Word problem toolkit for Math 30-1

The five-step method

The idea: do not start calculating on the first read. Most mistakes happen because the problem was solved before it was understood.

Read — twice. The first time, get the story. The second time, underline the numbers and circle the actual question. Write down what the answer should look like: a number of ways? A height in metres? A time in years?
Picture it. Draw something — boxes for choices, a quick sketch of one cycle, a timeline, a box with its dimensions labelled. A picture turns words into structure.
Plan. Name the tool before you use it: counting principle, permutation, combination, exponential model, sinusoidal model, polynomial. Say why in a few words.
Solve. Only now do the math. Define your variable in words first, such as "let t be the number of years".
Check. Is the answer sensible? Does it answer the question that was actually asked, with units, rounded the way the situation needs? Then write it as a sentence.

Estimate before you solve

A ten-second estimate catches most big mistakes. If a car loses 15% of its value each year, it will take a few years — not 0.7 years or 70 — to drop to a third of its price. If your answer lands far from the estimate, find out why before moving on.

Watch out: finding a correct number that answers a different question. "How many years until" and "what is it worth after" use the same equation but want different things.

Words that give it away

The idea: word problems follow patterns. Certain phrases almost always point to a particular tool.

When you see…	It usually means…
"and", "then", one choice after another	Multiply the choices (counting principle)
"or", separate cases	Add the cases
"arrange", "in a row", "ranked", "president and vice-president"	Order matters: permutation
"choose", "select", "committee", "team", "group"	Order does not matter: combination
"at least one"	Total minus "none" — or add up the cases
"no repeats", "different digits"	Choices drop by one each time
"increases by 6% per year", "loses 15% each year"	Exponential, with growth factor 1 + r or 1 − r
"doubles every", "half-life"	Exponential with base 2 or ½, exponent time ÷ period
"how long until", with growth or decay	Take the log of both sides
"times as intense" on the Richter, pH or decibel scale	Powers of 10
"repeats", "cycle", "high tide and low tide", "rotation"	Sinusoidal model
"box made by cutting squares from the corners"	Polynomial volume — check the domain

Watch out: keywords are clues, not rules. "Choose a president" is still a permutation, because the position matters. Always ask the underlying question: does order matter?

Which tool? Try it

The idea: deciding which tool to use is the real skill. For each one, decide before you open the answer — no calculating needed.

A. A band of 4 is formed from 9 musicians. How many bands are possible?

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Combination, ₉C₄. A band is a group; picking the same four people in a different order gives the same band.

B. A town of 12 000 people grows by 2.5% each year. When will it reach 20 000?

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Exponential model, solved with logs. "Grows by a percentage each year" means multiplying by 1.025 each year, and "when" means the unknown is the exponent.

C. A licence plate has 3 letters followed by 3 digits, and repeats are allowed. How many plates are possible?

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Fundamental counting principle. Six separate positions, one choice after another: 26 × 26 × 26 × 10 × 10 × 10.

D. The height of a point on a spinning fan blade is recorded over time. Find its height after 0.3 seconds.

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Sinusoidal model. Spinning means the height repeats every rotation. Find the amplitude, midline and period first.

E. Gold, silver and bronze medals are awarded among 8 finalists. How many results are possible?

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Permutation, ₈P₃. The three places are different, so the order of the winners matters.

F. A committee of 4 is chosen from 5 teachers and 6 students. How many committees have at least one teacher?

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Combinations with the complement. All committees, ₁₁C₄, minus the committees with no teachers, ₆C₄.

Strategies by unit

The idea: each unit has its own kind of word problem, and each kind has a setup that works almost every time.

Counting problems

Draw a box for every decision and write the number of choices in it before multiplying.
Handle restrictions first. If a PIN cannot start with 0, fill that box before the others.
"Together" means glue. Treat people who must sit together as one block, then multiply by the ways to arrange inside the block.
"At least" means complement. Total minus the case you do not want is usually far shorter than adding cases.

Review the counting concepts

Exponential and logarithm problems

Find three things: the starting amount, the factor it is multiplied by, and how often that happens.
Write the model: A = A₀(factor)^{t ÷ period}. Growth of 6% gives a factor of 1.06; a loss of 15% gives 0.85; a half-life gives ½.
If the unknown is in the exponent, isolate the power, then take the log of both sides.

Review exponents and logarithms

Sinusoidal problems

Pull out the maximum, minimum and period before writing anything else.
Sketch one cycle on a labelled time axis, starting where the situation starts.
Choose the function that starts in the right place: cosine at a maximum, negative cosine at a minimum, sine at the midline going up.

Review sinusoidal modelling

Polynomial problems

Label the diagram with a variable, then write each dimension in terms of it.
Write the domain straight away. Lengths must be positive, which usually rules out some answers.
Keep every answer that fits the domain — sometimes there really are two.

Review polynomial functions

Worked examples

The idea: try each one yourself using the five steps, then open the solution to compare — not just the answer, but the thinking.

Example 1 · Counting with "at least"

A student council of 5 is chosen from 6 Grade 11 students and 7 Grade 12 students. How many councils include at least one Grade 11 student?

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ReadWanted: a number of councils. "At least one Grade 11" is the condition.

Picture13 students in total, choose 5.

PlanA council is a group, so use combinations. "At least one" suggests the complement: all councils minus councils with no Grade 11 students.

SolveAll councils: ₁₃C₅ = 1287. No Grade 11 students means all 5 from the Grade 12s: ₇C₅ = 21. So 1287 − 21 = 1266.

CheckAdding up the cases gives 210 + 525 + 420 + 105 + 6 = 1266. It matches.

There are 1266 possible councils with at least one Grade 11 student.

Example 2 · Depreciation

A car bought for $30 000 loses 15% of its value every year. After how many full years is it first worth less than $10 000?

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ReadWanted: a whole number of years. "Less than" means we need the first year it drops below.

Picture$30 000 → × 0.85 → × 0.85 → … until under $10 000. Estimate: a few years.

PlanExponential decay with factor 0.85. The unknown is the exponent, so use logs.

Solve30 000(0.85)^t = 10 000, so 0.85^t = 1/3 and t = log(1/3) ÷ log(0.85) ≈ 6.76.

Check6.76 is not a whole number of years. After 6 years: 30 000(0.85)⁶ ≈ $11 314 — still above. After 7: ≈ $9 617 — below. So the answer is 7.

The car is first worth less than $10 000 after 7 years.

Example 3 · Tides

High tide of 9 m occurs at 3:00 a.m., and the next low tide of 1 m is at 9:00 a.m. Write a model for the water depth, and find the depth at 10:00 a.m.

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ReadWanted: an equation, then one depth in metres. Let t be hours after midnight.

PictureA maximum at t = 3, a minimum at t = 9, so a full cycle is 12 hours.

PlanSinusoidal. The data starts at a maximum, so cosine shifted 3 right is the natural choice.

SolveAmplitude (9 − 1) ÷ 2 = 4, midline 5, b = 2π ÷ 12 = π/6. So h(t) = 4 cos[π/6 (t − 3)] + 5. At t = 10: h = 4 cos(7π/6) + 5 = 4(−0.866) + 5 ≈ 1.54 m.

Check10:00 is one hour after low tide, so the water should be just a little above 1 m. It is.

h(t) = 4 cos[π/6 (t − 3)] + 5, and the depth at 10:00 a.m. is about 1.5 m.

Example 4 · An open box

An open-top box is made from a 30 cm by 20 cm sheet of cardboard by cutting equal squares from each corner and folding up the sides. What size squares give a volume of 1008 cm³?

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ReadWanted: the side length of the squares, in centimetres.

PictureCut squares of side x. The base becomes (30 − 2x) by (20 − 2x), and the height is x. Since 20 − 2x must be positive, 0 < x < 10.

PlanA polynomial volume equation, solved with the factor theorem.

Solvex(30 − 2x)(20 − 2x) = 1008 expands to 4x³ − 100x² + 600x − 1008 = 0, or x³ − 25x² + 150x − 252 = 0. Testing x = 3 gives 0, so x − 3 is a factor, leaving x² − 22x + 84 = 0 and x = 11 ± √37, about 4.92 or 17.08.

Check17.08 is outside 0 < x < 10, so reject it. Both 3 and 4.92 fit: 3 × 24 × 14 = 1008. ✓

Squares of 3 cm, or of about 4.92 cm, both give a volume of 1008 cm³.

Traps to avoid

The idea: most lost marks on word problems come from a short list of habits. Run through it before you move on.

Before you move on, check that you…

answered the question asked — the time, not the height; the number of councils, not the total
included units — metres, years, dollars
rounded for the situation — you cannot buy 6.76 cars; "less than" and "at least" change which way to round
wrote percents as decimals — 15% is 0.15, and a 15% loss is a factor of 0.85
checked the domain — lengths positive, time not negative, roots not extraneous
kept every valid answer — polynomial and trig problems often have more than one
used radians or degrees consistently — and set your calculator to match
wrote a final sentence — it forces you to check the answer makes sense

Try a mixed quiz Back to Math 30-1