Linear Equations Word Problems Worksheets with Answer Key - Free Printable

Let’s solve each problem one by one, step by step.

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Problem 1:
> There are two numbers whose sum is 72. One number is twice the other. What are the numbers?

Let the smaller number be x.
Then the larger number is 2x (since it’s twice the other).

Their sum is 72:
→ x + 2x = 72
→ 3x = 72
→ x = 24

So the numbers are:
→ Smaller: 24
→ Larger: 2 × 24 = 48

✔ Check: 24 + 48 = 72 ✔️ and 48 is twice 24 ✔️

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Problem 2:
> Clarke is 8 years older than John. In 5 years Clarke will be twice as old as John. How old are they now?

Let John’s current age = j
Then Clarke’s current age = j + 8

In 5 years:
→ John’s age = j + 5
→ Clarke’s age = (j + 8) + 5 = j + 13

At that time, Clarke is twice as old as John:
→ j + 13 = 2(j + 5)

Solve:
→ j + 13 = 2j + 10
→ 13 - 10 = 2j - j
→ 3 = j

So John is 3 now, Clarke is 3 + 8 = 11.

✔ Check in 5 years: John = 8, Clarke = 16 → 16 is twice 8 ✔️

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Problem 3:
> Find three consecutive odd integers such that the sum of the first, two times the second, and three times the third is 70.

Consecutive odd integers differ by 2. Let them be:
→ First: x
→ Second: x + 2
→ Third: x + 4

According to the problem:
→ First + 2×Second + 3×Third = 70
→ x + 2(x + 2) + 3(x + 4) = 70

Expand:
→ x + 2x + 4 + 3x + 12 = 70
→ (x + 2x + 3x) + (4 + 12) = 70
→ 6x + 16 = 70
→ 6x = 54
→ x = 9

So the numbers are:
→ 9, 11, 13

✔ Check: 9 + 2×11 + 3×13 = 9 + 22 + 39 = 70 ✔️

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Problem 4:
> Tim sold half of his story books and then bought 6 more. He now has 12. How many did he begin with?

Let original number = b

He sold half → left with b/2
Then bought 6 → now has b/2 + 6 = 12

Solve:
→ b/2 + 6 = 12
→ b/2 = 6
→ b = 12

✔ Check: Start with 12 → sell half (6 left) → buy 6 → 12 ✔️

Wait — if he started with 12, sold half (6), then bought 6, he’d have 12 again. That matches. But let’s double-check the math:

b/2 + 6 = 12 → subtract 6 from both sides → b/2 = 6 → multiply by 2 → b = 12. Correct.

But wait — does “sold half” mean he had 12 after selling half? No — he sold half, then added 6, ending at 12. So yes, starting with 12 works.

Actually, let me re-read: “Tim sold half of his story books and then bought 6 more. He now has 12.”

Yes — so equation is correct. Answer is 12.

BUT — hold on! If he started with 12, sold half → 6 left, then bought 6 → 12. Yes, that’s right.

Some might think it should be different, but math checks out.

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Problem 5:
> Sara can do a job in 12 hours. If Tom helps, it takes them 8 hours. Without help, how long would it take for Tom to do the same work?

This is a work rate problem.

Sara’s rate: 1 job / 12 hours → 1/12 per hour
Tom’s rate: 1 job / t hours → 1/t per hour (we need to find t)

Together, their combined rate: 1 job / 8 hours → 1/8 per hour

So:
→ Sara’s rate + Tom’s rate = Combined rate
→ 1/12 + 1/t = 1/8

Solve for t:

Subtract 1/12 from both sides:
→ 1/t = 1/8 - 1/12

Find common denominator (24):
→ 1/8 = 3/24
→ 1/12 = 2/24
→ 1/t = 3/24 - 2/24 = 1/24

So t = 24

✔ Check: Sara does 1/12 per hour, Tom does 1/24 per hour → together: 1/12 + 1/24 = 2/24 + 1/24 = 3/24 = 1/8 → which means 8 hours together ✔️

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Final Answers:

1. The numbers are 24 and 48.
2. John is 3, Clarke is 11.
3. The integers are 9, 11, 13.
4. Tim began with 12 story books.
5. Tom would take 24 hours alone.

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Final Answer:
1. 24 and 48
2. John is 3, Clarke is 11
3. 9, 11, 13
4. 12
5. 24