17 word problems covering increase, decrease, discounts, tax, tips, interest, mixtures, and area
The following 17 word problems cover percentages — from simple percent change through discounts, taxes, tips, and simple and compound interest, to mixture and geometric-area problems. Each problem includes a fully worked solution.
The original price of a shirt was \(\$ \)20. It was decreased to $15. What is the percent decrease?
Absolute decrease:
\[ 20 - 15 = 5 \]Percent decrease = decrease / original:
\[ \frac{5}{20} = 0.25 = 25\% \]Answer: 25% decrease.
Mary earns \(\$ \)1200 per month and spends $280 on food. What percent of her salary goes to food?
Fraction of salary spent on food:
\[ \frac{280}{1200} = \frac{7}{30} \approx 0.2333 \]As a percent (rounded to 2 decimal places):
\[ 0.2333 \times 100\% \approx 23.33\% \]Answer: ≈ 23.33% (exactly \(23.\overline{3}\%\)).
A pair of trousers is \(\$ \)30 after a 22% decrease. What was the original price?
Let \(x\) be the original price. After a 22% decrease, $30 = 78\%$ of \(x\):
\[ 0.78x = 30 \;\Rightarrow\; x = \frac{30}{0.78} \approx 38.46 \]Answer: approximately $38.46.
An item goes from \(\$ \)120 to \(\$ \)100, then from $100 to $80. Which decrease was larger in percentage terms?
First decrease:
\[ \frac{120-100}{120} = \frac{20}{120} \approx 16.67\% \]Second decrease:
\[ \frac{100-80}{100} = \frac{20}{100} = 20\% \]Answer: the second decrease (20%) is larger — the base is smaller, so the same absolute drop is a bigger fraction.
An item decreases 20% to \(\$ \)200, then decreases from \(\$ \)200 to \(\$ \)150. What is the total percent decrease from the original price?
Original price \(x\):
\[ 0.8x = 200 \;\Rightarrow\; x = 250 \]Total percent decrease:
\[ \frac{250 - 150}{250} = \frac{100}{250} = 40\% \]Answer: 40% decrease.
A number goes from 30 to 40, then back to 30. Compare the percent increase and percent decrease.
Percent increase (30 → 40):
\[ \frac{10}{30} \approx 33.33\% \]Percent decrease (40 → 30):
\[ \frac{10}{40} = 25\% \]Answer: the increase (≈33.33%) is larger than the decrease (25%) even though the absolute change is the same — the bases differ.
A family pays \(\$ \)30 for food, plus 9.5% sales tax and 10% tip. How much do they pay in total?
Answer: $35.85.
Shirts are \(\$ \)20 each. Buying two gives a 15% discount on both, plus an extra 10% off the second shirt. What is the total cost for two shirts?
First shirt after 15% off:
\[ 20 - 0.15(20) = 17 \]Second shirt after 15% off and an additional 10% off:
\[ 17 - 0.10(17) = 15.30 \]Total:
\[ 17 + 15.30 = 32.30 \]Answer: $32.30.
Smith invests $5000 for two years: 7% in year 1, 8.5% in year 2. How much interest is earned?
Year 1 interest:
\[ 0.07 \times 5000 = 350 \]Year 2 interest on new principal:
\[ 0.085 \times (5000 + 350) = 0.085 \times 5350 = 454.75 \]Total interest:
\[ 350 + 454.75 = 804.75 \]Answer: $804.75.
Janette invests \(\$ \)2000 at 5% compounded annually for 5 years. How much interest does she earn?
Final amount:
\[ 2000 \times (1.05)^5 \approx 2000 \times 1.27628 \approx 2552.56 \]Interest:
\[ 2552.56 - 2000 = 552.56 \]Answer: $552.56.
Tom borrows \(\$ \)600 at 10% simple interest for 3 years. How much does he repay?
Simple interest:
\[ I = 600 \times 0.10 \times 3 = 180 \]Total repayment:
\[ 600 + 180 = 780 \]Answer: $780.
World population: 6.6 billion, with 1.2 billion in richer countries growing at 0.25% per year and 5.4 billion in less developed countries growing at 1.5% per year. What is the total population in 5 years?
Richer countries:
\[ 1.2 \times (1.0025)^5 \approx 1.215 \text{ billion} \]Less developed countries:
\[ 5.4 \times (1.015)^5 \approx 5.817 \text{ billion} \]Total:
\[ 1.215 + 5.817 \approx 7.032 \text{ billion} \]Answer: ≈ 7.03 billion (3 sig figs).
Cassandra invests \(\$ \)10,000 in two parts, one at 7.5% and one at 8.5%. Total income is \(\$ \)820. How much is invested at each rate?
Let \(x\) and \(y\) be the amounts at 7.5% and 8.5%:
\[ \begin{cases} x + y = 10000 \\ 0.075x + 0.085y = 820 \end{cases} \]Substitute \(y = 10000 - x\):
\[ 0.075x + 0.085(10000 - x) = 820 \] \[ 850 - 0.01x = 820 \;\Rightarrow\; x = 3000, \quad y = 7000 \]Answer: $3000 at 7.5% and $7000 at 8.5%.
A shop assistant earns \(\$ \)500 plus 5% of monthly sales. What monthly sales are needed for her salary to reach \(\$ \)1500?
Let \(x\) be the monthly sales:
\[ 1500 = 500 + 0.05x \;\Rightarrow\; 1000 = 0.05x \;\Rightarrow\; x = 20000 \]Answer: $20,000 in monthly sales.
A chemist has 20% and 40% acid solutions. What amounts of each make 300 ml of 28% acid solution?
Let \(x\) and \(y\) be the amounts of 20% and 40% solutions (ml):
\[ \begin{cases} x + y = 300 \\ 0.20x + 0.40y = 0.28 \times 300 = 84 \end{cases} \]Substitute \(y = 300 - x\):
\[ 0.20x + 0.40(300 - x) = 84 \] \[ 120 - 0.20x = 84 \;\Rightarrow\; x = 180, \quad y = 120 \]Answer: 180 ml of the 20% solution and 120 ml of the 40% solution.
What percent of the total area of the circular disk is colored red?
The red sector has a central angle of \(360° - 120° = 240°\), i.e. \(t = \dfrac{4}{3}\pi\) radians.
Area of the disk:
\[ A_\text{disk} = \pi r^2 \]Area of the red sector:
\[ A_\text{red} = \tfrac{1}{2} t r^2 = \tfrac{1}{2} \cdot \tfrac{4}{3}\pi \cdot r^2 \]Percentage:
\[ P = \frac{A_\text{red}}{A_\text{disk}} = \frac{\tfrac{2}{3}\pi r^2}{\pi r^2} = \frac{2}{3} \approx 66.7\% \]Answer: ≈ 66.7%.
What percent of the total area of the rectangle is colored red?
Rectangle area:
\[ A_\text{rect} = L \times W \]Red triangle: base \(L\), height \(\tfrac{W}{2}\), so
\[ A_\text{triangle} = \tfrac{1}{2} \cdot L \cdot \tfrac{W}{2} = \tfrac{1}{4} LW \]Percentage:
\[ P = \frac{\tfrac{1}{4} LW}{LW} = \frac{1}{4} = 25\% \]Answer: 25%.
Percent decrease = (original − new) / original × 100. For example, $20 → $15 gives (20 − 15)/20 × 100 = 25%.
Percent increase = (new − original) / original × 100. For example, 30 → 40 gives (40 − 30)/30 × 100 ≈ 33.33%.
Simple interest is computed only on the principal: \(I = P \times r \times t\). Compound interest is computed on principal plus accumulated interest: \(A = P(1 + r)^t\). Compound grows faster over time.
Set up two equations: one for the total amount (\(x + y = \text{total}\)) and one for the total amount of the substance (percent₁ × x + percent₂ × y = final percent × total). Substitute and solve.
Discount amount = original price × discount rate. Final price = original price × (1 − rate). For successive discounts, apply them in order — each discount uses the price after the previous discount as the base.
Last updated: October 17, 2026