What must be added to this mixed fraction to produce this mixed fraction

Question

This may seem like a simple question which is fit for a Google search; however, I'm unable to find an answer.

What must be added to $6\frac{3}{5}$ to produce $3\frac{5}{6}$

I hope I'll finally understand. Step-by-step solution is much appreciated.

Answer 1

Noting that $3\frac{5}{6}$ is less than $6\frac{3}{5}$, it's clear that you'll have to add a negative. I would simply find the difference and put a negative out front $$ - \left( 6\frac{3}{5} - 3\frac{5}{6} \right) $$ $$ = - \left( \frac{33}{5} - \frac{23}{6} \right) $$ $$ = - \left( \frac{198}{30} - \frac{115}{30} \right) $$ $$ = - \left( \frac{198-115}{30} \right) $$ $$ = - \left( \frac{83}{30} \right) $$ $$ = - 2 \frac{23}{30} $$ Or you could have gone straight to the subtraction $$ 3\frac{5}{6} - 6\frac{3}{5} $$ $$ = \frac{23}{6} - \frac{33}{5} $$ $$ = \frac{115}{30} - \frac{198}{30} $$ $$ = -\frac{83}{30} $$ $$ = -2\frac{23}{30} $$

Answer 2

"What must be added to A to get B" means "B - A".

So you want $3 \frac 56 - 6 \frac 35$

$6 \frac 35$ is larger than $3 \frac 56$ so the answer will be negative.

First thing is to convert from mixed numbers to "improper" fractions by putting them over a common denominator.

$3 \frac 56 = 3 + \frac 56 = 3*\frac 66 + \frac 56 = \frac {18}{6} + \frac 56 = \frac {18 + 5}{6} = \frac {23}6$.

$6 \frac 35 = 6 * \frac 55 + \frac 35 = \frac {30}5 + \frac 35 = \frac {33}5$

So we need $\frac {23}6 - \frac {33}5$

We need to put them over a common denominator. We need to find the lowest common multiple of $5 $ and $6$. That is $30=5*6$.

So $\frac {23}6 - \frac {33}5=$

$\frac{23}6*\frac 55 - \frac{33}5*\frac 66 =$

$\frac {115}{30} - \frac {198}{30}=$

$\frac {115 - 198}{30}$

We do the subtraction knowing we will get a negative number.

$\frac{115-198}{30} = - \frac {198 - 115}{30} = -\frac {83}{30}$

For style points we should reduce to a "mixed number". So we divide by 30 and keep the remainder.

$83 \div 30 = 2 $ with $23$ remainder.

So $-\frac {83}{30} = -2 \frac {23}{30}$.

So $6 \frac 35 + (-2 \frac {23}{30}) = 3 \frac 56$.