Why $\frac{\frac{1}{2}\times\frac{k}{k+7}}{\frac{1}{2}\times\frac{k}{k+7} \ \ +\frac{1}{2}\times\frac{2}{5}} = \frac{k}{k+\frac{2}{5}(k+7)}$?

Question

Why $$\frac{\frac{1}{2}\times\frac{k}{k+7}}{\frac{1}{2}\times\frac{k}{k+7} \ \ +\frac{1}{2}\times\frac{2}{5}} = \frac{k}{k+\frac{2}{5}(k+7)}?$$

How did it become that form?

My attempt:

$$\frac{\frac{1}{2}\times\frac{k}{k+7}}{\frac{1}{2}\times\frac{k}{k+7}+\frac{1}{2}\times\frac{2}{5}} = \frac{\frac{1}{2}\times\frac{k}{k+7}}{\frac{1}{2}(\frac{k}{k+7}+\frac{2}{5})} = \frac{\frac{k}{k+7}}{\frac{k}{k+7}+\frac{2}{5}}$$

Answer 1

Hint, not a complete answer:

1. Your work so far is fine.

2. Now try multiplying both the numerator and denominator by $k+7$, i.e., multiply the whole fraction by $$ \frac{k+7}{k+7} = 1 $$ and see whether you can simplify a bit.

Answer 2

Simply multiply both the numerator and the denominator by $2(k+7)$.

Answer 3

Hint: $$\frac{\frac{k}{k+7}}{\frac{k}{k+7} \ \ +\frac{2}{5}}= \frac{\frac{k}{k+7}(k+7)}{(\frac{k}{k+7} \ \ +\frac{2}{5})(k+7)} $$

Answer 4

$\frac{\frac{1}{2}\times\frac{k}{k+7}}{\frac{1}{2}\times\frac{k}{k+7}+\frac{1}{2}\times\frac{2}{5}} = \frac{\frac{1}{2}\times\frac{k}{k+7}}{\frac{1}{2}(\frac{k}{k+7}+\frac{2}{5})} = \frac{\frac{k}{k+7}}{\frac{k}{k+7}+\frac{2}{5}}$

So far so good. But notice $\frac 25 = \frac 25*\frac {k+7}{k+7}=\frac 2{5(k+7)}(k+7)$ so we can do one more step.

$=\frac{\frac{k}{k+7}}{\frac{k}{k+7}+\frac{2}{5(k+7)}(k+7)}=\frac {\frac 1{k+7}\times k}{\frac 1{k+7}\times (k + \frac 25(k+7))}$

which does it.

Although... advice: Factoring fractions out and canceling is fine, but multiplying top and bottoms to simplify is easier.

$\frac{\frac{1}{2}\times\frac{k}{k+7}}{\frac{1}{2}\times\frac{k}{k+7}+\frac{1}{2}\times\frac{2}{5}} = \frac{(\frac{1}{2}\times\frac{k}{k+7})\times 2}{(\frac{1}{2}\times\frac{k}{k+7}+\frac{1}{2}\times\frac{2}{5})\times 2}= \frac{\frac{k}{k+7}}{\frac{k}{k+7}+\frac{2}{5}}$

$= \frac{\frac{k}{k+7}\times(k+7)}{(\frac{k}{k+7}+\frac{2}{5})\times (k+7)}=\frac k{k + \frac 25(k+7)}$

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Frankly, I'd take it one step further and

$\frac k{k + \frac 25(k+7)}= \frac {k\times 5}{(k + \frac 25(k+7))\times 5}= \frac {5k}{5k + 2k + 14} = \frac {5k}{7k + 14} = \frac {5k}{7(k+2)}$

but.... I guess you weren't asked to go that far. And you don't have to. It's just what I would do.