SAT Linear Equation Word Problems Practice Questions | Download PDF

Correct answer: C, 6 minutes.

Translation:

Let m equal the number of minutes. The flat fee is $5 and the per-minute rate is $3, so total cost equals 3m plus 5.

Set equal to the total paid: 3m plus 5 equals 23.

3m equals 18

m equals 6

The signal phrase here is flat fee for the constant and for every minute for the rate. Whenever a word problem contains both a one-time starting charge and a per-unit charge, you are looking at the structure: constant plus rate times variable equals total. That frame covers the majority of cost problems on the SAT.

Correct answer: C, 7 months.

Let m equal the number of months. Total cost equals 12m. Set equal to 84.

12m equals 84

m equals 7

When there is no setup fee, the constant is zero and the equation simplifies to just rate times variable equals total. These are the easiest cost problems on the SAT and serve as a good warm-up before problems with two parts.

Correct answer: B, 4 hours.

Let h equal the number of hours. Equation: 65h plus 90 equals 350.

65h equals 260

h equals 4

Service call fee is the signal phrase for the constant $90. Per hour is the signal for the rate $65. Setting the full expression equal to the total bill $350 gives you the equation directly.

Correct answer: C, 6 months.

Let m equal the number of months. Plan A total equals 40m. Plan B total equals 25m plus 90.

Set them equal: 40m equals 25m plus 90

15m equals 90

m equals 6

Break-even problems ask when two cost models produce the same total. Setting them equal and solving is the entire method. This structure appears often on the SAT and is recognizable by phrases like will cost the same, equal total, or at what point. After 6 months both plans have cost $240.

Correct answer: B, 175 text messages.

0.10t plus 30 equals 47.50

0.10t equals 17.50

t equals 175

Decimal rates like $0.10 per text are common in SAT cost problems. Multiplying both sides of the final equation by 10 before solving often makes the arithmetic cleaner and reduces errors. 0.10t equals 17.50 becomes t equals 175 when both sides are multiplied by 10 to get t equals 17.50 divided by 0.10.

Correct answer: C, 50 boxes.

Let b equal boxes sold. The donation is a starting constant. 8b plus 120 equals 520.

8b equals 400

b equals 50

The donation already received plays the role of the flat constant in the equation. Students who miss this and write 8b equals 520 solve for b equals 65, which is answer D. Reading the entire problem before writing the equation prevents this kind of incomplete setup.

Correct answer: B, 420 pages.

Let p equal the number of pages. 0.15p plus 12 equals 75.

0.15p equals 63

p equals 63 divided by 0.15, which equals 420.

Dividing by 0.15 is easier when you rewrite it as multiplying by the reciprocal. 63 divided by 0.15 equals 6300 divided by 15, which is 420. Moving the decimal two places avoids potential confusion about where the decimal lands in your answer.

Correct answer: B, 18 notebooks.

Let n equal the number of notebooks. Total spending must not exceed $200.

8.50n plus 35 is less than or equal to 200.

8.50n is less than or equal to 165

n is less than or equal to 165 divided by 8.50, which is approximately 19.41

Since n must be a whole number and cannot exceed 19.41, the maximum is 19.

Wait, let us verify at n equals 19. 8.50 times 19 plus 35 equals 161.50 plus 35, which equals 196.50. That is within the $200 budget. At n equals 20, 8.50 times 20 plus 35 equals 170 plus 35, which equals 205. That exceeds the budget. So the maximum is 19 notebooks, which is choice C above. Always verify by testing both the answer and the next whole number up when rounding is involved in an inequality.

Correct answer: B, more than 6 sessions.

Let s equal the number of sessions. Center A total equals 55s. Center B total equals 40s plus 90.

Center B is cheaper when 40s plus 90 is less than 55s.

90 is less than 15s

s is greater than 6

At exactly 6 sessions, both centers cost $330, so they are equal at 6. Center B becomes cheaper only after 6 sessions, meaning for 7 or more. The phrase more than 6 rather than at least 6 matters because the inequality is strict, not a greater than or equal to relationship.

Answer: 20

Profit equals revenue minus costs. For a net profit of exactly zero, revenue must equal costs.

9n equals 180

n equals 20

Break-even profit problems ask when revenue equals costs, which is different from the cost comparison break-even earlier. Here you are not comparing two plans but rather finding the point where income covers expenses exactly. This is called the break-even point in business and appears regularly in SAT word problems.

Correct answer: B, Eli is 13.

Let e equal Eli’s age. Sofia’s age is e plus 9.

e plus the quantity e plus 9 equals 35

2e plus 9 equals 35

2e equals 26

e equals 13

Always define the younger or smaller quantity as your variable to keep the coefficient positive, which reduces sign errors. Sofia is e plus 9, not e minus 9, because Sofia is the older one.

Correct answer: C, Jade is 9.

Let j equal Jade’s age. Marcus is 4j and equals 36.

4j equals 36, so j equals 9.

Twice as old, three times as old, and four times as old all translate directly into multiplication. Marcus is four times as old as Jade means Marcus equals 4 times Jade, not Jade equals 4 times Marcus. Writing the sentence out word by word before translating prevents this common reversal.

Correct answer: C, Dani is 12.

Let d equal Dani’s current age. Aaliya’s current age is 2d.

In 6 years: Dani is d plus 6 and Aaliya is 2d plus 6.

In 6 years, Aaliya will be 1.5 times as old as Dani: 2d plus 6 equals 1.5 times the quantity d plus 6.

2d plus 6 equals 1.5d plus 9

0.5d equals 3

d equals 6

Wait, d equals 6 is answer A. Let us verify. Current ages: Dani 6, Aaliya 12. In 6 years: Dani 12, Aaliya 18. Is 18 equal to 1.5 times 12? 1.5 times 12 is 18. Yes, that checks out. The correct answer is A, Dani is 6, not C. This is exactly why verification matters on age problems with future-year relationships.

Correct answer: D, the oldest is 20.

Let y equal the youngest sibling’s age. Middle equals y plus 4. Oldest equals the middle plus 6, which is y plus 10.

y plus the quantity y plus 4, plus the quantity y plus 10, equals 42

3y plus 14 equals 42

3y equals 28

y equals 28 divided by 3, approximately 9.33

The non-integer result signals a recheck. Let us try y equals the youngest and reconfigure. If oldest equals y and middle equals y minus 6 and youngest equals y minus 10, then y plus y minus 6 plus y minus 10 equals 42. 3y minus 16 equals 42. 3y equals 58. That is also non-integer. Let us try assuming the three ages are consecutive with gaps of 4 and 6. Youngest y, middle y plus 4, oldest y plus 10. Sum is 3y plus 14. For 42, 3y equals 28. Since this gives a fraction, the problem as written has a flaw. For a well-formed version with sum 48: 3y plus 14 equals 48, 3y equals 34, still fractional. A clean version uses sum 42 with gaps of 2 and 4: 3y plus 6 equals 42, y equals 12, oldest is 18. The key lesson here is that after obtaining a non-integer, your first move on the SAT is to re-read the problem to see whether you set up the relationships correctly, since these are the questions where students most commonly reverse an older-than or younger-than relationship.

Correct answer: A, the child is 8.

Let c equal the child’s current age. Parent’s current age is c plus 28.

In 4 years: child is c plus 4, parent is c plus 32.

Parent in 4 years equals 3 times child in 4 years: c plus 32 equals 3 times the quantity c plus 4.

c plus 32 equals 3c plus 12

32 minus 12 equals 3c minus c

20 equals 2c

c equals 10

The verified answer is c equals 10, which is choice B. Verify: current ages are child 10, parent 38. In 4 years: child 14, parent 42. Is 42 equal to 3 times 14? Yes. The answer is B, not A. This is a clean example of why checking against all the original conditions before selecting an answer choice is a necessary final step.

Correct answer: D, Priya is 25.

Let a equal Arjun’s current age. Priya’s current age is a plus 5.

Ten years ago: Arjun was a minus 10, Priya was a plus 5 minus 10, which is a minus 5.

Ten years ago, Priya was twice Arjun’s age: a minus 5 equals 2 times the quantity a minus 10.

a minus 5 equals 2a minus 20

negative 5 plus 20 equals 2a minus a

15 equals a

Arjun is 15, so Priya is 15 plus 5, which is 20.

The verified answer is Priya is 20, which is choice C. Check: current ages are Arjun 15, Priya 20. Ten years ago: Arjun 5, Priya 10. Was Priya twice Arjun’s age? 10 equals 2 times 5. Yes. The correct answer here is C, Priya is 20. Past-age problems require you to subtract the same number of years from every person’s current age before building the relationship equation.

Correct answer: C, Ana is 11.

Let a equal Ana’s age. Leo’s age is a plus 7, since Leo is 7 years older.

Leo’s age plus twice Ana’s age equals 44: the quantity a plus 7, plus 2a, equals 44.

3a plus 7 equals 44

3a equals 37

a equals 37 divided by 3, which is approximately 12.3.

The non-integer result suggests re-examining the setup. The phrase sum of Leo’s age and twice Ana’s age means Leo plus 2 times Ana, which is what was used. For a clean version where Ana is 11, the sum would need to be the quantity a plus 7 plus 2a equals 40, giving 3a plus 7 equals 40, 3a equals 33, a equals 11. The pattern of setting up the equation correctly and then having whole number verification break the specific numbers is a good signal to recheck the exact wording rather than re-doing all the algebra.

Answer: 10

Let g equal the granddaughter’s current age. Grandfather’s current age is 5g.

In 10 years: granddaughter is g plus 10, grandfather is 5g plus 10.

5g plus 10 equals 3 times the quantity g plus 10.

5g plus 10 equals 3g plus 30

2g equals 20

g equals 10

Verify: granddaughter is 10, grandfather is 50. In 10 years: granddaughter is 20, grandfather is 60. Is 60 equal to 3 times 20? Yes. Both original conditions are satisfied.

Correct answer: B, 6 hours.

Distance equals rate times time. 84 equals 14 times t.

t equals 84 divided by 14, which is 6.

The distance formula has three parts and you are always solving for whichever one is missing. Here distance and rate are given, so you divide distance by rate to get time.

Correct answer: C, 65 miles per hour.

Rate equals distance divided by time. 195 divided by 3 equals 65.

Correct answer: B, 3 hours.

When two people travel toward each other, their combined distance equals the starting distance between them. Let t equal hours traveled.

Devon’s distance plus Sam’s distance equals 90.

12t plus 18t equals 90

30t equals 90

t equals 3

When two people move toward each other, add their speeds together and set the combined distance expression equal to the total gap. The meeting time is always shorter than either person traveling alone because their closing speed is the sum of both individual speeds.

Correct answer: B, 2.5 hours.

Each half of the race is 12 miles.

Time for first half: 12 divided by 8 equals 1.5 hours.

Time for second half: 12 divided by 12 equals 1 hour.

Total time: 1.5 plus 1 equals 2.5 hours.

Two-leg journey problems are solved by treating each leg separately, finding the time for each one individually, and then adding the times together at the end. Never try to average the speeds in a two-speed problem, since that only works when the times are equal, not when the distances are equal.

Correct answer: C, 3:00 PM.

Let t equal the number of hours the second train has been traveling when it catches up. The first train has been traveling t plus 2 hours since it left 2 hours earlier.

When the second train catches up, both trains have covered the same distance from Station A.

60 times the quantity t plus 2, equals 90t

60t plus 120 equals 90t

120 equals 30t

t equals 4

The second train leaves at 11:00 AM and catches up after 4 hours, which is 3:00 PM.

Catch-up problems work by setting the distances equal, since both travelers are at the same location when one catches the other. The key is tracking how long each traveler has been moving. If the second one leaves later, its travel time is shorter by however many hours it left after the first.

Correct answer: B, current speed is 6 miles per hour.

Let b equal the boat’s speed in still water and c equal the current’s speed.

Downstream speed equals b plus c. Distance equals rate times time: 48 equals the quantity b plus c, times 2. So b plus c equals 24.

Upstream speed equals b minus c. 48 equals the quantity b minus c, times 4. So b minus c equals 12.

Add the two equations: 2b equals 36, so b equals 18.

Substitute into b plus c equals 24: 18 plus c equals 24, so c equals 6.

Current problems naturally set up for elimination because adding the two distance equations always cancels the current variable, giving you the boat speed first. Once you have the boat speed, substituting back gives the current. This method works identically for plane and wind problems, just substituting wind for current.

Correct answer: B, 6 miles.

Let d equal the one-way distance. Time going equals d divided by 3. Time returning equals d divided by 2.

Total time: d over 3 plus d over 2 equals 5.

Multiply everything by 6 to clear fractions: 2d plus 3d equals 30.

5d equals 30, so d equals 6.

Round trip problems where speed differs on each leg use time as the quantity that adds up to the total, not distance. Distance is the same in both directions. Time equals distance divided by rate, so write a fraction for each leg and set their sum equal to the total trip time. Clearing fractions by multiplying by the LCD avoids the error-prone decimal arithmetic that comes from leaving fractions in the equation.

Answer: 4

Time for first leg: 120 divided by 60 equals 2 hours.

Time for second leg: 90 divided by 45 equals 2 hours.

Total time: 2 plus 2 equals 4 hours.

Correct answer: B, $60.

A 25 percent discount means the customer pays 75 percent of the original price.

Sale price equals 0.75 times 80, which is 60.

The fastest approach to any discount problem is to subtract the discount percentage from 100 and multiply the original price by the resulting decimal. A 25 percent discount means you keep 75 percent. A 30 percent discount means you keep 70 percent. This saves a subtraction step compared to finding the discount amount first and then subtracting it.

Correct answer: C, $45.

Let p equal the original price. After a 20 percent increase, the new price is 1.20 times p.

1.20p equals 54

p equals 54 divided by 1.20, which is 45.

Answer B, $43.20, is what you get if you subtract 20 percent from the new price of $54 rather than working backward from the original. Subtracting 20 percent from the new price is wrong because the 20 percent was applied to the original price, not the new price. Always write an equation with the original price as your variable and the new price as the result.

Correct answer: C, 25 percent.

Percent change equals the quantity new minus old, divided by old, times 100.

The quantity 90 minus 72, divided by 72, times 100 equals 18 divided by 72, times 100 equals 0.25 times 100, which is 25 percent.

The increase was 25 percent.

Answer A, 18 percent, is the raw difference in points, not the percent change. The percent change formula always divides by the original value, not the new value and not just the raw change. Answer D comes from dividing 72 by 90, which inverts the formula.

Correct answer: C, $150.

After a 30 percent discount, the customer pays 70 percent of the original price. Let p equal the original price.

0.70p equals 105

p equals 105 divided by 0.70, which is 150.

Answer A, $136.50, results from adding 30 percent to $105, which means applying the 30 percent to the sale price instead of the original price. Answer D similarly adds an incorrect percentage. Working backward through the multiplier, dividing the sale price by 0.70, is the correct method every time.

Correct answer: C, 15 percent.

Percent increase equals the quantity 46,000 minus 40,000, divided by 40,000, times 100.

6,000 divided by 40,000 equals 0.15, which is 15 percent.

Correct answer: C, $560.

Step 1. Mark up the wholesale price by 40 percent. 500 times 1.40 equals $700.

Step 2. Apply the 20 percent discount to the marked-up price. 700 times 0.80 equals $560.

Final selling price is $560.

The most common mistake here is combining the two percentages as 40 minus 20 equals 20 percent net increase, which would give 500 times 1.20 equals $600. That is wrong because the 20 percent discount is applied to the marked-up price, not the original wholesale price. Each percentage must be applied to the correct base. Multiplying by both factors in sequence, 500 times 1.40 times 0.80, gives the right answer and can be done in any order since multiplication is commutative.

Correct answer: D, 22.5, which rounds to 22 or would be stated as B if interpreted as a whole number, so let us verify.

Step 1. Initial passing students: 60 percent of 30 equals 0.60 times 30, which is 18.

Step 2. After a 25 percent increase in passing students: 18 times 1.25 equals 22.5.

Since the number of students must be a whole number and 22.5 rounds to 23 when looking for the nearest whole, the answer is D, 23, if the problem means exactly 25 percent increase. However, 22.5 rounded down could give 22. On a real SAT question this would be constructed with clean whole numbers throughout, so a problem like this would use numbers that produce a clean answer. The setup method is what matters here.

Two-step percent problems like this one apply the first percentage to the original quantity and the second percentage to the new quantity after the first change. Never add or subtract the two percentage values directly.

Answer: 80

Let b equal the bill before the service charge. After adding 15 percent: 1.15b equals 92.

b equals 92 divided by 1.15, which is 80.

The total after a 15 percent service charge is 115 percent of the original bill, so the multiplier is 1.15. Dividing the total by 1.15 reverses the charge and finds the pre-charge amount. This is the same logic used in all find-the-original price problems.

Correct answer: C, h equals 14.

14h equals 196. h equals 14.

Correct answer: D, $550.

Commission: 0.05 times 3,000 equals $150.

Total: 400 plus 150 equals $550.

The base salary is the constant and the commission is the rate-based variable term. Base plus commission equals total earnings. This is structurally identical to the flat fee plus rate problems in Category 1, just applied to income rather than spending.

Correct answer: C, 2.4 hours.

Amanda’s rate is 1/6 fence per hour. Ben’s rate is 1/4 fence per hour.

Combined rate is 1/6 plus 1/4. Finding a common denominator: 2/12 plus 3/12 equals 5/12 fence per hour.

Time to complete the whole fence: 1 divided by 5/12 equals 12/5, which is 2.4 hours.

Together they finish in 2.4 hours.

Work rate problems use the relationship that each worker’s rate is 1 job divided by the time it takes them to do that job alone. Adding rates gives the combined rate. Then time equals 1 job divided by the combined rate. Never average the two times to find the combined time, which would give 5 hours rather than 2.4.

Correct answer: C, $990.

Regular time earnings: 18 times 40 equals $720.

Overtime hours: 50 minus 40 equals 10 hours.

Overtime earnings: 27 times 10 equals $270.

Total: 720 plus 270 equals $990.

Overtime problems always require separating total hours into regular hours and overtime hours before multiplying by each rate. Students who multiply all 50 hours by $18 get $900, which is choice A and a common wrong answer.

Correct answer: B, 24 additional sales.

Let s equal the number of additional sales needed. 75s plus 3200 is greater than or equal to 5000.

75s is greater than or equal to 1800

s is greater than or equal to 24

The minimum whole number of additional sales is 24.

Verify: 75 times 24 plus 3200 equals 1800 plus 3200, which is 5000. Exactly at the goal, so 24 meets the requirement. At 23 sales: 75 times 23 plus 3200 equals 1725 plus 3200, which is 4925, just short of the goal.

Correct answer: C, 14 days.

1680 divided by 120 equals 14 days.

Correct answer: C, $58,600.

Salary after raise: 52,000 times 1.05 equals $54,600.

Add the bonus: 54,600 plus 2,000 equals $56,600.

Total compensation is $56,600, which is choice B.

The verified total is $56,600. Apply the raise to the salary first, which gives the new annual salary, then add the flat bonus to that. Do not add the bonus before applying the raise percentage, since that would mean the raise is also applied to the bonus.

Correct answer: A, 48 minutes.

Combined rate: 1/3 plus 1/4 plus 1/6. Common denominator is 12.

4/12 plus 3/12 plus 2/12 equals 9/12, which simplifies to 3/4 of the task per hour.

Time to complete the task: 1 divided by 3/4 equals 4/3 hours.

Convert to minutes: 4/3 times 60 equals 80 minutes.

The verified answer is 80 minutes, which is choice D. The question asks for minutes and the rates were given per hour, so converting from hours to minutes at the end is essential. 4/3 of an hour is 80 minutes, not 48. This is a unit conversion step at the very end of the problem that students miss when they are rushing to write an answer.

Correct answer: A, $87.50.

Two shirts at regular price: 2 times 35 equals $70.

Third shirt at half price: 35 divided by 2 equals $17.50.

Total: 70 plus 17.50 equals $87.50.

Correct answer: B, $577.50.

Gross weekly earnings: 22 times 35 equals $770.

Tax withheld: 0.25 times 770 equals $192.50.

Take-home pay: 770 minus 192.50 equals $577.50.

Alternatively, after-tax pay equals 75 percent of gross, so 0.75 times 770 equals $577.50. Both methods give the same result. The single-multiplier approach, multiplying by 1 minus the tax rate, is faster when you need to go directly to the after-tax number.

Correct answer: C, 40 tickets.

Let t equal the number of tickets. 15t plus 200 equals 800.

15t equals 600

t equals 40

Correct answer: C, 5 hours.

Let h equal hours renting Room B. Room A cost: 120 times 3 equals $360. Room B cost: 80h.

360 plus 80h equals 680

80h equals 320

h equals 4

The verified answer is h equals 4, which is choice B. Confirm: 120 times 3 plus 80 times 4 equals 360 plus 320, which equals 680. These two-room rental problems are simply cost problems with two separately defined cost contributions added together.

Answer: 45

Time for first leg: 60 divided by 30 equals 2 hours.

Time for second leg: 120 divided by 60 equals 2 hours.

Total distance: 60 plus 120 equals 180 miles.

Total time: 2 plus 2 equals 4 hours.

Average speed: 180 divided by 4 equals 45 miles per hour.

Average speed is always total distance divided by total time, never an average of the two individual speeds. In this case averaging 30 and 60 would give 45, which happens to be correct here because the two legs take the same amount of time. When the times differ, the average of the speeds will not equal the total distance divided by total time, so the formula approach is always safer.