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How do you solve an equation with two fractions added to each other equaling a whole number?

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3 Answers

x/2+x/3=7

The general strategy is to get rid of those denominators altogether so that the x terms can be collected on one side of the equation. What you are looking for is the lowest common multiple (LCM) of the denominators, 2 and 3. If you cannot figure out the lowest common multiple, any multiple of the denominators will do.

For example, the LCM of 2 and 3 is 6. That is, 6 is a multiple of 2 and 6 is a multiple of 3, and it is the lowest multiple that 2 and 3 share.

Multiply both sides by the LCM

6*(x/2+x/3)=6 *7 (using * as multiplication)

Distribute the 6 over each term inside the parentheses

6x/2+6x/3=42

Reduce each fraction

(6/2)x +(6/3)x = 42

3x +2x = 42

Combine like terms.

5x = 42

Divide both sides by 5

x=42/5

Now if you had used another multiple of 2 and 3 instead of 6, it would work too. Try it with 12 or 18 or 24, all multiples of 2 and 3, and you will see that it is workable.

Comments

Yes, Robert, any multiple will work but the LCM is most efficient and they all should learn that concept, right? But I voted for your answer as best explanation.

- a PhD in math and long-time teacher of it, Fred S

Absolutely right, Fred. LCM is the way to go.

Comment

Hi John, here is a local answer from a U Wash PhD in math, who could help you on other questions there on Whidbey [by meeting in Port Townsend].

In three steps, just like adding ordinary fractions x/2 + x/3 can be added with the same denominator in each term, i.e. the LCD of 6 

 x/2 + x/3  = 3*x/3*2 + 2*x/2*3 = 3x/6 + 2x/6 = 5x/6 = 7 ,

or 5x = 42  which gives  x=42/5 = 8.4, the easiest answer to 'grid in.'

Comments

I was also inspired by Kathye's short answer, so I voted for it too! /FS

Comment

Hi, John.

The short answer to your question is to think of the Lowest Common Denominator (also called Least Common Multiple) for the fractions as if you were going to add them (in this case it would be 6). You do not have to rewrite the fractions using the LCD. Multiply each side of the equation by this LCD. So for your equation:  (the asterisk represents multiplication)

6(x/2+x/3)=7*6

Using distributive property, multiply the 6 by both terms on the left:

6 * x/2 + 6 * x/3 = 7*6

To multiply 6 by x/2, remember that the whole number is 6/1.

6x/2 + 6x/3 = 7*6

Written in lowest terms:

3x + 2x = 42

then   5x = 42

Divide both sides by 5, and you get

x = 42/5,  which can be written also as a mixed number 8 2/5   or as a decimal   8.4

 

Hope this helps!

Kathye