Addition And Subtraction Of Mixed Fractions

You read about fraction addition and subtraction .
Today we will look at a faster way to add/subtract mixed fractions.
Let us try to add two mixed fractions $16\dfrac{3}{4}$ and $23\dfrac{5}{8}$
If we try to do this by converting them into improper fractions and compute the sum we will get the following steps:

$16\dfrac{3}{4} + 23\dfrac{5}{8}$

$= \dfrac{16 \times 4 + 3 }{4} + \dfrac{23 \times 8 + 5}{8}$

$= \dfrac{67}{4} + \dfrac{189}{8}$

$= \dfrac{134}{8} + \dfrac{189}{8}$

$= \dfrac{134 + 189}{8}$

$= \dfrac{323}{8}$

$= 40\dfrac{3}{8}$

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As you saw in this case, the steps involve multiplications of quite large numbers. But if we take a different approach, and break up our mixed fractions into integer and proper fractions first, our calculations will be much easier.

$16\dfrac{3}{4} + 23\dfrac{5}{8}$

$= \left(16 + \dfrac{3}{4}\right) + \left(23 + \dfrac{5}{8}\right)$

$= 16 + 23 + \dfrac{3}{4} + \dfrac{5}{8}$

$= 39 + \dfrac{6}{8} + \dfrac{5}{8}$

$= 39 + \dfrac{6 + 5}{8}$

$= 39 + \dfrac{11}{8}$

$= 39 + 1\dfrac{3}{8}$

$= 40\dfrac{3}{8}$

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Now, let us look at how we can use the same technique to subtract one mixed fraction from another.
Let us try subtracting $17\dfrac{7}{12}$ from $29\dfrac{3}{8}$. As we can see that converting them to improper fraction will require large calculations, hence we will use the easier method we learnt above.

$29\dfrac{3}{8} - 17\dfrac{7}{12}$

$= \left(29 + \dfrac{3}{8}\right) - \left(17 + \dfrac{7}{12}\right)$

$= 29 + \dfrac{3}{8} - 17 - \dfrac{7}{12}$
(Observe that the $\unicode{0x2018}+\unicode{0x2019}$ sign before $\dfrac{7}{12}$ becomes $\unicode{0x2018}-\unicode{0x2019}$ sign because the sign outside the bracket is $\unicode{0x2018}-\unicode{0x2019}$).
Remember that whenever you open a bracket with a $\unicode{0x2018}-\unicode{0x2019}$ sign outside, all the terms inside the bracket will become the opposite sign from what they were before. That is, all $\unicode{0x2018}-\unicode{0x2019}$ signs become $\unicode{0x2018}+\unicode{0x2019}$ and all $\unicode{0x2018}+\unicode{0x2019}$ signs become $\unicode{0x2018}-\unicode{0x2019}$.
You will understand this concept better at a higher grade, for now just try to remember this rule).

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$29 - 17 + \dfrac{3}{8} - \dfrac{7}{12}$

$= 12 + \dfrac{9}{24} - \dfrac{14}{24}$ (here you can see that we are trying to subtract a larger fraction from a smaller one)

$= 11 + 1 + \dfrac{9}{24} - \dfrac{14}{24}$ (so we borrow $1$ from $12$)
 
$= 11 + \dfrac{24}{24} + \dfrac{9}{24} - \dfrac{14}{24}$ (we convert the borrow $1$ into a fraction with like denominator as the other two) 

$= 11 + \dfrac{24 +  9 - 14}{24}$ (now we should be able to simplify the term easily)

$= 11 + \dfrac{33 - 14}{24}$

$= 11 + \dfrac{19}{24}$

$= 11\dfrac{19}{24}$