# If the ratio of boys to girls is 3:2 and there are 25 students in a class, how do you make equal ratios to show how many students in the class are boys and how many are girls?

Feb 9, 2017

The explanation is much longer than doing the mathematics.

The ratio of 15 boys to 10 girls is equivalent to the ratio of 3:2

#### Explanation:

Consider the starting point:
We have 3 boys and 2 girls. This gives a total count of 5

So we need to see how many lots of 5 will fit into 25.

5 lots of 5 gives 25

So we have 5 lots of the ratio 3:2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Method 1}}$

5 lots of 3:2 $\to 5 \times \left(3 : 2\right) = \left(5 \times 3\right) : \left(5 \times 2\right) = 15 : 10$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Method 2}}$

Write the ratio in fractional form

$\textcolor{m a \ge n t a}{\text{We can do this as long as we do not view and treat it like a fraction}}$

It does not matter in this case which we put on the top. I chose:

$\left(\text{boys")/("girls}\right) \to \frac{3}{2}$

Multiply a value (or system) by 1 and you do not change the value. However, 1 comes in many forms.

$\textcolor{g r e e n}{\frac{3}{2} \textcolor{red}{\times 1} \text{ "->" } \frac{3}{2} \textcolor{red}{\times \frac{5}{5}}}$

" "=color(green)((3color(red)(xx5))/(2color(red)(xx5))

" "=15/10 =("boys")/("girls")

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The ratio of 15 boys to 10 girls is equivalent to the ratio of 3:2

$15 : 10 \equiv \left(15 \div 5\right) : \left(10 \div 5\right) = 3 : 2$

where $\equiv$ means equivalent to