off $80 saves $20.
  • Markup — a percent added to a store's cost to set the selling price.
  • Commission — pay that is a percent of sales. 5% of $4,000 in sales is $200.
  • Percent change — how much a value grew or shrank, as a percent of the original value.
  • Multiplier — the decimal you multiply by to apply a change in one step: 1.08 for +8%, 0.75 for −25%.
  • Simple interest — interest paid only on the original amount: I = Prt.
  • Principal (P) — the amount borrowed or invested.

All Math words in the glossary

Percent basics: one equation for three kinds of questions

Converting. Percent → decimal: move the decimal point two places left (8\% = 0.08, 12.5\% = 0.125, 150\% = 1.5). Decimal → percent: two places right (0.045 = 4.5\%). Handy fractions: 25\% = \tfrac{1}{4}, 50\% = \tfrac{1}{2}, 20\% = \tfrac{1}{5}, 10\% = \tfrac{1}{10}, 33\tfrac{1}{3}\% = \tfrac{1}{3}.

The percent equation:

\text{part} = \text{percent (as a decimal)} \times \text{base}

It answers three kinds of questions, depending on what's missing:

MissingExampleWork
partWhat is 15% of 60?0.15 \times 60 = 9
percent9 is what percent of 60?9 \div 60 = 0.15 = 15\%
base9 is 15% of what?9 \div 0.15 = 60
Percent bar model drawn to scale: a bar for the base 60 is marked from 0% to 100% in steps of 10%, with matching values 0, 6, 12, up to 60. The first 15% of the bar is shaded and equals 9.15% of 60 = 90%010%620%1230%1840%2450%3060%3670%4280%4890%54100%609base = 60 (100%)each 10% = 6, so 15% = 6 + 3 = 9

"Of" usually points to the base. "Is" usually sits next to the part.

Mental math with 10%. 10% of a number moves the decimal one place left: 10% of $64 is $6.40. Then 5% is half of that ($3.20), 20% is double ($12.80), and 15% is 6.40 + 3.20 = \$9.60. Great for tips and for estimating.

Adding or subtracting a percent: tax, tips, discounts, markups, commission

Two-step method. Find the percent of the base, then add or subtract it.

  • $45 shirt, 8% tax: tax = 0.08 \times 45 = \$3.60; total = 45 + 3.60 = \$48.60.
Bar model drawn to scale: a bar for the 45 dollars price is 100%; a small added piece of 8% is the 3.60 dollars tax. Together they make 108% of 45 dollars, which is 45 × 1.08 = 48.60 dollars.$45 shirt with 8% tax$45 (100%)$3.600%10%20%30%40%50%60%70%80%90%100%110%total = 108% of $45 = 45 × 1.08 = $48.60
  • $80 jacket, 25% off: discount = 0.25 \times 80 = \$20; sale price = 80 - 20 = \$60.
Bar model drawn to scale: the 80 dollars price is the whole bar, 100%. The last 25%, 20 dollars, is taken off; the remaining 75% is the 60 dollars sale price, 80 × 0.75.$80 jacket at 25% offpay $60 (75%)off $200%10%20%30%40%50%60%70%80%90%100%sale price = 75% of $80 = 80 × 0.75 = $60

One-step method: multipliers.

SituationMultiplierExample
Increase by r (tax, tip, markup, raise)1 + r$45 with 8% tax: 45 \times 1.08 = \$48.60
Decrease by r (discount, markdown, drop)1 - r$80 at 25% off: 80 \times 0.75 = \$60

Why it works: paying the price plus 8% means paying 100\% + 8\% = 108\% of the price.

Markup is a percent of the store's cost: a store buys a lamp for $40 and marks it up 35% → 40 \times 1.35 = \$54.

Commission is just "part = rate × sales": 6% of $3,250 is 0.06 \times 3{,}250 = \$195. With base pay: total pay = base + commission.

Read carefully: what is the base? A tip is a percent of the bill the question names (often the pre-tax amount); a discount is a percent of the original price; a markup is a percent of the cost.

Percent change, successive changes, and working backwards

Percent change:

\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%

A positive result is an increase; a negative result is a decrease. Always divide by the original (old) value.

  • Rent rises from $1,250 to $1,350: \dfrac{100}{1{,}250} = 0.08 → 8% increase.
  • A price drops from $250 to $210: \dfrac{-40}{250} = -0.16 → 16% decrease.

Successive percents multiply — they don't add. Up 20% then down 20%: 1.20 \times 0.80 = 0.96, a 4% decrease overall, not "no change." Markup 50% then 20% off: 1.5 \times 0.8 = 1.2, a 20% increase, not 30%.

Bar chart drawn to scale: a value of 100 rises 20% to 120, then falls 20% of 120 to 96. A dashed line at 100 shows the final 96 is below the start: 1.20 × 0.80 = 0.96, a 4% decrease.Up 20% then down 20% (start at 100)start100up 20%120then down 20%1.20 × 0.80 = 0.96 → a 4% decrease, not 0%96

Working backwards (find the original). If a price after a change is known, divide by the multiplier:

  • After 15% off, a coat costs $51. Original = 51 \div 0.85 = \$60.
Bar model drawn to scale for working backwards: the whole bar is the unknown original price, 100%. The 51 dollars sale price fills 85% of it and the 15% discount, 9 dollars, fills the rest. So the original is 51 divided by 0.85 = 60 dollars.After 15% off, the coat costs $51$51 = 85%$9 off0%10%20%30%40%50%60%70%80%90%100%original = 100% = 51 ÷ 0.85 = $60
  • After 6% tax, the total is $63.60. Price before tax = 63.60 \div 1.06 = \$60.

Don't add the percent back on: 51 \times 1.15 = \$58.65 is wrong, because the 15% was taken from the bigger original price, not from $51.

Simple interest

I = Prt
  • I = interest (dollars earned or owed)
  • P = principal (the starting amount)
  • r = yearly interest rate as a decimal (3.5\% = 0.035)
  • t = time in years
Labelled formula I = P × r × t: I is the interest in dollars, P is the principal in dollars, r is the yearly rate written as a decimal, and t is the time in years. Example: 5,000 × 0.04 × 3 = 600 dollars.Simple interestIinterest ($)=Pprincipal ($)×ryearly rateas decimal×ttime in yearsexample: 5,000 × 0.04 × 3 = $600

Time in months: divide by 12. 18 months = \tfrac{18}{12} = 1.5 years; 9 months = 0.75 year; 2 years 6 months = 2.5 years (not 2.6!).

  • $5,000 at 4% for 3 years: I = 5{,}000 \times 0.04 \times 3 = \$600.
  • $1,800 at 5% for 9 months: I = 1{,}800 \times 0.05 \times 0.75 = \$67.50.

Total amount (what you have or owe at the end) = P + I. Read the question: does it ask for the interest or the total?

Finding a missing value. Rearrange: r = \dfrac{I}{Pt}, t = \dfrac{I}{Pr}, P = \dfrac{I}{rt}. If $3,200 earns $360 in 2.5 years: r = \dfrac{360}{3{,}200 \times 2.5} = \dfrac{360}{8{,}000} = 0.045 = 4.5\%.

Formula

Formulas to know

\text{part} = \text{rate} \times \text{base} \qquad \text{new price} = \text{price} \times (1 \pm r) \qquad \text{original} = \frac{\text{new price}}{1 \pm r} \text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\% \qquad\qquad I = Prt
  • r — the percent written as a decimal.
  • P — principal, t — time in years, I — simple interest.

On the GED formula sheet? Only simple interest, I = Prt, is on the sheet. The percent equation, multipliers, and the percent-change formula are not — you must memorize them.

Worked example

Example 1 (easy): a restaurant tip

A dinner bill is $42. Ming leaves a 15% tip. What is the tip, and what is the total?

  1. 10% of $42 is $4.20; 5% is half of that, $2.10.
  2. Tip: 4.20 + 2.10 = \$6.30 \quad (\text{or } 0.15 \times 42 = 6.30)
  3. Total: 42 + 6.30 = \$48.30 \quad (\text{or } 42 \times 1.15 = 48.30)
Bar model drawn to scale: the 42 dollars bill is 100%. Added on are 10% = 4.20 dollars and 5% = 2.10 dollars, half of 10%. The tip is 6.30 dollars and the total, 115% of the bill, is 48.30 dollars.15% tip on a $42 billbill $42 (100%)$4.20$2.100%20%40%60%80%100%120%tip = 10% + 5% = 4.20 + 2.10 = $6.30; total = $48.30

Answer: tip $6.30; total $48.30.

Check: 6.30 \div 42 = 0.15 = 15\% ✓

Worked example

Example 2: percent increase in rent

Monthly rent went from $1,250 to $1,350. What was the percent increase?

  1. Change: 1{,}350 - 1{,}250 = 100
  2. Divide by the original: \frac{100}{1{,}250} = 0.08
  3. As a percent: 0.08 = 8\%.
Bar chart drawn to scale: old rent 1,250 dollars and new rent 1,350 dollars. The extra piece of 100 dollars is marked. Dividing the change by the original gives 100 ÷ 1,250 = 0.08, an 8% increase.Rent: $1,250 → $1,350old rent$1,250new rent$1,350change $100 ÷ original $1,250 = 0.08 = 8% increase+$100

Answer: an 8% increase.

Check: 1{,}250 \times 1.08 = 1{,}350 ✓

Worked example

Example 3: simple interest with years and months

(a) Ray invests $5,000 at 4% simple interest for 3 years. How much interest does he earn? (b) Dana borrows $1,800 at 5% simple interest for 9 months. How much interest does she owe?

  1. (a) I = Prt = 5{,}000 \times 0.04 \times 3 = \$600
  2. (b) Change months to years: t = \tfrac{9}{12} = 0.75. I = 1{,}800 \times 0.05 \times 0.75 = \$67.50
Two bar models. (a) 5,000 dollars at 4% earns 200 dollars each year: three equal 200 dollars blocks for 3 years make 600 dollars. (b) 1,800 dollars at 5% earns 90 dollars in a full year, shown as four 22.50 dollars quarters; 9 months is 3 of the 4 quarters, 0.75 year, so the interest is 67.50 dollars.Simple interest builds the same amount each year(a) $5,000 at 4%: $200 per year$200year 1$200year 2$200year 33 × $200 = $600(b) $1,800 at 5%: $90 per year$22.50$22.50$22.50$22.509 months¾ × $90 = $67.50(t = 9/12 = 0.75 year)

Answer: (a) $600; (b) $67.50.

Check: (a) 4% of $5,000 is $200 per year, and 3 years gives $600 ✓ (b) 5% of $1,800 is $90 per year, and \tfrac{3}{4} of $90 is $67.50 ✓

Worked example

Example 4 (multistep): markup, sale, and tax

A store pays $60 for a pair of boots and marks them up 50%. Later the boots go on sale for 20% off. A customer pays 6% sales tax on the sale price. What does the customer pay?

  1. Selling price after markup: 60 \times 1.50 = \$90
  2. Sale price (20% off): 90 \times 0.80 = \$72
  3. With 6% tax: 72 \times 1.06 = \$76.32
Bar chart drawn to scale of the boot price at each step: store cost 60 dollars, after a 50% markup 90 dollars, after 20% off 72 dollars, and after 6% tax 76.32 dollars. In one chain, 60 × 1.5 × 0.8 × 1.06 = 76.32.Boots: each step multiplies the pricecost$60.00+50% markup$90.00−20% sale$72.00+6% tax$76.3260 × 1.5 × 0.8 × 1.06 = $76.32

Answer: $76.32.

Check: One chain: 60 \times 1.5 \times 0.8 \times 1.06 = 76.32 ✓ (Note: +50% then −20% is not +30%; 1.5 \times 0.8 = 1.2, so it's +20% before tax.)

Worked example

Example 5 (GED-hard, work backwards): find the original price

Leah bought a jacket that was 25% off. She also paid 8% sales tax on the sale price. Her total was $97.20. What was the original price?

  1. Undo the tax — divide by 1.08: 97.20 \div 1.08 = \$90 \quad\text{(sale price)}
  2. Undo the 25% discount — the sale price is 75% of the original, so divide by 0.75: 90 \div 0.75 = \$120
Flowchart: going forward, the original 120 dollars times 0.75 gives the 90 dollars sale price, and times 1.08 gives the 97.20 dollars total. Working backwards, divide 97.20 dollars by 1.08 to get 90 dollars, then divide by 0.75 to get the original 120 dollars.Work backwards: undo each step by dividing$120original$90sale price$97.20total paid× 0.75× 1.08÷ 0.75÷ 1.0897.20 ÷ 1.08 = 90; 90 ÷ 0.75 = 120

Answer: the original price was $120.

Check: 120 \times 0.75 = 90, and 90 \times 1.08 = 97.20 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Enter percents as decimals. For 8%, type 0.08 (or .08). Avoid hunting for a % key — the decimal is always reliable.
  • Use multipliers in one line: 60 × 1.5 × 0.8 × 1.06 enter → 76.32.
  • Working backwards: 97.2 ÷ 1.08 ÷ 0.75 enter → 120.
  • Percent change: use parentheses: ( 1350 − 1250 ) ÷ 1250 enter → 0.08, then read it as 8%. For a decrease you'll see a negative number; use the (−) key only when entering a negative value, not for subtraction.
  • Interest: 2400 × 0.035 × 1.5 enter → 126.
  • When not to use it: 10%, 25%, and 50% of friendly numbers are faster in your head, and estimation with 10% is the quickest way to check any answer.
Common mistake

Common mistakes

  • Stopping after step one. 8% tax on $45 is $3.60, but the question asked for the total. ✔ Add it: $48.60 (or multiply by 1.08).
  • Dividing by the new value in percent change. $80 → $92 is not 12 \div 92 \approx 13\%. ✔ Divide by the original: 12 \div 80 = 15\%.
  • Adding successive percents. +20% then −20% is not 0%. ✔ Multiply: 1.2 \times 0.8 = 0.96, a 4% decrease.
  • Working backwards by adding the percent. A $51 sale price after 15% off is not 51 \times 1.15 = \$58.65. ✔ Divide: 51 \div 0.85 = \$60.
  • Months as years in I = Prt. 18 months is t = 1.5, not 18. And 2 years 6 months is 2.5, not 2.6. ✔ Divide months by 12.
  • Decimal slips. 8% is 0.08, not 0.8. 4.5% is 0.045. ✔ Move the decimal exactly two places.
Test strategy

GED test strategy

  • Estimate with 10%. 10% of $212 is $21.20, so 20% is about $42 and an 18% tip is a little less (about $38). Eliminate choices that are far off.
  • Ask "should the answer be bigger or smaller?" After a discount, the price drops; after tax or markup it rises; an original price before a discount is higher than the sale price.
  • Find the base for every percent: original price? sale price? cost? pre-tax bill? Underline it.
  • Plug answers back in for work-backwards questions: apply the percent(s) to the choice and see if you land on the given number.
  • Interest questions: circle the time unit and the word interest vs total amount. I = Prt is on the formula sheet — use it.
  • Expect traps: the first-step result, the "added percents" answer, and the "divided by the new value" answer usually all appear as choices.

Check yourself

Percent problems: interest, tax, markup, percent changeNo calculator

A shirt costs $45. The sales tax rate is 8%. What is the total cost of the shirt, including tax?

Answer choices
Hint

Find 8% of $45 first. Does the question want only the tax, or the price plus the tax?

Check yourself

Percent problems: interest, tax, markup, percent changeNo calculatorFill in

A salesperson earns a 6% commission on her sales. This week she sold $3,250 worth of furniture. How much commission did she earn?

$

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Commission is a percent of the sales. Change 6% to a decimal and multiply.

Summary

Key points

  • \text{part} = \text{rate} \times \text{base}; "of" points to the base.
  • Increase by r: multiply by 1 + r. Decrease by r: multiply by 1 - r.
  • Percent change = \dfrac{\text{new} - \text{original}}{\text{original}} — always divide by the original.
  • Successive percents multiply; to undo a change, divide by its multiplier.
  • Simple interest: I = Prt (on the formula sheet) with r as a decimal and t in years.
  • Reread the question: interest or total? tax or total? sale price or original?
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