Why the Addition of Dissimilar Fractions Trick Works

In the pervious post, we have learned a math trick on addition of two dissimilar fraction. This is done by “cross multiplying” the numerators and denominators of the fractions as shown below (read the previous post for more details). In this post, we discuss why this trick works.

Remember: It’s cool to know the math trick, but it’s even cooler, or should I say awesome, to know why the math trick works.

add fractions

Now, why does this trick works? Will it work every time? 

If we are going to use the standard method in adding fractions, we need to get the Least Common Multiple (LCM) of 4 and 3, the denominators of 2 fractions. The LCM of 3 and 4 is 12. We then convert 3/4 and 2/3 to fractions with denominator 12. This means that 3/4 becomes 9/12 and 2/3 becomes 8/12 making the sum 7/12.

But what about in general? How do we add a fraction a/b and c/d?

In adding a/b and c/d, we get the LCD of b and d which is bd. So,

\displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{bc}{bd} = \frac{ad + bc}{bd}.

Now, notice that if we use the trick above (see image), we have follow the following steps.

  1. Get the product of the numerator of the first fraction and the denominator of the second fraction which is (ad).
  2. Get the product of the denominator of the first fraction and the numerator of the second fraction (bc)
  3. Get the product of the denominators (bd).
  4. The sum of the two fractions: the sum of (1) and (2) is the numerator and (3) is the denominator.

So, the sum of a/b + c/d using the math trick is

\displaystyle \frac{ad + bc}{bd}

which is the same as the standard algorithm.

So this means that the trick will work each time.

Something to think about:

What if b and d are not relatively prime?

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