Turning a Paragraph of Words Into a Single Equation
Learn to choose the right variable, build equations for number and digit puzzles, solve age, money and sharing problems, and handle perimeter and angle questions.
How do you turn a word problem into an equation?
Name the unknown you know least about, write every other quantity in terms of it, then find the sentence that says two things are equal.
That last step is the one students skip. A word problem always contains one sentence that is really an equals sign in disguise — the total is, is the same as, gives, after five years he will be. Everything before it is just description.
Take this: a father is three times as old as his son; in twelve years he will be twice as old. Call the son's age . Then the father is . The equals sentence is the second half: in twelve years the father's age is twice the son's, so
One line of algebra, and the paragraph is gone. This page covers the second part of the ICSE Class 8 Mathematics chapter on linear equations in one variable.
That last step is the one students skip. A word problem always contains one sentence that is really an equals sign in disguise — the total is, is the same as, gives, after five years he will be. Everything before it is just description.
Take this: a father is three times as old as his son; in twelve years he will be twice as old. Call the son's age . Then the father is . The equals sentence is the second half: in twelve years the father's age is twice the son's, so
One line of algebra, and the paragraph is gone. This page covers the second part of the ICSE Class 8 Mathematics chapter on linear equations in one variable.
Which quantity should you call x?
Choose the smallest or most basic quantity — the one the others are described against.
Worked example 1 — a first translation. Five added to twice a number gives twenty-seven. Let the number be :
Verify against the words: twice is , and five added gives . Correct.
Worked example 2 — why the choice matters. A is three years older than B, and their ages total 31. Calling B's age makes A's age , and
So B is and A is . Had you called A's age , then B would be , giving and — the same pair, reached with one subtraction more. Either choice works, but naming the smaller quantity keeps the plus signs and avoids sign slips.
The phrases worth translating once and for all:
- more than, added to, increased by mean
- less than, reduced by mean , and the order reverses: *five less than * is
- of, times, product mean
- is, gives, equals, results in mean
The trap in that list. Five less than a number is , not . The words come in the opposite order to the algebra, and this one phrase probably costs more marks than any other in the chapter.
Always state what your variable stands for, with its unit. Writing *let be the son's age in years* takes four words and makes the whole answer readable — and if the examiner cannot tell what means, the working cannot be credited.
Worked example 1 — a first translation. Five added to twice a number gives twenty-seven. Let the number be :
Verify against the words: twice is , and five added gives . Correct.
Worked example 2 — why the choice matters. A is three years older than B, and their ages total 31. Calling B's age makes A's age , and
So B is and A is . Had you called A's age , then B would be , giving and — the same pair, reached with one subtraction more. Either choice works, but naming the smaller quantity keeps the plus signs and avoids sign slips.
The phrases worth translating once and for all:
- more than, added to, increased by mean
- less than, reduced by mean , and the order reverses: *five less than * is
- of, times, product mean
- is, gives, equals, results in mean
The trap in that list. Five less than a number is , not . The words come in the opposite order to the algebra, and this one phrase probably costs more marks than any other in the chapter.
Always state what your variable stands for, with its unit. Writing *let be the son's age in years* takes four words and makes the whole answer readable — and if the examiner cannot tell what means, the working cannot be credited.
How do you solve number, consecutive integer and digit problems?
**For digit problems, name the digits separately and remember that a two-digit number is .
Worked example 1 — consecutive integers.** Three consecutive integers have a sum of 72. Let them be , , :
The integers are , , , and . Correct.
Worked example 2 — consecutive odd numbers. Three consecutive odd numbers have a sum of 69. Consecutive odd numbers differ by , so they are , , :
They are , , , summing to . Correct. Consecutive even numbers work the same way — also differing by — so the only change is that comes out even.
Worked example 3 — a two-digit number. The digits of a two-digit number add to 9. Reversing the digits increases the number by 27. Find the number.
Let the tens digit be and the units digit be . Then the number is and the reversed number is .
From the first sentence, . From the second:
Adding the two results: , so and .
The number is . Verify: its digits add to , and . Correct.
**Why and not .** Writing the digits side by side means multiplication in algebra, which is not what a place-value number means. The number is , and until that is written down, no digit problem can be set up at all.
Notice what dropped out. The difference between a two-digit number and its reverse is always — a multiple of nine, whatever the digits. That is why these questions so often quote a difference like , or : any other value would make the problem impossible.
Worked example 1 — consecutive integers.** Three consecutive integers have a sum of 72. Let them be , , :
The integers are , , , and . Correct.
Worked example 2 — consecutive odd numbers. Three consecutive odd numbers have a sum of 69. Consecutive odd numbers differ by , so they are , , :
They are , , , summing to . Correct. Consecutive even numbers work the same way — also differing by — so the only change is that comes out even.
Worked example 3 — a two-digit number. The digits of a two-digit number add to 9. Reversing the digits increases the number by 27. Find the number.
Let the tens digit be and the units digit be . Then the number is and the reversed number is .
From the first sentence, . From the second:
Adding the two results: , so and .
The number is . Verify: its digits add to , and . Correct.
**Why and not .** Writing the digits side by side means multiplication in algebra, which is not what a place-value number means. The number is , and until that is written down, no digit problem can be set up at all.
Notice what dropped out. The difference between a two-digit number and its reverse is always — a multiple of nine, whatever the digits. That is why these questions so often quote a difference like , or : any other value would make the problem impossible.
How do you solve age, money and sharing problems?
For ages, add the same number of years to everybody. For money, count the value, not the number of items.
Worked example 1 — ages. Return to the father and son. Let the son be years old now, so the father is :
The son is and the father . Verify: in twelve years they will be and , and is twice . Correct.
The mistake this question is built to catch. Only the father's age is tripled; both ages gain the same twelve years. Writing forgets that the son ages too, and gives a wrong answer that looks tidy.
Worked example 2 — a simpler age pair. Two sisters' ages total 28, and the elder is 4 years older. With the younger as :
They are and , totalling . Correct.
Worked example 3 — notes in a purse. A purse holds ₹100 and ₹50 notes. There are 4 more fifty-rupee notes than hundred-rupee notes, and the total is ₹1400. How many of each?
Let the number of ₹100 notes be , so there are fifty-rupee notes. Now count value, in rupees:
So hundred-rupee notes and fifty-rupee notes. Verify: . Correct.
Worked example 4 — coins in a ratio. A box has ₹5 and ₹2 coins in the ratio 3 : 5, worth ₹125 altogether.
Ratio means the counts are and for some :
So there are five-rupee coins and two-rupee coins. Verify: , and does reduce to . Correct.
Worked example 5 — distributing articles. Pens are shared among some students. Giving 4 each leaves 6 pens over; giving 5 each falls 4 short. How many students and pens?
Let there be students. The number of pens can be written twice:
So students and pens. Verify: as well. Correct.
The idea that makes this type easy. Falls 4 short means the pens needed exceed the pens available by , so the available pens are . Expressing one fixed quantity in two different ways and setting them equal is the whole technique — and it works for sweets, chairs, rows of trees and money alike.
Worked example 1 — ages. Return to the father and son. Let the son be years old now, so the father is :
The son is and the father . Verify: in twelve years they will be and , and is twice . Correct.
The mistake this question is built to catch. Only the father's age is tripled; both ages gain the same twelve years. Writing forgets that the son ages too, and gives a wrong answer that looks tidy.
Worked example 2 — a simpler age pair. Two sisters' ages total 28, and the elder is 4 years older. With the younger as :
They are and , totalling . Correct.
Worked example 3 — notes in a purse. A purse holds ₹100 and ₹50 notes. There are 4 more fifty-rupee notes than hundred-rupee notes, and the total is ₹1400. How many of each?
Let the number of ₹100 notes be , so there are fifty-rupee notes. Now count value, in rupees:
So hundred-rupee notes and fifty-rupee notes. Verify: . Correct.
Worked example 4 — coins in a ratio. A box has ₹5 and ₹2 coins in the ratio 3 : 5, worth ₹125 altogether.
Ratio means the counts are and for some :
So there are five-rupee coins and two-rupee coins. Verify: , and does reduce to . Correct.
Worked example 5 — distributing articles. Pens are shared among some students. Giving 4 each leaves 6 pens over; giving 5 each falls 4 short. How many students and pens?
Let there be students. The number of pens can be written twice:
So students and pens. Verify: as well. Correct.
The idea that makes this type easy. Falls 4 short means the pens needed exceed the pens available by , so the available pens are . Expressing one fixed quantity in two different ways and setting them equal is the whole technique — and it works for sweets, chairs, rows of trees and money alike.
How do you use equations in perimeter and angle problems?
Write down the geometric fact that gives you the equals sign — a perimeter formula, or an angle sum.
Worked example 1 — a rectangle. The length of a rectangle is 5 cm more than its breadth, and the perimeter is 50 cm.
Let the breadth be cm, so the length is cm. The perimeter of a rectangle is :
Breadth cm, length cm. Verify: cm. Correct. The area is then .
Worked example 2 — angles of a triangle. *Two angles of a triangle are in the ratio 2 : 3 and the third is .*
The angles are , and , and they sum to :
The angles are , and , summing to . Correct.
Worked example 3 — complementary angles. *One of two complementary angles is more than the other.* Complementary means they total :
The angles are and . Correct.
Worked example 4 — an isosceles triangle. Each base angle of an isosceles triangle is twice the vertex angle.
Let the vertex angle be , so each base angle is :
The angles are , and , summing to . Correct.
Always sanity-check a geometric answer. A breadth of cm or an angle of in a triangle means the equation was set up wrongly, not that geometry has been broken. Unlike a pure number problem, a geometry answer must also be physically possible, and that extra check is free.
Worked example 1 — a rectangle. The length of a rectangle is 5 cm more than its breadth, and the perimeter is 50 cm.
Let the breadth be cm, so the length is cm. The perimeter of a rectangle is :
Breadth cm, length cm. Verify: cm. Correct. The area is then .
Worked example 2 — angles of a triangle. *Two angles of a triangle are in the ratio 2 : 3 and the third is .*
The angles are , and , and they sum to :
The angles are , and , summing to . Correct.
Worked example 3 — complementary angles. *One of two complementary angles is more than the other.* Complementary means they total :
The angles are and . Correct.
Worked example 4 — an isosceles triangle. Each base angle of an isosceles triangle is twice the vertex angle.
Let the vertex angle be , so each base angle is :
The angles are , and , summing to . Correct.
Always sanity-check a geometric answer. A breadth of cm or an angle of in a triangle means the equation was set up wrongly, not that geometry has been broken. Unlike a pure number problem, a geometry answer must also be physically possible, and that extra check is free.
Exam tip
Exam tip: state your variable, then verify against the words
Write one line naming the variable with its unit: *let be the breadth in cm*. Unmarked working cannot be credited.
Name the smallest quantity so the other expressions use plus signs.
Five less than a number is , not . The words reverse the order.
A two-digit number is , and its reverse is . Their difference is always .
In age problems, add the same number of years to every person.
In money problems, form the equation from value in rupees, not the count of coins. With a ratio, write the counts as and .
For distribution questions, express the same total two ways and equate them: .
Quote the geometric fact you are using — perimeter , angle sum or for complementary angles.
Verify against the original words, not your own equation, and check the answer is physically sensible — no negative lengths, no angle above in a triangle.
And answer the question actually asked: if it wants the number, give , not .
Name the smallest quantity so the other expressions use plus signs.
Five less than a number is , not . The words reverse the order.
A two-digit number is , and its reverse is . Their difference is always .
In age problems, add the same number of years to every person.
In money problems, form the equation from value in rupees, not the count of coins. With a ratio, write the counts as and .
For distribution questions, express the same total two ways and equate them: .
Quote the geometric fact you are using — perimeter , angle sum or for complementary angles.
Verify against the original words, not your own equation, and check the answer is physically sensible — no negative lengths, no angle above in a triangle.
And answer the question actually asked: if it wants the number, give , not .
Did you know
Why the hard part is never the algebra
Look back at every problem on this page. The algebra was two or three lines — collect, divide, done. What took the thinking was deciding what to call and spotting which sentence was the equals sign.
That imbalance is the real lesson. Once a situation has been written as an equation, solving it is mechanical, and a machine could finish it. Getting it into that form cannot be automated, because it requires understanding what the words describe.
Notice, too, how many different situations reduced to the same equation shape. The father and son, the two sisters, the notes in the purse and the pens among students are four unrelated stories, and all four came down to collecting terms and dividing once. The algebra does not know or care whether is a boy's age, a count of coins or a breadth in centimetres.
That indifference is the whole reason algebra is worth learning. A single method, learned once, handles every problem that shares a structure — and the skill you are actually building in this chapter is the skill of seeing that shared structure through the words.
It is also why the advice to verify against the words matters more than verifying against your own equation. Checking the equation only proves your arithmetic. Checking the words proves you understood the problem.
That imbalance is the real lesson. Once a situation has been written as an equation, solving it is mechanical, and a machine could finish it. Getting it into that form cannot be automated, because it requires understanding what the words describe.
Notice, too, how many different situations reduced to the same equation shape. The father and son, the two sisters, the notes in the purse and the pens among students are four unrelated stories, and all four came down to collecting terms and dividing once. The algebra does not know or care whether is a boy's age, a count of coins or a breadth in centimetres.
That indifference is the whole reason algebra is worth learning. A single method, learned once, handles every problem that shares a structure — and the skill you are actually building in this chapter is the skill of seeing that shared structure through the words.
It is also why the advice to verify against the words matters more than verifying against your own equation. Checking the equation only proves your arithmetic. Checking the words proves you understood the problem.
Key takeaways
Word problems with linear equations: quick revision
- Name the unknown you know least about, with its unit, then find the sentence that means equals.
- Five added to twice a number gives 27 is , so .
- A is 3 years older than B and their ages total 31 gives , so B is and A is .
- Watch the order: five less than a number is , never .
- Consecutive integers: summing to gives , so .
- Consecutive odd or even numbers differ by : summing to gives .
- A two-digit number is ; its reverse is . With and a difference of : so , giving , and the number . Check .
- The difference between a two-digit number and its reverse is always a multiple of nine.
- Ages: father , son , and in twelve years gives , so and . Both people age.
- Money by value: gives , so hundreds and fifties, totalling ₹1400.
- Ratio counts as and : gives , so five-rupee and two-rupee coins.
- Distribution: gives students and pens — one total written two ways.
- Perimeter: gives breadth cm, length cm and area .
- Angles: gives ; complementary angles differing by are and ; base angles twice the vertex give .
- Verify against the words, and reject impossible answers such as a negative length.
Pick three word problems from different topics — a digit puzzle, an age question and a perimeter question — and write only the equation for each, without solving. Getting the setup right is the skill being tested.
- Five added to twice a number gives 27 is , so .
- A is 3 years older than B and their ages total 31 gives , so B is and A is .
- Watch the order: five less than a number is , never .
- Consecutive integers: summing to gives , so .
- Consecutive odd or even numbers differ by : summing to gives .
- A two-digit number is ; its reverse is . With and a difference of : so , giving , and the number . Check .
- The difference between a two-digit number and its reverse is always a multiple of nine.
- Ages: father , son , and in twelve years gives , so and . Both people age.
- Money by value: gives , so hundreds and fifties, totalling ₹1400.
- Ratio counts as and : gives , so five-rupee and two-rupee coins.
- Distribution: gives students and pens — one total written two ways.
- Perimeter: gives breadth cm, length cm and area .
- Angles: gives ; complementary angles differing by are and ; base angles twice the vertex give .
- Verify against the words, and reject impossible answers such as a negative length.
Pick three word problems from different topics — a digit puzzle, an age question and a perimeter question — and write only the equation for each, without solving. Getting the setup right is the skill being tested.