Linear Equations • Topic 4 of 4

Word Problems

This is where linear equations earn their keep in CAT. The skill is translation: name the unknown, write each English relationship as an equation, then solve. Anchor on the smallest unknown to keep numbers clean. Watch the standard phrasings — "is" means equals, "of" means multiply, "more than"/"less than" reverse the order you might expect, and "twice", "thrice", "half" are coefficients. Age problems hinge on the fact that every person ages by the same amount, so a difference in ages stays constant while ratios change; write present ages, then add or subtract the time gap to each. Coin and note problems give two equations — a total count and a total value. Digit problems use 10t + u for a two-digit number with tens digit t and units digit u. Always sanity-check the answer against the story: a negative age or a fractional number of coins means a setup error, not a real answer. Choosing variables well is half the marks.

✅ Solved examples

1. The sum of two numbers is 50 and their difference is 8. Find the numbers.
x + y = 50, x − y = 8. Add: 2x = 58 ⇒ x = 29, y = 21.
2. A father is 4 times as old as his son. In 20 years he will be twice as old. Find their present ages.
Let son = s, father = 4s. (4s + 20) = 2(s + 20) ⇒ 4s + 20 = 2s + 40 ⇒ 2s = 20 ⇒ s = 10. Son 10, father 40.
3. A box has ₹1 and ₹2 coins, 30 coins in all worth ₹46. How many of each?
Let x be ₹1 coins, y be ₹2 coins. x + y = 30, x + 2y = 46. Subtract: y = 16, x = 14.
4. The sum of the digits of a two-digit number is 9. Reversing the digits increases it by 27. Find the number.
Number = 10t + u, t + u = 9. Reversed − original = 27 ⇒ (10u + t) − (10t + u) = 27 ⇒ 9(u − t) = 27 ⇒ u − t = 3. With t + u = 9: u = 6, t = 3. Number = 36.

✏️ Practice — try these, take hints as needed

1. Two numbers add to 36 and one is twice the other. Find them.
Let the smaller be x.
x + 2x = 36.
3x = 36.
12 and 24
2. A is 5 years older than B; in 5 years their ages add to 45. Find present ages.
Let B = b, A = b + 5.
(b + 5) + (b + 5) + 5 = 45.
2b + 15 = 45.
B = 15, A = 20
3. Tickets cost ₹40 (adult) and ₹25 (child). 12 tickets cost ₹390. How many adult tickets?
Let a adults, c children, a + c = 12.
40a + 25c = 390.
Substitute c = 12 − a.
6 adult tickets
4. A two-digit number is 7 times the sum of its digits. The tens digit exceeds the units digit by 4. Find it.
10t + u = 7(t + u).
That gives 3t = 6u, so t = 2u.
Use t − u = 4.
84
5. The cost of 3 pens and 2 books is ₹260; 2 pens and 5 books cost ₹485. Find the price of one book.
3p + 2b = 260, 2p + 5b = 485.
Multiply to match p-coefficients.
6p + 4b = 520 and 6p + 15b = 1455; subtract.
₹85 per book

📝 Topic test — 8 questions

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