Solving Equations with Rational Numbers

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∕₂f = ²∕₁ . ⁵∕₈

f = ²∕₁ . ⁵∕₈

f = ¹⁰∕₈ or ⁵∕₄

Exercise:

1. 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?
2. Solve the following equation.
   ⁵∕₉y = ¼
3. Solve the following equation.
   s + ¼ = 12 ½
4. Solve the following equation.
   a – 4 ³∕₈ = 2 ½
5. Solve the following equation.
   ²∕₇ q = 3 ³∕₅
6. Solve the following equation.
   ⁷∕₁₀ = x – ³∕₅ 
7. Solve the following equation.
   9 = ³∕₈y
8. Solve the following equation.
   x/3 = ⁶∕₉
9. Solve the following equation.
   ²∕₇ = y/12
10. 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.

Concept map: