Linear Equations in One Variable (Word Problems) worksheet | Live ... - Free Printable

Let's solve each of the word problems from the worksheet step by step. We'll use linear equations in one variable and provide answers in proper fractions, as instructed.

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1. Four-fifth of a number is more than three-fourth of the number by 4. Find the number.

Let the number be $ x $.

Given:
$$
\frac{4}{5}x = \frac{3}{4}x + 4
$$

Subtract $ \frac{3}{4}x $ from both sides:
$$
\frac{4}{5}x - \frac{3}{4}x = 4
$$

Find common denominator (20):
$$
\frac{16}{20}x - \frac{15}{20}x = 4 \Rightarrow \frac{1}{20}x = 4
$$

Multiply both sides by 20:
$$
x = 80
$$

✔ Answer: $ \boxed{80} $

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2. The difference between the squares of two consecutive numbers is 31. Find the numbers.

Let the smaller number be $ x $. Then the next consecutive number is $ x+1 $.

Given:
$$
(x+1)^2 - x^2 = 31
$$

Expand:
$$
x^2 + 2x + 1 - x^2 = 31 \Rightarrow 2x + 1 = 31
$$

Solve:
$$
2x = 30 \Rightarrow x = 15
$$

So the numbers are $ 15 $ and $ 16 $

✔ Answer: $ \boxed{15, 16} $
(Option: $ \boxed{16,15} $ or $ \boxed{15,16} $ — both are correct depending on order.)

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3. Find a number whose double is 45 greater than its half.

Let the number be $ x $.

Given:
$$
2x = \frac{1}{2}x + 45
$$

Subtract $ \frac{1}{2}x $ from both sides:
$$
2x - \frac{1}{2}x = 45 \Rightarrow \frac{3}{2}x = 45
$$

Multiply both sides by $ \frac{2}{3} $:
$$
x = 45 \times \frac{2}{3} = 30
$$

✔ Answer: $ \boxed{30} $

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4. Find a number such that when 5 is subtracted from 5 times that number, the result is 4 more than twice the number.

Let the number be $ x $.

Given:
$$
5x - 5 = 2x + 4
$$

Subtract $ 2x $ from both sides:
$$
3x - 5 = 4
$$

Add 5 to both sides:
$$
3x = 9 \Rightarrow x = 3
$$

✔ Answer: $ \boxed{3} $

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5. A number whose fifth part increased by 5 is equal to its fourth part diminished by 5. Find the number.

Let the number be $ x $.

Given:
$$
\frac{x}{5} + 5 = \frac{x}{4} - 5
$$

Bring like terms together:
$$
\frac{x}{5} - \frac{x}{4} = -5 - 5 = -10
$$

Common denominator (20):
$$
\frac{4x - 5x}{20} = -10 \Rightarrow \frac{-x}{20} = -10
$$

Multiply both sides by -20:
$$
x = 200
$$

✔ Answer: $ \boxed{200} $

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6. A number consists of two digits whose sum is 9. If 27 is subtracted from the number the digits are reversed. Find the number.

Let the tens digit be $ x $, units digit be $ y $. So number is $ 10x + y $.

Given:
- $ x + y = 9 $
- $ 10x + y - 27 = 10y + x $

Simplify second equation:
$$
10x + y - 27 = 10y + x \Rightarrow 9x - 9y = 27 \Rightarrow x - y = 3
$$

Now solve:
- $ x + y = 9 $
- $ x - y = 3 $

Add equations:
$$
2x = 12 \Rightarrow x = 6 \Rightarrow y = 3
$$

Number is $ 10x + y = 60 + 3 = 63 $

Check: $ 63 - 27 = 36 $ → digits reversed ✔

✔ Answer: $ \boxed{63} $

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7. Divide 184 into two parts such that one-third of one part may exceed one-seventh of another part by 8.

Let one part be $ x $, then other is $ 184 - x $

Given:
$$
\frac{1}{3}x = \frac{1}{7}(184 - x) + 8
$$

Multiply both sides by 21 (LCM of 3 and 7) to eliminate denominators:
$$
21 \cdot \left( \frac{1}{3}x \right) = 21 \cdot \left( \frac{1}{7}(184 - x) + 8 \right)
\Rightarrow 7x = 3(184 - x) + 168
$$

Expand:
$$
7x = 552 - 3x + 168 = 720 - 3x
$$

Add $ 3x $:
$$
10x = 720 \Rightarrow x = 72
$$

Other part: $ 184 - 72 = 112 $

Check:
- $ \frac{1}{3} \times 72 = 24 $
- $ \frac{1}{7} \times 112 = 16 $
- $ 24 - 16 = 8 $ ✔

✔ Answer: $ \boxed{72 \text{ and } 112} $

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8. Twice the number decreased by 22 is 48. Find the number.

Let the number be $ x $.

Given:
$$
2x - 22 = 48
$$

Add 22:
$$
2x = 70 \Rightarrow x = 35
$$

✔ Answer: $ \boxed{35} $

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9. Seven times the number is 36 less than 10 times the number. Find the number.

Let the number be $ x $.

Given:
$$
7x = 10x - 36
$$

Subtract $ 10x $:
$$
-3x = -36 \Rightarrow x = 12
$$

✔ Answer: $ \boxed{12} $

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10. 4/5 of a number is more than 3/4 of the number by 5. Find the number.

Let the number be $ x $.

Given:
$$
\frac{4}{5}x = \frac{3}{4}x + 5
$$

Subtract $ \frac{3}{4}x $:
$$
\frac{4}{5}x - \frac{3}{4}x = 5
$$

Common denominator (20):
$$
\frac{16}{20}x - \frac{15}{20}x = 5 \Rightarrow \frac{1}{20}x = 5
$$

Multiply by 20:
$$
x = 100
$$

✔ Answer: $ \boxed{100} $

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✔ Final Answers Summary:

| Question | Answer |
|--------|--------|
| 1 | $ 80 $ |
| 2 | $ 15, 16 $ |
| 3 | $ 30 $ |
| 4 | $ 3 $ |
| 5 | $ 200 $ |
| 6 | $ 63 $ |
| 7 | $ 72 $ and $ 112 $ |
| 8 | $ 35 $ |
| 9 | $ 12 $ |
| 10 | $ 100 $ |

> Note: All answers are whole numbers, so they are already in simplest form (or can be written as improper fractions if required, e.g., $ \frac{80}{1} $, but not necessary unless specified).

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