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. Every word problem on the PAF GD Pilot test fits 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).