Arithmetic word problems: a method

Word problems test the same arithmetic as every other quant question, wrapped in a story about a shopkeeper, a cashier or a salary. The maths is rarely hard. What slows candidates down is turning three lines of English into one line of algebra, and doing it under a 20-minute sectional timer.

That translation is a skill you can train. Once you have a fixed routine, a long question stops looking long. This guide gives you the routine, a phrase-by-phrase translation table, and practice on the kinds of situations the clerk exam likes: stock sold in stages, notes of two denominations, fractions of a salary, prices of two items.

A five-step method

1. Read the whole question once before writing anything, and underline what is being asked. Many mistakes come from answering a different question, such as finding the ₹100 notes when the question asked for ₹500 notes.
2. Name the unknown. Let x be the thing asked for, or the thing everything else depends on (the original stock, the salary, the number of adult tickets).
3. Write the relation, one sentence at a time. Each sentence of the question usually becomes one expression.
4. Solve. Most clerk-level word problems give one linear equation or a pair of them.
5. Check the answer against every fact in the question, not just the last one. It takes five seconds and catches most slips.

Translating words into symbols

Two ways of handling "the rest"

Questions that take a fraction of the whole and then a fraction of what is left are the most common trap. Track what remains at each step as a fraction of the original. If 40% is sold, 60% remains. If half of the remaining is sold next, that is half of 60%, which is 30% of the original, and 30% remains.

Worked examples

Example 1. A shopkeeper sells 40% of the stock on the first day and half of the remaining stock on the second day. 150 items are left. How many items were there at the start?

1. After day 1: 60% left. Day 2 sells half of that, which is 30%. Left: 30%.
2. 30% of the stock = 150, so the stock = 500.
3. Check: 500 → sell 200 → 300 → sell 150 → 150 left.

Example 2. The sum of three consecutive odd numbers is 63. Find them.

1. x + (x + 2) + (x + 4) = 63, so 3x + 6 = 63 and x = 19.
2. The numbers are 19, 21 and 23.

Example 3. A cashier has 30 notes of ₹500 and ₹100, worth ₹7,000 in all. How many ₹500 notes are there?

1. Let x be the ₹500 notes. Then there are 30 − x notes of ₹100.
2. 500x + 100(30 − x) = 7,000, so 400x + 3,000 = 7,000 and x = 10.
3. 10 notes of ₹500 and 20 of ₹100. Check: 5,000 + 2,000 = 7,000.

Example 4. A number increased by 20% becomes 300. Find the number.

1. 1.2x = 300, so x = 250.
2. Adding 20% of 300 to 300 would be wrong: the 20% is of the original number, not of 300.

Example 5. A candidate spends a quarter of their monthly income on rent and a third of the remainder on food, and saves the rest, which is ₹12,000. Find the monthly income.

1. After rent: ¾ of the income remains.
2. Food = ⅓ of ¾ = ¼ of the income. Left = ¾ − ¼ = ½ of the income.
3. ½ of the income = 12,000, so the income = ₹24,000.
4. Check: rent 6,000, remainder 18,000, food 6,000, savings 12,000.

Example 6. A fair sells adult tickets at ₹50 and child tickets at ₹30. It sells 120 tickets for ₹5,000. How many adult tickets were sold?

1. Let a be the adult tickets. Children = 120 − a.
2. 50a + 30(120 − a) = 5,000, so 20a + 3,600 = 5,000 and a = 70.
3. Check: 70 × 50 + 50 × 30 = 3,500 + 1,500 = 5,000.

Working backwards from the options

When the equation is awkward, put each option into the story. In Example 6, suppose the options are 60, 65, 70, 75 and 80. Try 70: 3,500 + 1,500 = 5,000. It fits, so you are done. Start with the middle option: if it gives too much or too little, you know which way to move.

Common mistakes

• Taking a fraction of the original when the question says "of the rest" or "of the remaining".
• Finding x and then reporting it, when the question asked for 30 − x or for the total.
• Treating "increased by 20% becomes 300" as 300 − 20% of 300.
• Skipping the check. A thirty-second question can cost a quarter mark if you answer the wrong unknown.

Practice

Set an eight-minute timer.

1. 25% of a stock is sold, then a third of the rest. 200 items remain. What was the original stock?
2. Two numbers add up to 60 and differ by 14. Find them.
3. A cashier has 25 notes of ₹200 and ₹50, worth ₹3,050. How many ₹200 notes are there?
4. A number decreased by 15% becomes 170. Find the number.
5. The sum of four consecutive even numbers is 100. Find them.
6. A candidate spends 30% of their income on rent and 25% of the remainder on travel, and saves the rest, which is ₹21,000. Find the income.
7. 3 pens and 2 notebooks cost ₹110. 2 pens and 3 notebooks cost ₹115. Find the price of one pen.
8. Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Find the numbers.

Answers:

1. 400. After 25% is sold, 75% is left. A third of that (25%) is sold, leaving 50%. 50% = 200.
2. 37 and 23. Adding: 2a = 74.
3. 12 notes of ₹200 (and 13 of ₹50). 200x + 50(25 − x) = 3,050 gives 150x = 1,800.
4. 200. 0.85x = 170.
5. 22, 24, 26 and 28. 4x + 12 = 100 gives x = 22.
6. ₹40,000. Left after rent = 0.7. Travel = 0.25 × 0.7 = 0.175. Savings = 0.7 − 0.175 = 0.525 of the income, and 0.525 × 40,000 = 21,000.
7. ₹20. Adding the two equations: 5(p + n) = 225, so p + n = 45. Subtracting: p − n = −5. So p = 20 and n = 25. Check: 60 + 50 = 110.
8. 15 and 25. (3x + 10) ÷ (5x + 10) = 5/7 gives 21x + 70 = 25x + 50, so x = 5. Check: 25 : 35 = 5 : 7.

What to do next

• For one week, write the five steps in the margin of every word problem you solve.
• Keep a list of phrases that tripped you up and add them to your own translation table.
• Most word problems are one of the standard chapters in disguise; revise percentage, ratio and proportion and problems on ages.
• Mix five word problems into every quant sectional so you practise spotting the chapter quickly.