2.5 One-Step and Two-Step Word Problems with Unknown Quantities
Key Takeaways
- TABE word problems are classified by structure — join, separate, part-part-whole, and comparison — and each structure maps to a predictable equation.
- Represent the unknown with a letter or a box before computing; writing 47 + n = 82 prevents the single most common error, which is choosing the wrong operation.
- A two-step problem requires an intermediate quantity that the question never names; identify it explicitly before solving.
- Always close with a reasonableness check using rounding and mental math, which is an assessed skill in its own right at Levels E and M.
One-Step and Two-Step Word Problems with Unknown Quantities
Computation is only half of what Levels E and M assess. The other half is deciding which computation to perform. The DRC content specification asks candidates to use addition, subtraction, multiplication, and division to solve word problems "in situations involving equal groups, arrays, and measurement quantities," representing the problem "with equations with a letter standing for the unknown quantity," and to "assess the reasonableness of answers."
The Four Problem Structures
Almost every one-step item on TABE is one of four types. Learn to spot the structure and the equation writes itself.
| Structure | What is happening | Unknown can be… | Sample equation |
|---|---|---|---|
| Join (add to) | A quantity increases | result, change, or start | $47 + n = 82$ |
| Separate (take from) | A quantity decreases | result, change, or start | $150 - n = 92$ |
| Part–part–whole | Two parts make a whole | either part, or the whole | $p + 38 = 91$ |
| Compare | Two quantities set side by side | difference, larger, or smaller | $b = 24 + 15$ |
The important insight: the operation in the story is not always the operation you perform. A story about adding inventory can require subtraction if the starting amount is unknown.
A technician used 47 feet of conduit and had 82 feet total at the start of the day. Wait — read again: she had 47 feet, then received a delivery, and now has 82 feet. How much was delivered?
This is a join with the change unknown: $47 + n = 82$, so $n = 82 - 47 = \mathbf{35}$ feet. The story says "received," but the arithmetic is subtraction.
Step 1: Name the Unknown
Before touching numbers, write a sentence: "Let $n$ = the number of feet delivered." This one habit converts guessing into algebra and it is the same habit Levels D and A will demand of you later.
Step 2: Write the Equation, Then Solve
| Story | Equation | Solve | Answer |
|---|---|---|---|
| Bus had 34 riders, 12 got off. How many now? | $34 - 12 = r$ | direct | 22 riders |
| Bus had some riders, 12 got off, 22 remain. How many started? | $s - 12 = 22$ | add | 34 riders |
| A shelf holds 6 rows of cans, 144 cans total. Cans per row? | $6 \times c = 144$ | divide | 24 cans |
| Boxes hold 15 files each. How many boxes for 210 files? | $15 \times b = 210$ | divide | 14 boxes |
Two-Step Problems: Find the Hidden Quantity
A two-step problem contains a number the question never asks for but that you cannot proceed without. Name it.
Problem. A caterer buys 9 trays of rolls with 24 rolls per tray. She sets aside 36 rolls for staff and divides the rest equally among 6 tables. How many rolls does each table get?
| Step | Hidden quantity | Work |
|---|---|---|
| 1 | Total rolls | $9 \times 24 = 216$ |
| 2 | Rolls for tables | $216 - 36 = 180$ |
| 3 | Rolls per table | $180 \div 6 = \mathbf{30}$ |
Notice that the answer to Step 1 and the answer to Step 2 are both plausible-looking numbers, and TABE will put 216 and 180 in the answer choices. Finishing the problem is as important as starting it correctly.
A second pattern: two-step with a comparison
A part-time aide earns $16 per hour. A full-time aide earns $4 more per hour and works 38 hours. What does the full-time aide earn in a week?
- Hidden quantity — full-time hourly rate: $16 + 4 = $20$.
- Weekly pay: $20 \times 38 = \mathbf{$760}$.
The distractor $608 comes from using $16 instead of $20 — skipping the hidden step.
Step 3: Assess Reasonableness
DRC lists reasonableness checking as its own skill. Round every number to something friendly and redo the problem mentally:
- Caterer problem: about 9 × 25 = 225 rolls, minus about 36 leaves roughly 190, divided among 6 tables is roughly 30. The exact answer 30 is confirmed.
- Aide problem: $20 × 40 = $800, so $760 is sensible and $608 is not.
If your exact answer and your estimate disagree by more than a little, you made an arithmetic slip or chose the wrong operation. Estimating takes ten seconds and catches both.
Cue Words — Useful but Not Decisive
| Cue | Usually means | But beware |
|---|---|---|
| in all, total, altogether | add | "How many more in all?" can be subtraction |
| left, remaining, fewer | subtract | "3 fewer than 4 times x" is a two-step |
| each, per, every | multiply or divide | "cost per unit" divides; "cost for each of 6" multiplies |
| share, split, distribute | divide | check whether groups or group size is unknown |
Cue words are a first read, never the final decision. The structure of the sentence outranks any single word.
A hardware store had some boxes of screws in stock. It received a shipment of 148 boxes, and now has 403 boxes. Which equation represents the situation, and how many boxes were in stock before the shipment?
A food pantry packs 12 cases of rice with 18 bags per case. It reserves 40 bags for an emergency shelf and divides the remainder equally among 4 distribution sites. How many bags does each site receive?