Turning a Word Problem Into an Equation You Can Solve
Learn to frame expressions from words, solve linear equations by balancing and transposition, verify your answer, and handle word problems without losing the unit.
How do you turn a sentence into an algebraic expression?
You choose a letter for the unknown quantity, then translate the sentence phrase by phrase into symbols. "Five more than a number" becomes ; "three times a number, less two" becomes .
This page covers everything in the ICSE Class 6 Mathematics chapter on framing expressions and solving simple linear equations: building an expression from words, evaluating it, the two standard solving methods, and turning word problems into equations.
This page covers everything in the ICSE Class 6 Mathematics chapter on framing expressions and solving simple linear equations: building an expression from words, evaluating it, the two standard solving methods, and turning word problems into equations.
How do you frame an expression from a real situation?
Name the unknown first, write down what the letter means, then build the expression from the words in order.
If a pen costs Rs and a notebook costs Rs 15 more, the notebook costs . The cost of 3 pens and 2 notebooks is then
Some standard phrases are worth learning as translations:
- "sum of and " is
- " exceeds by 4" is
- " is twice " is
- "half of " is
- "the number decreased by 7" is
For example, if your age now is years, your age five years ago was and your age in eight years will be .
The order of words matters for subtraction and division. "7 less than " is , but "7 less " is — these are different expressions.
If a pen costs Rs and a notebook costs Rs 15 more, the notebook costs . The cost of 3 pens and 2 notebooks is then
Some standard phrases are worth learning as translations:
- "sum of and " is
- " exceeds by 4" is
- " is twice " is
- "half of " is
- "the number decreased by 7" is
For example, if your age now is years, your age five years ago was and your age in eight years will be .
The order of words matters for subtraction and division. "7 less than " is , but "7 less " is — these are different expressions.
How do you evaluate a framed expression?
Once framed, an expression becomes a reusable rule: substitute any value and read off the answer.
Take the pens and notebooks above, . If a pen costs Rs 12, then
If the pen costs Rs 20 instead, the same expression gives without any rework.
Always interpret the result back in the language of the question. Here 90 is not just a number — it is the total cost in rupees of 3 pens and 2 notebooks.
A sensible check: does the answer's size make sense? If a pen costs Rs 12 and you buy five items, a total of Rs 90 is plausible, whereas an answer of Rs 9 or Rs 900 would signal an arithmetic slip.
Take the pens and notebooks above, . If a pen costs Rs 12, then
If the pen costs Rs 20 instead, the same expression gives without any rework.
Always interpret the result back in the language of the question. Here 90 is not just a number — it is the total cost in rupees of 3 pens and 2 notebooks.
A sensible check: does the answer's size make sense? If a pen costs Rs 12 and you buy five items, a total of Rs 90 is plausible, whereas an answer of Rs 9 or Rs 900 would signal an arithmetic slip.
How do you solve an equation by balancing and by transposition?
An equation states that two expressions are equal. Solving it means finding the value of the variable that makes the statement true.
The balancing method treats the equation like a pair of scales: whatever you do to one side, you must do to the other. Solve :
Subtract 4 from both sides: . Divide both sides by 3: .
The transposition method is the same process written faster. Move a term across the equals sign and change its sign; a multiplier becomes a divisor.
Always verify by substituting back into the original equation: , which matches the right-hand side, so is correct.
Verification is not optional padding — it is the only step that catches a sign error, and it costs one line.
The balancing method treats the equation like a pair of scales: whatever you do to one side, you must do to the other. Solve :
Subtract 4 from both sides: . Divide both sides by 3: .
The transposition method is the same process written faster. Move a term across the equals sign and change its sign; a multiplier becomes a divisor.
Always verify by substituting back into the original equation: , which matches the right-hand side, so is correct.
Verification is not optional padding — it is the only step that catches a sign error, and it costs one line.
How do you translate a word problem into an equation?
Work in four steps: let a letter stand for the unknown, express the other quantities in terms of it, write the equation from the condition given, then solve and verify.
The sum of three consecutive numbers is 48. Find them.
Let the smallest be . The next two are and . The condition gives
So the numbers are 15, 16 and 17. Check: .
A second example with units. A rectangle is 4 cm longer than it is wide, and its perimeter is 28 cm. Let the width be , so the length is :
The width is 5 cm and the length 9 cm. Note that the question asked for lengths, so the answer must carry cm — a bare 5 would be incomplete.
The sum of three consecutive numbers is 48. Find them.
Let the smallest be . The next two are and . The condition gives
So the numbers are 15, 16 and 17. Check: .
A second example with units. A rectangle is 4 cm longer than it is wide, and its perimeter is 28 cm. Let the width be , so the length is :
The width is 5 cm and the length 9 cm. Note that the question asked for lengths, so the answer must carry cm — a bare 5 would be incomplete.
Exam tip
The mistake most students make when framing equations
The commonest error is solving for the letter and then stopping, when the question asked for something else.
In the rectangle problem, is not the final answer if the question asked for the length, which is 9 cm. In the consecutive-numbers problem, is only the smallest of the three.
Build the habit of writing "Let = …" at the start and then re-reading that line at the end. Ask: did the question want , or something built from ? Then state the answer in a full sentence with its unit. Those two lines protect marks that are otherwise thrown away after the algebra was done correctly.
In the rectangle problem, is not the final answer if the question asked for the length, which is 9 cm. In the consecutive-numbers problem, is only the smallest of the three.
Build the habit of writing "Let = …" at the start and then re-reading that line at the end. Ask: did the question want , or something built from ? Then state the answer in a full sentence with its unit. Those two lines protect marks that are otherwise thrown away after the algebra was done correctly.
Did you know
Why does a term change sign when it crosses the equals sign?
Transposition is not a separate rule — it is the balancing method with the middle step left out.
When you "move" across and write , what actually happened is that 4 was subtracted from both sides. On the left it cancelled, and on the right it appeared as a subtraction. The sign change is simply the visible half of an operation done to both sides at once.
When you "move" across and write , what actually happened is that 4 was subtracted from both sides. On the left it cancelled, and on the right it appeared as a subtraction. The sign change is simply the visible half of an operation done to both sides at once.
Key takeaways
Framing and solving equations in 30 seconds
- Choose a letter for the unknown, state what it means, and translate the sentence phrase by phrase, watching the word order in subtraction and division.
- A framed expression is reusable: substitute any value and interpret the result in the context of the question.
- The balancing method does the same operation to both sides; transposition moves a term across the equals sign and changes its sign.
- Always verify by substituting your answer back into the original equation.
- In word problems, express every quantity in terms of your chosen letter, then check whether the question wanted that letter or something built from it, and state the answer with its unit.
You will remember all of this far better after answering five questions on it than after reading it twice.
- A framed expression is reusable: substitute any value and interpret the result in the context of the question.
- The balancing method does the same operation to both sides; transposition moves a term across the equals sign and changes its sign.
- Always verify by substituting your answer back into the original equation.
- In word problems, express every quantity in terms of your chosen letter, then check whether the question wanted that letter or something built from it, and state the answer with its unit.
You will remember all of this far better after answering five questions on it than after reading it twice.