The first challenge in many ACT Math word problems is deciding what to calculate. Before reaching for a formula, name the unknown, attach units to the numbers, and describe the relationship in a short sentence. Then build an equation that represents that sentence.
The current ACT Math section has 45 delivered questions in 50 minutes. It includes mathematical modeling and problems that combine foundational skills. ACT's section overview describes those expectations. The examples here are original practice explanations; they are not taken from official tests.
Use a four-line setup
For an unfamiliar problem, write four small pieces of information:
- Find: the quantity the question asks for, including its unit.
- Know: the relevant values and relationships.
- Model: an equation, table, diagram, or inequality.
- Check: whether the answer matches the requested quantity and the situation.
This does not mean copying the whole prompt. “Find total dollars; know fixed fee plus per-ticket fee” is enough for a straightforward cost model.
A correct equation for the wrong unknown still produces the wrong answer. Watch whether the question asks for one item's price, a group total, a difference, or a percentage.
Model a fixed cost plus a variable cost
An original scenario: a school club pays a $45 setup fee and $6 for each printed shirt. Its printing budget is at most $225. What is the greatest number of shirts it can order?
Let n be the number of shirts. The total cost is:
45 + 6n ≤ 225
Subtract 45 to obtain 6n ≤ 180, so n ≤ 30. Because shirts are counted in whole numbers, the greatest allowed order is 30 shirts.
Check it in context: 45 + 6 × 30 = 225. An order of 31 costs 231 and exceeds the budget.
Two common traps are dividing 225 by 6 without subtracting the setup fee, and treating “at most” as a strict less-than condition. The words define the model.
Keep rates attached to units
A tap fills a tank at 7 liters per minute. A drain removes 2 liters per minute. If the tank initially contains 15 liters and both rates remain constant, how long until it contains 60 liters?
The net increase is 7 − 2 = 5 liters per minute. The required increase is 60 − 15 = 45 liters, so time is 45 ÷ 5 = 9 minutes.
The unit calculation provides a check:
liters ÷ (liters/minute) = minutes
Adding the two rates would describe both pipes filling the tank. Subtracting them represents one filling and one draining. Do not combine rates until you understand the direction of each process.
If units differ, convert first. A rate in miles per hour cannot be multiplied directly by a time expressed in minutes without accounting for that difference.
Distinguish a percent change from a percent of the original
A jacket priced at $80 is discounted by 25%, then the discounted price increases by 10%. What is the final price before tax?
The discount leaves 75% of the original price:
80 × 0.75 = 60
The increase is applied to that new price:
60 × 1.10 = 66
The final price is $66. It is not $68, which would result from subtracting a net 15% from the original $80.
Percent changes are applied to specific bases. Write the base beside each percentage when the problem contains more than one step.
| Wording | Model |
|---|---|
| 15% of x | 0.15x |
| 15% more than x | 1.15x |
| 15% less than x | 0.85x |
| x is 15% of y | x = 0.15y |
| x is 15% greater than y | x = 1.15y |
Do not memorize these as word-matching tricks. Read which quantity is being compared with which.
Turn averages into totals
Four laboratory measurements have an average of 18 units. What must a fifth measurement be to make the five-measurement average 20 units?
The first four measurements total 4 × 18 = 72. The desired five-measurement total is 5 × 20 = 100. The fifth measurement must therefore be 100 − 72 = 28 units.
Averages often hide totals. Recovering the total makes the operation clearer than trying to manipulate the average directly.
For a weighted average, the groups do not necessarily contribute equally. If one group contains 10 observations and another contains 30, averaging the two group averages without weighting them treats unequal groups as equal. Use each group's total and divide by the combined count.
Translate a comparison in the right direction
A common error is reversing expressions such as “three fewer than twice the number.”
If x is a number, “three fewer than twice x” is 2x − 3, not 3 − 2x. Substitute an easy value to check. If x = 5, twice the number is 10 and three fewer is 7.
For an age or quantity comparison, label each variable explicitly. “Ava has four more tokens than Noah” can be written A = N + 4. If Noah has 6 tokens, Ava should have 10. The test value confirms the direction.
You do not need to retain the test value when solving the actual problem. It is a quick way to catch a translation error.
Choose between algebra, a table, and answer choices
Algebra is efficient when the relationship is clear. A table can help when quantities change in repeated steps. Substituting answer choices can be useful when the question asks for a specific unknown and each choice can be checked directly.
Consider a membership plan with a $20 fee plus $4 per visit, compared with another plan charging $6 per visit and no fee. The equal-cost equation is 20 + 4v = 6v, giving v = 10. A small table for 5, 10, and 15 visits would also reveal the relationship.
Choose the representation that reduces your chance of losing track of the situation. Using a calculator to evaluate a wrong model only produces that wrong answer faster.
Recognize when the answer needs an additional step
Suppose you solve for a circle's radius, but the question asks for its diameter. Or you find the number of adult tickets, but the question asks how many tickets were sold in total. The algebra may be finished before the question is answered.
Underline or briefly rewrite the requested quantity. After solving, compare your answer's label with that request.
Check size as well as units. A discounted price should be below the original price if no later charge changes it. The average of several positive values should lie between their minimum and maximum. A count cannot be negative in a situation involving physical objects.
Review the point where the model went wrong
An error log for word problems should distinguish:
- Misidentified unknown.
- Reversed comparison.
- Wrong percentage base.
- Incompatible units.
- Incorrect equation despite correct reading.
- Correct equation followed by an arithmetic error.
- Correct intermediate result reported as the final answer.
Those categories lead to different practice tasks. If your equations are sound but fractions cause mistakes, more reading strategy is unlikely to fix the main problem.
Use ACT's mathematics standards to identify skills, and use the ACT score calculator only with the appropriate official practice form. A raw total cannot tell you which modeling habit needs attention.
A short practice routine
Choose three unfamiliar problems involving different relationships. Work untimed and show your four-line setup. Check the solutions, then rewrite one problem with a changed number and predict how the answer should move before recalculating.
On a later day, attempt fresh problems under a modest time limit. If speed improves while your explanations remain correct, the model-building process is becoming more reliable. If explanations disappear and guessing returns, shorten the set and rebuild the setup habit.
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