Worksheet titled "Word Problems Involving Linear Equations" featuring five math problems related to real-life scenarios, including sums of numbers, age comparisons, consecutive odd integers, book counts, and work rates.

Let's solve each of the word problems step by step using linear equations.

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Problem 1:

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

Step 1: Let the smaller number be $ x $.
Then the other number (twice the first) is $ 2x $.

Step 2: Their sum is 72:
$$
x + 2x = 72
$$
$$
3x = 72
$$
$$
x = 24
$$

So, the numbers are:
- $ x = 24 $
- $ 2x = 48 $

✔ Answer: The numbers are 24 and 48.

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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?

Step 1: Let John’s current age be $ x $.
Then Clarke’s current age is $ x + 8 $.

Step 2: In 5 years:
- John’s age: $ x + 5 $
- Clarke’s age: $ x + 8 + 5 = x + 13 $

According to the problem:
$$
x + 13 = 2(x + 5)
$$

Solve:
$$
x + 13 = 2x + 10
$$
$$
13 - 10 = 2x - x
$$
$$
3 = x
$$

So:
- John is $ 3 $ years old
- Clarke is $ 3 + 8 = 11 $ years old

✔ Answer: John is 3, Clarke is 11.

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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.

Step 1: Let the three consecutive odd integers be:
- First: $ x $
- Second: $ x + 2 $
- Third: $ x + 4 $

Step 2: Set up the equation:
$$
x + 2(x + 2) + 3(x + 4) = 70
$$

Simplify:
$$
x + 2x + 4 + 3x + 12 = 70
$$
$$
6x + 16 = 70
$$
$$
6x = 54
$$
$$
x = 9
$$

So the numbers are:
- $ x = 9 $
- $ x + 2 = 11 $
- $ x + 4 = 13 $

Check:
$$
9 + 2(11) + 3(13) = 9 + 22 + 39 = 70 ✔
$$

✔ Answer: The integers are 9, 11, 13.

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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?

Step 1: Let the original number of books be $ x $.

He sold half: $ \frac{1}{2}x $ left.

Then he bought 6 more:
$$
\frac{1}{2}x + 6 = 12
$$

Solve:
$$
\frac{1}{2}x = 6
$$
$$
x = 12
$$

✔ Answer: Tim began with 12 books.

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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.

Step 1: Let’s define rates:
- Sara’s rate: $ \frac{1}{12} $ jobs per hour
- Tom’s rate: $ \frac{1}{t} $ jobs per hour (we want to find $ t $)
- Combined rate: $ \frac{1}{8} $ jobs per hour

Step 2: Add their rates:
$$
\frac{1}{12} + \frac{1}{t} = \frac{1}{8}
$$

Solve for $ t $:

Subtract $ \frac{1}{12} $ from both sides:
$$
\frac{1}{t} = \frac{1}{8} - \frac{1}{12}
$$

Find common denominator (24):
$$
\frac{1}{8} = \frac{3}{24},\quad \frac{1}{12} = \frac{2}{24}
$$
$$
\frac{1}{t} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24}
$$

So:
$$
t = 24
$$

✔ Answer: It would take Tom 24 hours to do the job alone.

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✔ Final Answers Summary:

1. 24 and 48
2. John: 3 years, Clarke: 11 years
3. 9, 11, 13
4. 12 books
5. 24 hours

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