1

2.2 Equations Involving Fractional Forms

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Procedure: To solve a fractional equation:Step 1. Determine LCM of denominators.

Step 2: Multiply all terms by LCM (clearing denominators).Step 3: Solve equation as in previous lessons.

Step 4: Check

To solve equations that involve fractions, it is easiest to begin by clearing the equation of all fractions. This can be accomplished by multiplying both sides of the equation by the least common multiple of all the denominators (LCD) in the equation.

Example: Find the LCM of 12 and 15.

The least common multiple of two or more whole numbers is actually the smallest whole number that is divisible by each of the numbers.

Option 1. List multiples of each number until a common multiple is found.

Multiples of 12 are 12, 24, 36, 48, 60, 72,... Multiples of 15 are 30, 45, 60, 75, 90,....

Since 60 is the smallest number that appears in each list, the least common multiple of 12 and 15 is 60.

Option 2. Write the prime factorization of each number.

12 = 2 × 2 × 315 = 3 × 5

For each prime number from the group, write down the largest number of times it appears in either factorization. The least common multiple is the product of all the prime numbers written down.

The LCM will have two 2’s, one 3, and one 5.LCM = 2 × 2 × 3 x 5 = 60

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2.2 Equations Involving Fractional Forms

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2a 1 a 1 Solve:

9Exam

6ple 1

6 .

9

Solution: 1. Find the LCD

2. Multiply all fractions by the LCD.

LCD=18

(18) (18) (2a 1 a 1

9 6 9

18) (18)

6

3. Clear the denominators by reducing.

2 2 33

4. Simplify and solve.

4a 3 2a 3

2a 6

a 3

5. Check.

2(3) 1 3 19 6 9 6

6 19 6

3 19 6

12 318 18

6 318 18

9 9

18 18

Your Turn Problem #1

x 1 1

Solve and Check: 18 6 2

Answer: x=12

3

2.2 Equations Involving Fractional Forms

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Example 2. Solve:

3x 4 1 5x 2 112 3 12 2

Solution: 1. Find the LCD

2. Multiply all fractions by the LCD.

LCD=12

(12) (12) (13x 4 1 5x 2 1

12 3

2) (1

1

2)

2 2

3. Clear the denominators by reducing.

64

4. Simplify and solve.

3x 4 4 5x 2 6

3x 5x 4

x 2

5. Check.

3(2) 4 1 5(2) 2 1

12 3 12 2

10 412 12

12 612 12

6 612 12

Your Turn Problem #2

x 5 1 x 2 1Solve and Check:

9 3 4 4

Answer: x = 7

4

2.2 Equations Involving Fractional Forms

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Example 3. Solve:

x 4 x 2 24 6 3

Solution:1. Find the LCD

2. Multiply all fractions by the LCD.

LCD=12

(12) (12)x 4 x 2 2

4 6 3

(12)

3. Clear the denominators by reducing.

4. Simplify and solve.

3x 12 2x 4 8

x 16 8

x 8

5. Check.

8 4 8 2 2

4 6 3

4 104 6

23

2 23 3

43 2

Your Turn Problem #3

2x 1 3x 1 3Solve:

2 4 1021

Answer: 5

5

2.2 Equations Involving Fractional Forms

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Solution: 1. Define the variable. x = the number.

2. Translate.

1 3 4

x x 2 x2 4 3

3. Solve by using LCD 1 3 4

x x(12) (12) (12) (12

24

)x

2 3

6 3 4

6x 9x 24 16x4. Answer the question.

15x 24 16x

x 24The number is 24.

Example 4. One-half a number plus three-fourths of the number is 2 more than four-thirds of the number. Find the number.

Recall the procedure for translating sentences into equations and solving.

Step 1. Assign a variable to the unknown quantity (What is the problem asking you to find?)Step 2. Translate the sentence into a mathematical expression.

Step 3. Solve the equation using the steps from the preceding slides.

Your Turn Problem #4

Three-eighths of a number minus one-fourth of the number is six less than one-half of the number. Find the number.

Answer: 16

6

2.2 Equations Involving Fractional Forms

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Example 5. Suppose that the width of a rectangle is 3 feet less two-thirds of its length. The perimeter of the rectangle is 114 feet. Find the length and width of the rectangle.

1. Define the variable. Whenever possible use a picture or chart.

x = length(being referred to)

L =x

2

W x 33

2. Use formula for perimeter to obtain an equation. 2L+2W=P

2

2 x 2 x 3 1143

3. Solve

4

2x x 6 1143

4

2x x 1203

6x 4x 360

x 36

4. Answer the question.

(3W3

6)2

3

W=21

The length is 36 feet and the width is 21 feet.

(3)(3) (3)

Your Turn Problem #5

Suppose that the width of a rectangle is 1 foot less than three-fourths of its length. The perimeter of the rectangle is 40 feet. Find the length and width of the rectangle.

Answer: The width is 8 feet and the length is 12 feet.

The End.B.R.12-09-06