Equations with Fractions & Simplifying Complex Fractions

Equations With Fractions & Simplifying Complex Fractions

Example: Solve -1/3 = x – ¾

Add ¾ to both sides, resulting in

-1/3 + 3/4 = x – 3/4 + 3/4 and now simplify to

-1/3 + 3/4 = x

Now you need to write the fractions with the common denominator of 12. Multiply –1/3 by 4/4 and multiply ¾ by 3/3 and the result is

-4/12 + 9/12 = x which simplifies to 5/12 = x

Example: Solve -2/3 = x + 5/11

Subtract 5/11 from both sides, resulting in

-2/3 - 5/11 = x + 5/11 – 5/11 and now simplify to

-2/3 - 5/11 = x

Now you need to write the fractions with the common denominator of 33. Multiply –2/3 by 11/11 and multiply 5/11 by 3/3 and the result is

-22/33 - 15/33 = x which simplifies to -37/33 = x

Example: Solve -36 = 4x

Divide both sides by 4, resulting in

-36/4 = (4x)/4 but (4x)/4 simplifies to 1x = x, so

-36/4 = x

And –36 divided by 4 is –9, so x = -9

Example: Negative three-eighths is equal to ten-thirds of a number. Find the number.

When we translate this to words, we get

–3/8 = (10/3) x, where x is our number. We want x all by itself, so to get rid of the factor (10/3) that it is multiplied by, we multiply both sides by 3/10 to get

(3/10)  (-3/8) = (3/10)  (10/3)x

The left multiplies out to –9/80 and on the right side of =, the fractions 3/10 times 10/3 multiply to 1, so our equation becomes

-9/80 = 1x which simplifies to –9/80 = x.

Complex Fractions

Example: Simplify the complex fraction shown below.

3 ¼ - 1/6 .

5/8 – 1 1/3

To simplify fractions containing fractions, you first simplify so you have a single fraction divided by a single fraction. Then you invert and multiply. And first we could rewrite the mixed numbers as improper fractions. So we would get

13/4 - 1/6 .

5/8 – 4/3

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Now, we subtract 13/4 -1/6 and 5/8 – 4/3 but we need common denominators of 12 and 24, respectively. We multiply 13/4 by 3/3 and multiply 1/6 by 2/2. Also, we multiply 5/8 by 3/3 and multiply 4/3 by 8/8 and we have

39/12 - 2/12 .

15/24 – 32/24 and this simplifies to

37/12 .

-17/24 and this means we are dividing 37/12 by (-17/24), so we invert and multiply and write as

37  - 24

12 17

and this simplifies to

37  2 = -74 .

1 17 17