SHSAT Word problems
The core skill is translation — turning English into math. Once translated, the math is often straightforward. Here's the translation strategy, the common problem types, and how to study.
- How many word problems are on the SHSAT?
- Roughly 15–20% of math questions are explicit word problems, but many other questions are framed in word-problem form. Functionally, translation skill affects performance across most of the math section.
- What's the core skill?
- Translating English statements into algebraic expressions or equations. Once translated, the math itself is often straightforward — the difficulty is in the translation step.
- What are the common word-problem types?
- Distance/rate/time, work/rate, mixture, age problems, percent and ratio applications, geometry word problems, and counting/probability problems. Each has a recognizable structure once you've seen a few examples.
- Should I write equations or just think it through?
- For two-step problems, mental math is often fine. For three-or-more-step problems, write the equation. The time saved by skipping setup is often lost to errors on complex problems.
The translation skill
Word-problem performance comes down to one core skill: turning English into math. Some common translation patterns:
| English | Math |
|---|---|
| "is", "equals", "results in" | = |
| "more than", "increased by", "sum of" | + |
| "less than", "fewer", "decreased by" | − |
| "of" (with a fraction or percent) | × |
| "per", "out of", "ratio of" | ÷ |
| "a number", "an unknown" | x (or any variable) |
| "twice a number", "three times" | 2x, 3x |
Word-order matters too: "three less than a number" is x − 3 (not 3 − x). "Five more than twice a number" is 2x + 5. Practice noticing the order, not just the operations.
Common word-problem types
Distance / rate / time
Core formula: distance = rate × time (d = rt). Rearranged: r = d/t, t = d/r.
Example: A car travels 240 miles in 4 hours. What is its average speed? r = d/t = 240/4 = 60 mph.
Common complications:
- Two trains, different speeds: Set up separate equations for each. If they meet at the same point: d_train1 = d_train2 (often) or d_train1 + d_train2 = total distance.
- Different units: Convert before computing. 30 minutes = 0.5 hours.
- Average speed for round trip: NOT the average of the two speeds. Use total distance / total time.
Work / rate
Core idea: combined work rate = sum of individual rates. If person A does a job in 6 hours, their rate is 1/6 per hour. If person B does it in 4 hours, their rate is 1/4. Together: 1/6 + 1/4 = 2/12 + 3/12 = 5/12. Time together: 1 ÷ (5/12) = 12/5 = 2.4 hours.
Mixture problems
Combining two solutions of different concentrations. Use a weighted average or a table:
Example: How much 30% saline must be mixed with 70% saline to make 10 liters of 50% saline? Setup: 0.30x + 0.70(10−x) = 0.50(10). Solve: 0.30x + 7 − 0.70x = 5, so −0.40x = −2, x = 5. Answer: 5 liters of each.
Age problems
Translate carefully — "in 5 years" means add 5 to ages; "5 years ago" means subtract.
Example: Sarah is 3 times as old as Tom. In 4 years, Sarah will be twice as old as Tom. How old are they now? Let Tom = t, Sarah = 3t. In 4 years: 3t + 4 = 2(t + 4). Solve: 3t + 4 = 2t + 8, so t = 4. Tom is 4, Sarah is 12.
Counting and probability
Basic counting principle: if event A has m outcomes and event B has n outcomes, the combined event has m × n outcomes. (For example: 3 shirts × 4 pants = 12 outfits.)
Probability: P(event) = favorable outcomes / total outcomes. P(rolling a 4 on a die) = 1/6.
For "or" problems with mutually exclusive events: add probabilities. P(rolling a 4 or 5) = 1/6 + 1/6 = 1/3. For "and" problems with independent events: multiply. P(heads on first AND second flip) = 1/2 × 1/2 = 1/4.
A general approach to any word problem
- Read the entire problem before doing anything. Identify what's being asked. Many students start solving for the wrong quantity because they jumped in halfway through.
- Define variables explicitly. Write "Let x = the number of apples" before doing any math. This makes errors easier to spot.
- Translate sentence by sentence. Each English sentence usually becomes one mathematical statement.
- Solve the math. Once you have an equation or expression, the algebra is usually straightforward.
- Verify the answer makes sense. If the problem asks for an age and you got −7, you made an error. If you got 250 for a small number of objects, you made an error.
- Re-read the question. Make sure you answered what was actually asked, not an intermediate quantity. (E.g., the problem asked for "the total cost," but you solved for the price per item.)
How to study word problems
- Practice translation isolated from solving. Take 10 word problems and just write the equations — don't solve. This builds the translation skill directly.
- Build a library of types. Once you've seen a few mixture problems, you'll recognize the pattern instantly. Most SHSAT word problems use familiar structures.
- Time yourself on translation specifically. Aim to read a problem and set up the equation within 30 seconds. The solving step is usually fast once the setup is right.
Common questions
How are word problems different from regular math problems?
The math itself is usually the same — algebra, arithmetic, geometry. The difference is the translation step: turning an English statement into an equation. Strong test-takers find the translation faster than the solving, but students often struggle with translation even when they can do the underlying math.
Should I always write equations?
For multi-step problems, yes. Writing the equation makes errors easier to spot and forces you to be explicit about what you're solving for. For one-step or two-step problems, mental math is often fine. As a default: if the problem has more than two pieces of information, write the equation.
What's the most common word-problem mistake?
Answering the wrong question. Students sometimes solve for an intermediate quantity (like "x") when the problem asked for "the total," or "the price per item" rather than "the number of items." Always re-read the final question after solving to verify your answer matches what was asked.
How long should word problems take?
Aim for about 90 seconds for a straightforward word problem and up to 2 minutes for a multi-step one. If you're consistently over 2 minutes, the issue is usually translation speed — practice that specifically.