Fraction Addition & Subtraction

Let us recall our chocolate sharing problem to understand fraction addition and subtraction. Like before, we will use the chocolate bar below with 18 small tablets to understand our fraction.

Let us say, this time, you have two friends visiting you. You give $5$ of the $18$ tablets to the first friend and $7$ tablets to the second friend. So totally you have given out $12$ out of $18$ tablets to your friends. Let us see this in fraction.

You gave $\dfrac{5}{18}$ of your chocolate to the first friend and $\dfrac{7}{18}$ to the second friend, and totally you have given away $\dfrac{12}{18}$ of your chocolate bar.

So, clearly:

$\dfrac{5}{18} + \dfrac{7}{18} = \dfrac{12}{18}$.

We can see that the denominator is not changing in this case, only the numerators are getting added.

We can surely reduce $\dfrac{12}{18}$ to get a smaller numerator and denominator. But we will not do it for now, just to keep the denominators same.

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Now we will change the problem a little bit. Let us say you gave the first friend $6$ tablet out of $18$ and the second friend $5$ tablets out of $18$. So, in all, you gave away $11$ out of $18$ tablets. In terms of fraction, you gave $\dfrac{6}{18}$ of your chocolate to your first friend. Reducing this fraction by 2 and 3, and you get

$\dfrac{\cancel{6}\raise{5px}{\cancel{3}}\raise{10px}{1}}{\cancel{18}\lower{5px}{\cancel{9}}\lower{10px}{3}} = \dfrac{1}{3}$

You gave $\dfrac{5}{18}$ to your second friend. As you can see this fraction cannot be reduced further.

So you gave $\dfrac{1}{3}$ of your chocolate bar to your first friend, and you gave $\dfrac{5}{18}$ to your second friend. But you also know that in all you gave away $\dfrac{11}{18}$ of your chocolate bar to your friends.

So, how does $\dfrac{1}{3}$ and $\dfrac{5}{18}$ add up to give $\dfrac{11}{18}$? Let us find out.

We will calculate the value of:

$\dfrac{1}{3} + \dfrac{5}{18}$

Earlier we saw that if the denominators of two fractions are same, then addition is easy, we just need to add the numerators and keep the denominator same as the two given fractions. But in this case the fractions have different denominators, so what do we do?

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No problem, we will $make$ the denominators same. We saw in our earlier lesson that multiplying or dividing the both the numerator and the denominator by the same, non-zero, number does not change the value of the fraction. So we multiply the numerator and the denominator of the fraction $\dfrac{1}{3}$ by $6$. And we get $\dfrac{1}{3} = \dfrac{1 \times 6}{3 \times 6} = \dfrac{6}{18}$

Keep in mind, that fractions with the same denominators are called $Like\ fractions$ and fractions with different denominators are called $Unlike\ fractions$.

Our sum becomes:

$\dfrac{1}{3} + \dfrac{5}{18}$

$\dfrac{1 \times 6}{3 \times 6} + \dfrac{5}{18}$

$= \dfrac{6}{18} + \dfrac{5}{18}$

$= \dfrac{6 + 5}{18}$

$=\dfrac{11}{18}$

This matches our earlier result, where we saw that you gave a total of $11$ out of $18$ tablets to your friends. We will take one more example to understand the concept of fraction addition.

Let us see how we can add the fractions $\dfrac{1}{9}$ and $\dfrac{1}{6}$.

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First we have to convert both the fractions to fractions with the same denominator. Here we need to find out a number number that is divisible by $9$ and $6$. We know that the $LCM$ of $9$ and $6$ is $18$.

We can convert the first fraction to a denominator of 18 by multiplying both the denominator and the numerator by $2$.

So, we get:

$\dfrac{1}{9} = \dfrac{1 \times 2}{9 \times 2} = \dfrac{2}{18}$

To convert the second fraction to a denominator of $18$, we have to multiply both the numerator and the denominator by 3.

So, we get:

$\dfrac{1}{6} = \dfrac{1 \times 3}{6 \times 3} = \dfrac{3}{18}$

Now since both fractions have the same denominator, we can add them easily, as $\dfrac{2}{18} + \dfrac{3}{18} = \dfrac{5}{18}$.

Let us do a recap of the steps once more:

$\dfrac{1}{9} + \dfrac{1}{6}$

$= \dfrac{1 \times 2}{9 \times 2} + \dfrac{1 \times 3}{6 \times 3}$

$= \dfrac{2}{18} + \dfrac{3}{18}$

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$= \dfrac{2 + 3}{18}$

$= \dfrac{5}{18}$

In case you are not familiar with $LCM$ you can use any common multiple of $6$ and $9$ and you should get the same result. The easiest way of finding out a common multiple of the two denominators, is to take the product of the two denominators. We can solve the same sum, by converting the denominators to $6 \times 9 = 54$ in the following way:

$\dfrac{1}{9} + \dfrac{1}{6}$

$= \dfrac{1 \times 6}{9 \times 6} + \dfrac{1 \times 9}{6 \times 9}$

$= \dfrac{6}{54} + \dfrac{9}{54}$

$= \dfrac{6 + 9}{54}$

$= \dfrac{15}{54}$

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as we can see, we can cancel out $3$ from the result's numerator and denominator,

$= \dfrac{\cancel{15} \raise{5px}{5}}{\cancel{54} \lower{5px}{18}}$

$= \dfrac{5}{18}$ , which is the same answer as before.

If you are familiar with $LCM$, use the $LCM$, that will keep both the numerator and denominator small, and will make calculations easier.

Fraction Subtraction:

Fraction subtraction is done is exactly the same way as addition, that is, by converting the fractions to $Like\ Fractions$ with the same denominator and by subtracting the numerators.

Let us try to evaluate $\dfrac{5}{8} - \dfrac{7}{12}$

We know that the $LCM$ of $8$ and $12$ is $24$. So, we will use $24$ as a denominator for converting the fractions to like fractions.

$\dfrac{5}{8} - \dfrac{7}{12}$

$= \dfrac{5 \times 3}{8 \times 3} - \dfrac{7 \times 2}{12 \times 2}$

$= \dfrac{15}{24} - \dfrac{14}{24}$

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$= \dfrac{15 - 14}{24}$

$= \dfrac{1}{24}$