7.3 Translating Word Problems into Math

7.3 Translating Word Problems into Math

Quick Answer: Find what is asked → mark the knowns → turn cue words into +, −, ×, or ÷ → write one clear expression or equation → solve → check against the story. The hard part is translation; the arithmetic is usually standard.

Word problems test whether you can move from English to mathematics. On SCD Mathematics items for the Fire Fighter entrance pathway, expect ordinary civil-service arithmetic stories—not advanced algebra theory and not technical firefighting calculations.

This section trains translation using everyday Trinidad and Tobago contexts: market purchases, transport fares, workshop attendance, simple wages, and rainfall-style totals presented as general practice data.

The translation pipeline

Use the same pipeline every time:

1. Read once for the story (who, what setting).
2. Read again for the question (underline the ask).
3. List known quantities with units.
4. Name the unknown (e.g. let C = change in dollars).
5. Translate cue words into operations.
6. Write the expression/equation.
7. Compute.
8. Check by reverse operation or substitution.

Skipping steps 4–6 is how candidates add when they should multiply, or answer a subtotal when the stem asked for change.

Cue words → operations

Careful: “more than” structures

• “7 more than a number n” → n + 7.
• “7 is more than a number by …” needs full sentence context.
• “How many more A than B?” → A − B (assuming A ≥ B).

Careful: “of”

• “Half of 40” → (1/2) × 40 = 20.
• “25% of 80” → 0.25 × 80 = 20.
• “Product of 6 and 9” → 6 × 9 = 54.

Defining the unknown

Write a short definition sentence:

• Let T = total cost in TTD.
• Let n = number of passengers.
• Let d = distance in km.
• Let p = percentage attendance.

If the question already gives the total and asks for one part, the unknown is the part, not the total.

Worked example — change

“A customer buys 2 shirts at $45 each and a belt for $30. She pays with $150. What change should she receive?”

• Known: 2 × 45, + 30, paid 150.
• Unknown: C = change.
• Translation: C = 150 − (2 × 45 + 30).
• Inside: 90 + 30 = 120; C = 150 − 120 = $30.

Wrong translations that match distractors: 2 × 45 = 90 (ignored belt and change); 150 − 45 = 105 (used one shirt only); 150 − 30 = 120 (ignored shirts).

Expressions vs equations

• An expression is a recipe without “equals answer” yet: 3 × 28 + 2 × 42.
• An equation sets expressions equal: 3 × 28 + 2 × 42 + 35 = T, or T = 203.

Multiple-choice items often only need a correct expression evaluated. Algebra “solve for x” still appears in simple form: “five more than twice a number is 27.”

Simple unknown number

“Five more than twice a number is 27. What is the number?”

Translation: 2n + 5 = 27.
2n = 22; n = 11.

Check: twice 11 is 22; plus 5 is 27. Good.

Multi-clause stories

Break long stems into clauses:

1. Compute intermediate results.
2. Feed them into the next clause.
3. Only then answer the final ask.

Worked example — bus fares and group cost

“Four adults pay $8 each and three children pay $5 each for a maxi-taxi ride. They also tip the driver $6. What is the total amount spent?”

• Adults: 4 × 8 = 32.
• Children: 3 × 5 = 15.
• Fares: 32 + 15 = 47.
• Total with tip: 47 + 6 = $53.

Expression: T = 4 × 8 + 3 × 5 + 6.

Order of operations: multiply first, then add—same as the clause order above.

Worked example — remaining after use

“A school receives 120 notebooks. It gives 8 notebooks to each of 12 classrooms and keeps the rest in storage. How many notebooks remain in storage?”

• Given out: 12 × 8 = 96.
• Remaining: 120 − 96 = 24.

Expression: 120 − (12 × 8).

Distractors: 12 × 8 = 96 (answered “given out” instead of remaining); 120 − 12 = 108 (ignored 8 each); 120 − 8 = 112 (ignored 12 classrooms).

Percent language in words

Worked example — discount

“A phone case marked $80 is sold at 25% off. What is the sale price?”

Discount amount: 0.25 × 80 = 20.
Sale price: 80 − 20 = $60.
Or: 80 × 0.75 = 60.

Distractors: $20 (discount only); $25 (confused percent with money); $80 (ignored discount).

Worked example — “what percent”

“In a Couva workshop, 21 of 28 trainees completed a module. What percent completed it?”

21 ÷ 28 = 0.75 → 75%.

Ratio and “for every” language

• “3 parts water for every 1 part concentrate” → ratio 3 : 1.
• Total parts = 4; if mixture is 8 litres, one part = 2 litres; water = 6 litres.

“For every 5 adults there are 2 children; 35 adults → children?”
Adults in groups of 5: 35 ÷ 5 = 7 groups; children = 7 × 2 = 14.

Rate language (without advanced formulas)

• “km per hour” → distance ÷ time.
• “pages per day” → pages ÷ days.
• “cost per kg” → total cost ÷ kilograms.

Worked example — simple rate

“A driver covers 150 km in 2.5 hours. What is the average speed in km/h?”

Speed = 150 ÷ 2.5.
2.5 × 60 = 150, so 150 ÷ 2.5 = 60 km/h.

Translation: s = d / t.

Parentheses from English grouping

Words that group quantities map to parentheses:

• “Twice the sum of 7 and 5” → 2 × (7 + 5) = 24, not 2 × 7 + 5 = 19.
• “Five more than the product of 3 and 4” → (3 × 4) + 5 = 17.

Mis-grouping is a major MCQ trap.

Building a translation table on scrap paper

For messy stems, sketch:

Then write T = 4(8) + 3(5) + 6.

Reverse checks

If reverse fails, the translation or arithmetic is wrong.

Common translation errors

1. Answering a middle step (subtotal) instead of the final ask.
2. Using numbers in the order they appear without operators (“12 8 3” jammed).
3. Ignoring “each.”
4. Mixing units (minutes with hours).
5. Percent as raw add (adding 25 instead of 25% of the base).
6. Wrong comparison direction (B − A when asked how many more A than B).

Full worked story — market and transport

“A vendor spends $240 on produce. She sells it all for $360. She then pays $45 for transport. What is her profit after transport?”

Translation:

• Gross profit on goods: 360 − 240 = 120.
• After transport: 120 − 45 = $75.
• Or: P = 360 − 240 − 45.

Check: 240 + 45 + 75 = 360. Yes—costs plus profit rebuild the sales figure.

Distractors: $120 (forgot transport); $45 (transport only); $360 (sales only).

Connecting to tables and estimation

• If data come from a table, translation still applies: the “knowns” are cells you extract first (Section 7.1).
• After solving, apply a reasonableness filter (Section 7.2): is profit larger than sales? Impossible—retranslate.

Study routine

1. Take any practice word problem and only write the expression before computing.
2. Cover the options; predict the answer type (money, people, percent, km/h).
3. Solve; uncover options; match.
4. For each wrong option you almost picked, name the translation mistake it represents.

Boundaries (again)

• No invented official item counts, times, or pass marks.
• No pump pressures, hose hydraulics, or Fire Service School technical word problems.
• Keep skills general: read → translate → compute → check.

Master translation and the arithmetic you already practise in earlier chapters becomes usable under exam wording.