SOLVING FRACTION EQUATIONS

Solve for x:

Problem 1 :

Solution :

Find the least common multiple of the denominator 2 and 7.

LCM of (2, 7) = 14

Multiply each side of the fraction by 14.

14(x/2) = 14(3/7)

7x = 6

x = 6/7

So, the value of x is 6/7.

Problem 2 :

3/5 = x/6

Solution :

Find the least common multiple of the denominator 5 and 6.

LCM of (5, 6) = 30

Multiply each side of the fraction by 30.

30(3/5) = 30(x/6)

18 = 5x

5x = 18

x = 18/5

So, the value of x is 18/5.

Problem 3 :

Solution :

Find the least common multiple of the denominator 5 and 3.

LCM of (5, 3) = 15

Multiply each side of the fraction by 15.

15(x/5) = 15[(x-2)/3]

3x = 5(x - 2)

3x = 5x - 10

3x - 5x = -10

-2x = -10

x = 5

So, the value of x is 5.

Problem 4 :

Solution :

Find the least common multiple of the denominator 3 and 8.

LCM of (3, 8) = 24

Multiply each side of the fraction by 24.

24 [(x+1)/3] = 24 [(2x-1)/8]

Use distributive property.

8x + 8 = 6x – 3

Subtract 6x from each side.

2x + 8 = -3

2x = -3 - 8

2x = -11

x = -11/2

So, the value of x is -11/2.

Problem 5 :

Solution :

Find the least common multiple of the denominator 3 and 4.

LCM of (3, 4) = 12

Multiply each side of the fraction by 12.

12(2x/3) = 12[(5-x) /4]

Use distributive property.

8x = 3(5 - x)

8x = 15 – 3x

8x + 3x = 15

11x = 15

x = 15/11

So, the value of x is 15/11.

Problem 6 :

Solution :

Find the least common multiple of the denominator 3 and 2.

LCM of (3, 2) = 6

Multiply each side of the fraction by 6.

6[(3x+2)/3] = 6[(2x-5)/2]

Use distributive property.

2(3x + 2) = 3(2x - 5)

6x + 4 = 6x - 15

6x - 6x = -15 - 4

0 ≠ -19

There is no solution.

Problem 7 :

Solution : 

Find the least common multiple of the denominator 3 and 12.

LCM of (3, 12) = 12

Multiply each side of the fraction by 12.

12[(2x-1)/3] = 12[(4-x)/12]

Use distributive property.

4(2x - 1) = 4 - x

8x – 4 = 4 – x

8x + x = 4 + 4

9x = 8

x = 8/9

Problem 8 :

Solution :

Find the least common multiple of the denominator 6 and 2.

LCM of (6, 2) = 6

Multiply each side of the fraction by 6.

6[(3x+7)/6] = 6[(4x-1)/(-2)]

Use distributive property.

3x + 7 = -3(4x - 1)

3x + 7 = -12x + 3

3x + 12x = 3 - 7

15x = - 4

x = - 4/15

Problem 9 :

An alloy is formed by mixing two or more metals. Sterling silver is an alloy composed of 92.5% silver and 7.5% copper by weight. You have 15 ounces of 800 grade silver, which is 80% silver and 20% copper by weight. How much pure silver should you mix with the 800 grade silver to make sterling silver?

Solution :

Let x be the amount of silver to be added.

Original quantity of silver = 15 ounces.

= 20% of 15

After adding x ounces of silver, the new weight of silver is 15 + x

7.5/100 = (20% of 15)/(15 + x)

7.5(15 + x) = 20% of 15 (100)

112.5 + 7.5x = 0.2(15)(100)

112.5 + 7.5x = 300

7.5x = 300 - 112.5

7.5x = 187.5

x = 187.5/7.5

x = 25

You should mix 25 ounces of pure silver with the 15 ounces of 800 grade silver.

Problem 10 :

So far in your volleyball practice, you have put into play 37 of the 44 serves you have attempted. Solve the equation

90/100 = (37 + x)/(44 + x)

to find the number of consecutive serves you need to put into play in order to raise your serve percentage to 90%.

Solution :

90/100 = (37 + x)/(44 + x)

90(44 + x) = 100(37 + x)

3960 + 90x = 3700 + 100x 

3960 - 3700 = 100x - 90x

260 = 10x

x = 260/10

x = 26

So, the required number of consective serves is 26.