12 SAT Math Word-Problem Types and Translation Examples

The fastest safe word-problem method is define quantities and units, translate one relationship, solve, then interpret the requested value. These 12 types cover common SAT models.

Type	Translation example	Result
Constant rate	180 miles in 3 hours: (r=180/3)	60 mph
Linear start + rate	25 fee plus 8/hour: (C=25+8h)	intercept 25, slope 8
Percent increase	80 increased 15%: (80(1.15))	92
Percent decrease	250 discounted 20%: (250(0.80))	200
Ratio parts	red:blue = 3:5, total 64: (3k+5k=64)	red 24
Direct variation	(y=kx), y=18 at x=6	(k=3)
Exponential growth	500 grows 4%/year: (500(1.04)^t)	model
Exponential decay	90% remains each period: (A=A_0(0.90)^t)	model
System/break-even	(20+5x=8x)	(x=20/3)
Geometry scale	lengths ×3	area ×9
Weighted mean	10 items avg 8 and 5 avg 14	((80+70)/15=10)
Probability	3 favorable of 12 equally likely	(1/4)

Translation rules that prevent errors

• “Per” signals division or a rate; attach units.
• “Of” in percent problems usually signals multiplication.
• “More than” adds to the reference; “times as many” multiplies it.
• A percent change divides by the original, not the new value.
• An exponential percent rate becomes a factor (1\pm r), not an additive slope.
• If length scales by (k), area scales by (k^2) and volume by (k^3).

A worked model

A gym charges 40 to join and 18 per month. Another charges no joining fee and 23 per month. Let (m\) be months. The costs are (40+18m\) and (23m\). Break-even occurs when (40+18m=23m\), so (m=8\). At eight months, both cost 184. The answer is a time, not the common cost, unless the question asks for cost.

College Board’s SAT Math overview organizes questions into Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry. Use our 25 word-problem types, math shortcuts, and question-type recognition guide to practice.

Before calculating, write the requested quantity and unit. Many “hard” word problems are correct equations followed by answering the wrong variable.

1–4: rates, linear models, and percent

1. Constant rate

A machine produces 420 parts in 7 hours. Divide output by time: (420/7=60) parts per hour. At the same rate, 12 hours produces (60(12)=720) parts.

Cue: “per,” “each,” and constant change. Trap: mixing minutes and hours.

2. Distance, rate, and time

A car travels 150 miles at 50 miles per hour. Since distance = rate × time, time is (150/50=3) hours.

Cue: travel or work completed at a steady rate. Trap: multiplying when the unknown is time.

3. Linear fee plus rate

A service charges 25 once plus 8 per hour: (C=25+8h). The intercept is dollars; the slope is dollars per hour. A $73 total gives (25+8h=73), so (h=6).

4. Percent change and reverse percent

A price rises from 80 to 92. Percent increase is ((92-80)/80=15%). If a price after a 20% discount is $72, the original satisfies (0.80p=72), so (p=90).

Trap: dividing change by the new value or adding 20% back to $72.

5–8: ratios, systems, and exponential models

5. Ratio parts

Red and blue marbles are in a (3:5) ratio with 64 total. Write (3k+5k=64), so (k=8). Red is 24 and blue is 40.

6. Ticket or mixture system

Adult tickets cost 12, student tickets 8, and 30 tickets total $300. Use (a+s=30) and (12a+8s=300). Substitution gives (a=15) and (s=15).

Cue: total count plus total value. Trap: reporting the wrong variable.

7. Exponential growth

A population begins at 500 and grows 6% each year: (P=500(1.06)^t). The 1.06 is a multiplier, while 0.06 is the rate.

8. Exponential decay

A 640-milligram sample loses 12% each hour: (A=640(0.88)^t). After three hours, use (640(0.88)^3), not (640-3(0.12)(640)).

Cue: same percentage each interval. Trap: treating percent change as linear subtraction.

9–12: averages, geometry, density, and probability

9. Weighted average

Eight scores average 70 and two scores average 90. Convert averages to totals: (8(70)+2(90)=740). Divide by 10 to get 74. Do not average 70 and 90 directly because group sizes differ.

10. Geometry dimensions and scale

A rectangle has perimeter 50 and length 15. (2(15)+2w=50), so (w=10), and area is 150. If the question asks for area, width is only an intermediate answer.

For similar figures, a length scale factor (k) produces area factor (k^2) and volume factor (k^3).

11. Density and unit conversion

Density = mass/volume. A 240-gram sample occupying 80 cubic centimeters has density (3\text{ g/cm}^3). Convert units before dividing when mass or volume uses different scales.

12. Probability and counts

A bag has 5 red, 3 blue, and 2 green marbles. The probability of blue is (3/10). For two draws without replacement, update both the favorable count and total after the first draw.

A fast translation template

For every problem, write:

1. Asked: the final quantity and unit.
2. Let: variable definitions with units.
3. Relationship: the equation in words.
4. Solve: algebra or Desmos.
5. Interpret: a sentence answering the original question.

Example: “Asked: months. Let (m) = months. One-time fee plus monthly charge equals total: (30+18m=156). Solve (m=7). The membership lasted seven months.”

Translation cue table

Language	Likely operation or model
per, each	rate or slope
of	multiplication
increased by (r%)	multiply by (1+r)
decreased by (r%)	multiply by (1-r)
total count + total cost	system
same percent each period	exponential
average	total/count
concentration	part/whole

These cues start the translation; context decides the exact equation.

A mixed-practice routine

Remove type labels and solve six questions. For each miss, tag:

• recognition: wrong model chosen;
• translation: right model, wrong equation;
• execution: algebra or calculator error; or
• interpretation: right calculation, wrong requested quantity.

Recognition improves with mixed examples. Translation improves by defining variables and units. Execution needs targeted algebra. Interpretation improves by restating the task after solving.

Estimate before submitting

A discounted price should be below the original. A combined average must lie between the group averages. Positive exponential growth should exceed the starting amount. A probability must be between 0 and 1.

Estimation does not replace exact work, but it catches incorrect operations and calculator entries in seconds.