SAT Math quiz

SAT Math Word Problems and Modeling

Practice 10 SAT Math word problems involving equations, functions, rates, percentages, units, systems, exponential growth, geometry, ratios, and multi-step mathematical modeling. Use hints when needed and review the explanation after each answer.

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1 Question 1 of 10

Not answered

A bike rental shop charges a fixed fee of $45 plus $12 for each hour a bike is rented. If a customer pays $117, for how many hours was the bike rented?

Hint: Model the total cost as 45 + 12h = 117.

Explanation: Let h be the number of rental hours. Then 45 + 12h = 117. Subtract 45 to get 12h = 72, so h = 6.

2 Question 2 of 10

Not answered

A printer produces 420 pages in 7 minutes at a constant rate. At this rate, how many pages will it produce in 18 minutes?

Hint: Find the unit rate in pages per minute, then multiply by 18.

Explanation: The printer rate is 420 ÷ 7 = 60 pages per minute. In 18 minutes it produces 60 × 18 = 1,080 pages.

3 Question 3 of 10

Not answered

After a 15% increase, the price of a service is $184. What was the price before the increase?

Hint: After a 15% increase, the new price is 115% of the original.

Explanation: Let P be the original price. Then 1.15P = 184, so P = 184 ÷ 1.15 = 160. The original price was $160.

4 Question 4 of 10

Not answered

At a theater, adult tickets cost $15 and student tickets cost $9. A total of 140 tickets are sold for $1,740. How many adult tickets were sold?

Hint: Let a be adult tickets and use 140 - a for student tickets.

Explanation: Let a be the number of adult tickets. Then 15a + 9(140 - a) = 1,740. This simplifies to 15a + 1,260 - 9a = 1,740, so 6a = 480 and a = 80.

5 Question 5 of 10

Not answered

A bacteria culture starts with 500 cells and increases by 20% every 3 hours. How many cells are expected after 9 hours?

Hint: Nine hours contains three 3-hour growth periods. Multiply by 1.20 for each period.

Explanation: There are 9 ÷ 3 = 3 growth periods. The model is 500(1.20)³ = 500 × 1.728 = 864 cells.

6 Question 6 of 10

Not answered

A machine dispenses liquid at 2.4 liters per minute for 35 minutes. How many milliliters of liquid does it dispense?

Hint: First find liters dispensed, then convert liters to milliliters.

Explanation: The machine dispenses 2.4 × 35 = 84 liters. Since 1 liter = 1,000 milliliters, 84 liters = 84,000 milliliters.

7 Question 7 of 10

Not answered

A circular garden has radius 6 feet. Edging costs $4.50 per foot. Using π = 3.14, what is the cost to place edging around the entire garden?

Hint: Find the circumference first, then multiply by the cost per foot.

Explanation: Circumference = 2πr = 2 × 3.14 × 6 = 37.68 feet. Cost = 37.68 × $4.50 = $169.56.

8 Question 8 of 10

Not answered

A 40-liter drink mixture contains concentrate and water in a ratio of 3:7. How many liters of concentrate must be added so that the new concentrate-to-water ratio is 1:2?

Hint: Find the original amounts of concentrate and water. The amount of water stays unchanged.

Explanation: The original mixture has 10 ratio parts, so each part is 4 liters. Concentrate = 12 liters and water = 28 liters. For a 1:2 ratio with 28 liters of water, concentrate must be 14 liters. Therefore, 2 liters of concentrate must be added.

9 Question 9 of 10

Not answered

A rectangular poster has an area of 84 square inches. Its length is 5 inches greater than its width. What is the length of the poster?

Hint: Let the width be w. Then the length is w + 5 and w(w + 5) = 84.

Explanation: The equation is w² + 5w - 84 = 0, which factors as (w + 12)(w - 7) = 0. The positive width is 7 inches, so the length is 7 + 5 = 12 inches.

10 Question 10 of 10

Not answered

A club has a $1,200 event budget. It spends 35% of the budget on equipment. The remaining money is divided between supplies and travel in a ratio of 3:2. How much money is allocated to travel?

Hint: Find the amount remaining after equipment, then take 2 of the 5 ratio parts.

Explanation: Equipment costs 0.35 × $1,200 = $420, leaving $780. The supplies-to-travel ratio is 3:2, so travel receives 2/5 of the remaining amount: (2/5) × $780 = $312.

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SAT Modeling Workshop

SAT word problems test whether you can turn a real situation into the right mathematical object

These 10 questions move across equations, rates, percentages, systems, exponential growth, unit conversion, geometry, ratios, quadratics, and multi-step budgets. The central skill is translation: deciding what the variables mean and which relationship connects them.

EquationsFunctionsRatesPercentages UnitsSystemsGeometryMulti-step models

Story-to-equation translation

Separate fixed quantities from variable quantities before solving

Situation $45 fixed fee plus $12 for each rental hour; total $117

Model 45 + 12h = 117 → h = 6

Unit-rate engine

Normalize to one unit when a constant rate is hidden

A printer produces 420 pages in 7 minutes. Convert that relationship into pages per minute before scaling to 18 minutes.

420 ÷ 7 = 60 pages/minute

60 × 18 = 1,080 pages

Check: longer time should produce more than 420 pages

Reverse percent vs exponential growth

Percent language can describe a one-time multiplier or repeated growth

Reverse percentage

$184 is the final price after a 15% increase, so it represents 115% of the original.

1.15P = 184 → P = 160

Repeated growth

A 20% increase every 3 hours over 9 hours produces three growth periods.

500(1.20)^3 = 864

Constraint model

When a story gives both a total count and a total value, use both constraints

Ticket count

If a is the number of adult tickets, student tickets are 140 − a.

a + s = 140

Revenue

Adult tickets contribute $15 each and student tickets contribute $9 each.

15a + 9(140 − a) = 1740

Unit conversion chain

In multi-unit problems, conversion is part of the model rather than an afterthought

Rate2.4 L/min

Time35 min

Volume84 L

Convert84,000 mL

Geometry inside a financial context

First find the geometric quantity; then apply the contextual rate

Edging is priced per foot, so the needed geometric quantity is circumference, not area.

circumference = 2 × 3.14 × 6 = 37.68 ft

cost = 37.68 × $4.50

final cost = $169.56

Multi-step model chain

Intermediate results should stay labeled so they can be reused correctly

The event-budget problem mixes a percentage with a ratio, so it cannot be solved in a single operation.

1Budget$1,200 total

2Equipment35% = $420

3Remaining$780

4Travel share2/5 × 780 = $312

Modeling diagnostic

When an answer is wrong, identify which translation step failed

Variable meaningDid the variable represent the requested quantity?

RelationshipDid fixed, per-unit, percent, and total quantities enter correctly?

Units & scaleWere conversions and rates applied in compatible units?

Context checkDoes the numerical answer make sense back in the original story?

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