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Word problems into equations

Turn a story about ages, numbers in a row or a rectangle into an equation, then solve it and check the answer.

Word problems are as old as mathematics. An Egyptian scroll called the Rhind Papyrus, copied around 1550 BC, already asks for "a quantity" which, added to one seventh of itself, makes 19. The hard part of a word problem is almost never the solving. It is turning the story into an equation. Once the equation is written down, it is solved with the same steps as any other. Below is one method that works for every story. It is used on number puzzles, numbers in a row, ages and rectangles.

The method

  1. Say what x is. Write it down: "Let x be the age of Sam." Usually x is the thing the question asks for, or the smallest amount in the story.
  2. Write every other amount in the story using x.
  3. Find the sentence that says two things are equal, and write it as an equation.
  4. Solve the equation, one step per line.
  5. Check your answer in the story, not only in the equation. Then answer the question that was asked.

These words come up again and again. Each one turns into a piece of algebra.

In the storyIn the equationExample
is, makes, equals=a number is 9: x = 9
plus, more than, older than+4 years older than Sam: x + 4
minus, less than, younger than−8 less than the number: x − 8
times, twice, three times asa number in frontthree times as old: 3x
half ofdivided by 2half of the number: x/2
together, the sum, add up toadd every parttogether they are 30: x + (x + 4) = 30

A number puzzle

Example 1

5 times a number is 45. What is the number?

  1. Let x be the number. "5 times a number is 45" becomes 5x = 45.
  2. Divide both sides by 5: x = 45/5
  3. x = 9
  4. Check: 5 times 9 is 45. Correct.

Answer: 9

Example 2

Half of a number is 9. What is the number?

  1. Let x be the number. Then x/2 = 9.
  2. Multiply both sides by 2: x = 9(2)
  3. x = 18
  4. Check: half of 18 is 9. Correct.

Answer: 18

Numbers in a row

Numbers that follow each other, like 7, 8, 9, are called consecutive numbers. Each one is 1 more than the one before. If the first is x, the next is x + 1, and the one after is x + 2.

Example 3

Two consecutive numbers add up to 41. What is the smaller one?

  1. Let x be the smaller number. The next one is x + 1.
  2. Together they make 41: x + x + 1 = 41
  3. Collect the x terms: 2x + 1 = 41
  4. Take 1 away from both sides: 2x = 40
  5. Divide both sides by 2: x = 20
  6. Check: the numbers are 20 and 21, and 20 + 21 = 41. Correct.

Answer: 20

Example 4

Three consecutive numbers add up to 48. What is the smallest one?

  1. Let x be the smallest. The numbers are x, x + 1 and x + 2.
  2. Together: x + x + 1 + x + 2 = 48
  3. Collect like terms: 3x + 3 = 48
  4. Take 3 away from both sides: 3x = 45
  5. Divide both sides by 3: x = 15
  6. Check: 15 + 16 + 17 = 48. Correct.

Answer: 15

Age problems

In an age problem, call the youngest person x. Then write everyone else using x.

Example 5

Tom is 4 years older than Sam. Together they are 30. How old is Sam?

  1. Let x be Sam's age. Tom is x + 4.
  2. Together: x + x + 4 = 30
  3. Collect like terms: 2x + 4 = 30
  4. Take 4 away from both sides: 2x = 26
  5. Divide both sides by 2: x = 13
  6. Check: Sam is 13, Tom is 17. 13 + 17 = 30, and Tom is 4 years older. Correct.

Answer: Sam is 13.

Example 6

A father is 3 times as old as his son. Together they are 48. How old is the son?

  1. Let x be the son's age. The father is 3x.
  2. Together: x + 3x = 48
  3. Collect like terms: 4x = 48
  4. Divide both sides by 4: x = 12
  5. Check: the son is 12, the father is 36. 12 + 36 = 48. Correct.

Answer: The son is 12.

Some age problems talk about the past and the future. Write each age with x first, in a small table.

AnaAge
nowx
2 years agox − 2
in 6 yearsx + 6

Example 7

In 6 years, Ana will be twice as old as she was 2 years ago. How old is Ana now?

  1. "In 6 years she will be twice her age 2 years ago": x + 6 = 2(x − 2)
  2. Multiply out the bracket, each term by 2: 2(x) − 2(2) = 2x − 4
  3. So the equation is x + 6 = 2x − 4
  4. Take x away from both sides: 6 = x − 4
  5. Add 4 to both sides: 10 = x
  6. Check: in 6 years Ana is 16. Two years ago she was 8. And 16 is twice 8. Correct.

Answer: Ana is 10.

The perimeter of a rectangle

The perimeter is the distance all the way around a shape. For a rectangle you add all four sides. The two long sides are equal and the two short sides are equal.

Example 8

The perimeter of this rectangle is 26. What is x?

  1. Add the four sides: x + 3 + x + 3
  2. Collect like terms: 2x + 6
  3. The perimeter is 26: 2x + 6 = 26
  4. Take 6 away from both sides: 2x = 20
  5. Divide both sides by 2: x = 10
  6. Check: 10 + 3 + 10 + 3 = 26. Correct.

Answer: x = 10

Try it yourself

  1. 1.A number minus 13 is 29. What is the number?

    Show the answer

    42

    x − 13 = 29. Add 13 to both sides: x = 42.

  2. 2.Three times a number, plus 4, is 31. What is the number?

    Show the answer

    9

    3x + 4 = 31. Take 4 away: 3x = 27. Divide by 3: x = 9.

  3. 3.Two consecutive numbers add up to 87. What is the larger one?

    Show the answer

    44

    x + x + 1 = 87, so 2x = 86 and x = 43. The larger one is 44.

  4. 4.Maria is 6 years younger than her brother. Together they are 40. How old is Maria?

    Show the answer

    17

    Maria is x and her brother is x + 6. So 2x + 6 = 40, 2x = 34, x = 17. Her brother is 23.

  5. 5.One side of a rectangle is x and the other is 7. The perimeter is 30. What is x?

    Show the answer

    8

    x + 7 + x + 7 = 2x + 14 = 30. So 2x = 16 and x = 8.

  6. 6.In 4 years, Leo will be three times as old as he was 4 years ago. How old is Leo now?

    Show the answer

    8

    x + 4 = 3(x − 4). Multiply out the bracket: x + 4 = 3x − 12. Take x away: 4 = 2x − 12. Add 12: 16 = 2x, so x = 8. Check: in 4 years Leo is 12, 4 years ago he was 4, and 12 is three times 4.

Ready to test yourself?

Play Algebra: 20 seconds per question, and it gets harder as you go.