## Key Concepts:

- Solve addition equations with fractions.
- Solve subtraction, multiplication, and division equations with fractions.

## Introduction:

**Solving equations with rational numbers: **

You can solve equations with fractions and mixed numbers the same way that you solve equations with whole numbers: using inverse relationships and properties of equality to isolate the variable.

For example, let us see a subtraction equation.

**4.5.1.1 Solve addition equations with fractions**

**Example 1:**

Joyce needs to swim a total of 8 miles this week. So far, she swam 5 ⅜ miles. Find how many more miles Joyce needs to swim.

**Solution:**

Use a bar diagram to show how the qualities are related and to write an equation.

5⅜ + *m* = 8

Solve for *m*.

5 ⅜ + *m* = 8

5 ⅜ + *m* – 5 ⅜ = 8 – 5 ⅜

*m* = 8 – 5 ⅜

*m* = 7 ⁸∕₈ – 5 ⅜

*m* = 2 ⁵∕₈

**Example 2:**

Billy carved 3 ⁵∕₉

feet of a totem pole of 6 feet. Find the remaining length of the totem pole.

**Solution:**

Use a bar diagram to show how the qualities are related and to write an equation.

3 ⁵∕₉ + *h* = 6

Solve for *h*.

3 ⁵∕₉+ *h* = 6

3 ⁵∕₉ + *h* – 3 ⁵∕₉ = 6 – 3 ⁵∕₉

*h* = 6 – 3 ⁵∕₉

*h* = 5 ⁹∕₉ – 3 ⁵∕₉

*h* = 2 ⁴∕₉

**4.5.1.2 Solve subtraction, multiplication, and division equations with fractions**

**Example 3:**

Use inverse relationship to solve the following subtraction equation.

*a* – 4 ⅜ = 2 ½

**Solution:**

*a* – 4 ⅜ = 2 ½

*a* – 4 ⅜ + 4 ⅜ = 2 ½ + 4 ⅜

*a* = 2 ½ + 4 ⅜

*a* = 6 ⁷∕₈

**Example 4:**

Use inverse relationship to solve the following multiplication equation.

²∕₇ *x* = 18/5

**Solution:**

²∕₇ *x* = 18/ 5

(7/2) ²∕₇ x = (7/2) ¹⁸∕₅

*x* = ⁷∕₂ × ¹⁸∕₅

*x* = ¹²⁶∕₁₀ or ⁶³∕₅ or 12 ³∕₅

**Example 5:**

Use inverse relationship to solve the following division equation.

f/2 = ⁵∕₈

**Solution:**

f/2 = ⁵∕₈

²∕₁.¹∕₂*f* = ²∕₁ . ⁵∕₈

*f* = ²∕₁ . ⁵∕₈

*f* = ¹⁰∕₈ or ⁵∕₄

## Exercise:

- Henry worked at a car wash for 6 hours. For 3 hours, he vacuumed the interiors of the cars. For the other part of his shift, he collected money from customers. For how many hours did Henry collect money?
- Solve the following equation.

⁵∕₉*y*= ¼ - Solve the following equation.
*s*+ ¼ = 12 ½ - Solve the following equation.

a – 4 ³∕₈ = 2 ½ - Solve the following equation.

²∕₇*q*= 3 ³∕₅ - Solve the following equation.

⁷∕₁₀ =*x*– ³∕₅ - Solve the following equation.

9 = ³∕₈*y* - Solve the following equation.

x/3 = ⁶∕₉ - Solve the following equation.

²∕₇ = y/12 - Is the solution of
*w*×¹¹∕₁₂ greater than or less than 19? How can you tell without solving the equation?

### What have we learned:

- Solve addition equations with fractions by using inverse relationships and properties of equality.
- Solve subtraction, multiplication, and division equations with fractions by using inverse relationships and properties of equality.