Confusion Dividing A Fraction with a Whole Number...

Question

In the lesson I am doing, I divide fractions. Here is my problem:
28/55 / 7
I had to look up how to do this problem. According to Math Is Fun, you divide the denominator by the whole number and then simplify if possible. I did this, and got:
28/385
I couldn't simplify it, and the online quiz would only let me enter 2 digits in the denominator textbox. I couldn't figure it out, so I put in a random answer. They said it was:
4/55
How did they get this answer, and where did I go wrong? Please help, because I can't figure it out.

Answer 1

Just divide both $28$ and $385$ by $7$. The number $7$ is a common factor of both of those numbers:

$$ \frac{28}{385}=\frac{28\div7}{385\div7}=\frac{4}{55}. $$

Or if you want you can think of it like this. $7$ goes into $24$ 4 times, therefore, $28=7\cdot 4$. $7$ goes into $385$ 55 times, therefore, $7\cdot 55=385$. One very importnat thing you must know about fractions is that if you divide both the top and the bottom of a fraction by the same number, the fraction itself (a fraction is basically a ratio between two numbers) is not going to change ($\frac{1}{2}$ is the same as $\frac{2}{4}$ which is the same as $\frac{6}{12}$—the ratio 1 to 2 is still preserved despite the fact that it can take on many different forms):

$$ \frac{7\cdot 4}{7\cdot 55}=\frac{7\cdot 4\div7}{7\cdot 55\div7}= \frac{7\div 7\cdot 4}{7\div 7\cdot 55}=\frac{1\cdot4}{1\cdot55}= \frac{4}{55}. $$

Answer 2

To divide a fraction $\frac{a}{b}$ by $c$, you can either divide $a$ by $c$ or multiply $b$ with $c$. It's up to you which one you use.

Answer 3

For a general purpose, suppose we need to find $y=\frac{\frac{a}{b}}{\frac{c}{d}}$ where $\frac{c}{d}\not=0.$ Then we can do the following - $$y\frac{c}{d}=\frac{a}{b}\implies \frac{c}{d}=\frac{\frac{a}{b}}{y}\\ \implies \frac{d}{c}=\frac{y}{\frac{a}{b}}\\ \implies y=\frac{d}{c} \times \frac{a}{b}.$$

For your problem, take $\frac{a}{b}=\frac{28}{55}$ and $\frac{c}{d}=\frac{7}{1}$.