Word problems: from a story to an equation

Four steps for any word problem

Word problems become easy when you follow the same four steps every time:

1. Name the unknown with a letter, and say what it stands for.
2. Write the facts of the problem using that letter.
3. Make the equation from the two amounts that are equal, and solve it.
4. Answer the question that was asked, and check it in the story.

A pattern of tiles

Ranjana makes T shapes with square tiles: Step 1 uses 4 tiles, Step 2 uses 7 and Step 3 uses 10. At Step k the top row has k tiles on each side of a centre tile, and a stem of k tiles hangs below it, so the total is k + k + k + 1 = 3k + 1. Can she use exactly 100 tiles? Solve 3k + 1 = 100: 3k = 99, so k = 33. Step 33 uses exactly 100 tiles.

Money: a party and two savers

Madhubanti may spend ₹500 on snacks for a party. Each plate costs ₹25, and the shop charges ₹50 to deliver. If p people come in all, the cost is 25p + 50, so 25p + 50 = 500. Then 25p = 450 and p = 18. Her family has 5 members, so she can invite 13 friends. Another way: let f be the number of friends; then 25(f + 5) = 450, so f + 5 = 18 and f = 13. Both ways agree.

Jahnavi starts with ₹4,000 and saves ₹650 a month, while Sunita starts with ₹5,050 and saves ₹500 a month. After m months they have 4000 + 650m and 5050 + 500m. Setting these equal gives 150m = 1050, so m = 7. Check: after 7 months each has ₹8,550.

Number tricks

Riyaz asks Akash to think of a number, take away 3, multiply by 4 and add 8; Akash says 24. With x as the number, the steps give x − 3, then 4x − 12, then 4x − 4. So 4x − 4 = 24, 4x = 28 and x = 7. The rule works for any answer: the starting number is the answer divided by 4, plus 1. A letter also explains a magic trick in which everyone ends with 8, whatever number they start with.

Two unknowns, one letter

Ramesh and Suresh have 60 marbles together, and Ramesh has 30 more than Suresh. There seem to be two unknowns, but one letter is enough: if Suresh has y, Ramesh has y + 30. Then y + (y + 30) = 60, so 2y = 30 and y = 15. Suresh has 15 marbles and Ramesh 45. The same trick solves “two numbers add up to 76 and one is three times the other”: x + 3x = 76 gives x = 19, so the numbers are 19 and 57.

Making equations from an answer

We can also work backwards: start from a solution and build equations. From y = 5, add 2 to both sides to get y + 2 = 7, multiply by 3 to get 3y + 6 = 21, and take away 6 to get 3y = 15. Each new equation has the same solution, because the same step was done to both sides. Going back up the chain uses the opposite steps in reverse order.

An equation can also be read as a story. 100x + 75 = 250 could mean 100 pencils at ₹x each with ₹75 for delivery and a bill of ₹250, so each pencil costs ₹1.75.

Why it matters

With these steps, a question about tiles, money or marbles becomes one small equation. Practise the solving itself in solving simple equations, and see how Bhaskaracharya posed problems like these in bijaganita: algebra in ancient India. Answers to the textbook questions are in the NCERT solutions for Part 2 Chapter 7; see all of Part 2 Chapter 7 notes.