In short
Every word problem in algebra starts the same way: name the unknown. Write "Let x = \ldots" on the very first line, before any arithmetic, before any equation, before anything. This single sentence is the bridge between English and algebra — without it, the paragraph stays a paragraph; with it, the paragraph becomes a solvable equation. NCERT graders in Class 7 and 8 give a separate mark for this step, and JEE-level word problems become trivial once it is done well.
You read a word problem. Three sentences of English about ages, or trains, or rectangles. Your eyes glaze. You vaguely know the answer should pop out somewhere, but where do you even start?
Here is the secret that every textbook hides in plain sight: the first line you write is always "Let x = something." Always. Not sometimes. Not "if you feel like it." Always.
That single line is the conversion bridge. English on one side, algebra on the other. Until you cross it, no formula in your head can help you, because there is no symbol to plug into a formula. Once you cross it, the rest is just solving a linear equation — a skill you already have.
Why: A word problem is a story. An equation is a calculation. The "Let x = \ldots" line is the translator — it tells you which character in the story corresponds to which symbol in the equation. Skip it and you are doing the rest of the problem in your head, hoping nothing collides.
The flowchart of every word problem
Whether the problem is about Ria's age or a train from Mumbai, the steps never change.
Notice the highlighted box. Every other step is mechanical. Reading is mechanical. Translating is mechanical once x is named. Solving is mechanical once the equation is written. The single act of naming the unknown is where thought happens.
Worked examples
Ria's age
Ria's age is twice her brother's. Together they are 24 years old. How old is Ria?
Step 1 — Name the unknown.
Let x = the brother's age (in years).
That is it. One sentence. Now everything else in the problem can be written in terms of x.
Step 2 — Translate.
Ria is twice her brother's age, so Ria's age is 2x. Their sum is 24:
Step 3 — Solve.
So the brother is 8 years old, and Ria is 2x = 16.
Check. 8 + 16 = 24. ✓
Why call x the brother's age and not Ria's? Because the brother's age is the smaller quantity, and Ria's age is described in terms of his (2x). If you had set x = Ria's age, the brother would be \frac{x}{2}, which works but introduces a fraction. Both are valid; one is cleaner.
A rectangle's dimensions
A rectangle's length is 5 cm more than its width. Its perimeter is 30 cm. Find the dimensions.
Step 1 — Name the unknown.
Let w = the width (in cm).
(You can call it x if you prefer. Picking a letter that hints at the meaning — w for width, t for time, b for bus speed — makes your work readable to a checker, and to your future self.)
Step 2 — Translate.
Length is 5 more than width, so length = w + 5. Perimeter is twice the sum of length and width:
Step 3 — Solve.
Width is 5 cm, length is w + 5 = 10 cm.
Check. Perimeter = 2(5 + 10) = 30. ✓
Train and bus
A train travels 5 km/h faster than a bus. Both cover the same 200 km route. The train arrives 1 hour earlier than the bus. Find the bus's speed.
This problem looks scary. It is not — once you name the unknown.
Step 1 — Name the unknown.
Let b = the bus's speed (in km/h).
Step 2 — Translate.
Train's speed is b + 5. Time taken = distance ÷ speed, so:
- Bus's time: \dfrac{200}{b} hours.
- Train's time: \dfrac{200}{b+5} hours.
The bus takes 1 hour longer:
Step 3 — Solve.
Multiply both sides by b(b+5) to clear denominators:
Using the quadratic formula (or factoring nearby): b = \dfrac{-5 + \sqrt{25 + 4000}}{2} = \dfrac{-5 + \sqrt{4025}}{2} \approx \dfrac{-5 + 63.44}{2} \approx 29.2 km/h.
(In a tidied textbook version, the numbers would land on a clean b = 40 or similar — but the method is identical.)
Check. Bus time \approx 200/29.2 \approx 6.85 h. Train time \approx 200/34.2 \approx 5.85 h. Difference: 1 h. ✓
How to choose what to call x
You always have a choice. Two guidelines:
1. Call x what the question asks for. If the problem says "find Ria's age," it is tempting to set x = Ria's age. This works and is the safe default — your final answer is just x.
2. Call x the quantity that makes the other unknowns easiest to write. In the Ria problem, calling x the brother's age made Ria's age 2x (clean) instead of the brother being x/2 (a fraction). In the rectangle problem, calling x the width made the length x+5 (clean) instead of the width being x-5.
When the two guidelines clash, pick whichever keeps fractions and negative numbers out of your equation. Why: every unnecessary fraction is an extra place to make an arithmetic mistake. The "let x = \ldots" choice is the only chance to engineer cleanliness before the algebra starts.
What NCERT graders look for
In the CBSE Class 7 and Class 8 NCERT textbooks, the chapter on linear equations explicitly grades word-problem solutions in three parts:
- Naming the variable. ("Let x = ...") — typically 1 mark.
- Setting up the equation. — typically 1 mark.
- Solving and answering. — typically 1–2 marks.
A student who jumps straight to the equation without writing "Let x = ..." loses the first mark even if the final answer is correct. This is not pedantry — it is the marker checking that you can do the thinking, not just the arithmetic. The arithmetic is the easy part.
Common mistakes
- Skipping the line entirely. Most students try to "see" the equation in their head. For one-step problems this works; for anything with two related quantities (ages, rectangles, mixtures), the unwritten x becomes ambiguous halfway through.
- Forgetting the units. "Let x = age" is weaker than "Let x = age in years." Units anchor the meaning.
- Naming two unknowns when one will do. If the problem can be reduced to one variable (because the second unknown is described in terms of the first), do not introduce a second letter — that turns a linear equation in one variable into a system in two variables for no reason.
- Solving for x but forgetting the question. If x = brother's age and the question asks for Ria's age, your final answer is 2x, not x. Always re-read the question after solving.
The bigger picture
Word problems do not get harder in higher classes — they get longer. A JEE problem on motion under gravity is just a Class 8 word problem with three or four "Let x = ..." lines stacked at the top. Mixture problems in chemistry, time-and-work problems, geometry optimization in calculus — all of them open with the same first move.
Master the move now and the rest of school maths becomes a sequence of solved equations with translation-overhead. Skip it, and every word problem will feel like a fresh puzzle for the rest of your life.
The line is one sentence. Write it.
References
- NCERT Class 7 Mathematics — Chapter 4: Simple Equations — explicit "Let x = \ldots" framing for every word problem.
- NCERT Class 8 Mathematics — Chapter 2: Linear Equations in One Variable — graded word problems with the variable-naming convention.
- Polya, How To Solve It (Princeton University Press) — the four-step framework whose first step is "understand the problem," operationalised here as "name the unknown."
- Khan Academy — Two-step equation word problems — worked translations from English to algebra.