Math CentralQuandaries & Queries


Question from Stuart, a parent:



When dealing with equations involving fractions, it is always a good idea to get a common denominator first. Each of 40 and 80 is a good choice for the equation in question. Two fractions with the same denominator are equal to each other exactly when their numerators are equal. So after getting a common denominator and writing each side as a single fraction (rather than a sum or difference of fractions), you know the numerators must be equal. This leads to an equation involving x, but no fractions. Solve for x.

For example, look at the equation is

3/7 + 2x/5 = x/3 + 1/2.

A good choice for a common denominator is 7 times 5 times 3 times 2, or 210. Writing all of the fractions with denominator 210 makes the equation look like this:

90/210 + 84x/210 =70x/210 + 105/210.

Next, write each side as a single fraction. This gives you the equation

(90 + 84x)/210 = (70x + 105)/210.

These two fractions have the same denominator, so they are equal exactly when the numerators are equal, that is, when 90+84x = 70x+105. After rearranging, this becomes 14x = 15, or x = 15/14.

It is always wise to check the solution by substituting it back into the original equation. This gives

3/7 + 30/70 = 15/42 + 1/2, or 90/210 + 90/210 = 75/210 + 105/210.

That is, 180/210 = 180/210, which is true. Thus, our solution checks out.

Victoria West


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