14.2 Word Problems

Word Problems

Quick Answer: Translate English to an equation, assign a variable, set up the relationship, solve. Common templates: age, mixture, work, pipes, profit/loss, interest, probability.

Translating English to Equations

The core skill is reading the problem, naming the unknown x, and writing what the words say as mathematics. Many practice word problems fit one of a small set of templates; recognising the template is half the battle.

Age Problems

Worked example: Ali is 3 times as old as Bilal. In 5 years, Ali will be twice as old as Bilal. Find their ages now.

Let Bilal = x now, so Ali = 3x now. In 5 years: Ali = 3x + 5, Bilal = x + 5. Equation: 3x + 5 = 2(x + 5) 3x + 5 = 2x + 10 x = 5 So Bilal is 5 and Ali is 15.

Mixture Problems

Worked example: How much water must be added to 40 L of a 30% salt solution to dilute it to a 20% solution?

The amount of salt stays constant. Salt = 40 x 0.30 = 12 L. Let water added = w. New total volume = 40 + w. 0.20 x (40 + w) = 12 40 + w = 60 w = 20 L.

Work Problems (Combined Rate)

If worker A finishes a job in a days and worker B in b days, their combined rate is 1/a + 1/b (the fraction of the job done per day). Time working together is:

T = 1 / (1/a + 1/b) = ab / (a + b)

Worked example: A fills a tank in 6 hours, B in 3 hours. Together: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2. So T = 1 / (1/2) = 2 hours.

Pipes and Cisterns

Identical to work problems, but a draining pipe has a negative rate.

Worked example: Pipe A fills a cistern in 4 h; pipe B empties it in 6 h. With both open, the net rate is 1/4 - 1/6 = 3/12 - 2/12 = 1/12 per hour. Time to fill = 12 h.

Profit, Loss, and Discount

• Cost price (CP) is what the seller paid; selling price (SP) is what the buyer pays.
• Profit = SP - CP (when positive); Loss = CP - SP (when SP < CP).
• Profit % = Profit / CP x 100. Loss % = Loss / CP x 100. Both are computed on the cost price.
• Marked price (MP) minus discount equals selling price. Discount % is computed on the marked price.

Worked example: A shopkeeper marks an item 40% above cost (Rs 500). MP = 500 x 1.40 = Rs 700. He gives a 10% discount on the marked price. SP = 700 x 0.90 = Rs 630. Profit = 630 - 500 = Rs 130. Profit % = 130/500 x 100 = 26%.

Simple and Compound Interest

• Simple Interest (SI) = P x R x T / 100. Total amount = P + SI.
• Compound Interest (CI) = P x (1 + R/100)^T - P. Total amount = P x (1 + R/100)^T.

Worked example: Rs 10,000 at 10% per year for 3 years.

SI = 10,000 x 10 x 3 / 100 = Rs 3,000. Total = Rs 13,000. CI = 10,000 x (1.1)^3 - 10,000 = 10,000 x 1.331 - 10,000 = Rs 3,310. Total = Rs 13,310.

The CI exceeds the SI by Rs 310 because each year's interest earns interest in the next year.

Probability

Probability of an event = favourable outcomes / total outcomes.

Worked example: Probability of rolling a prime number on a fair six-sided die. The primes on a die are 2, 3, and 5 - three favourable outcomes out of six. P = 3/6 = 1/2.

Factorials, Permutations, Combinations

• 0! = 1 by definition (the empty product). This is a favourite test fact.
• nPr = n! / (n - r)! - order matters (arrangements).
• nCr = n! / (r! x (n - r)!) - order does not matter (selections).

Worked example: In how many ways can 3 cadets be selected from 5 for a flight? Order does not matter, so use combinations: 5C3 = 5! / (3! x 2!) = (120) / (6 x 2) = 10.

Common Word-Problem Templates

Strategy Checklist

1. Read the whole problem once before writing anything.
2. Name the unknown with a single variable.
3. Write the equation that mirrors the English sentence.
4. Solve and check the answer makes sense in the story (a negative age is a red flag).