# Math Help - ratio

1. ## ratio

The ratio of boys to girls is 6:5 in one first-period class and 3:5 in a second first period class. If the school does not permit class sizes to be less than 10 students, what is the minimum number of total students in the two first-period classes so the ratio for the combined classes is 1:1?

2. I stand corrected. Sorry for any confusion on my account.

3. Hello, sri340!

The ratio of boys to girls is 6:5 in Class #1, and 3:5 in Class #2.

If the school does not permit class sizes to be less than 10 students,
what is the minimum number of total students in the two classes
so the ratio of boys to girls for the combined class is 1:1 ?
Class #1 has: . $6a$ boys and $5a$ girls.
Class #2 has: . $3b$ boys and $5b$ girls.

We have this table: . $\begin{array}{|c||c|c|} \hline
& \text{Boys} & \text{Girls} \\ \hline \hline
\text{Class 1} & 6a & 5a \\ \hline
\text{Class 2} & 3b & 5b \\ \hline \hline
\text{Total} & 6a+3b & 5a+5b \\ \hline\end{array}$

We want: . $(6a+3b):(5a+5b) \:=\:1:1 \quad\Rightarrow\quad 6a+3b \:=\:5a+5b \quad\Rightarrow\quad a \:=\:2b$

The table becomes: . $\begin{array}{|c||c|c|} \hline
& \text{Boys} & \text{Girls} \\ \hline \hline \text{Class 1} & 12b & 10b \\ \hline \text{Class 2} & 3b & 5b \\ \hline \hline \text{Total} & 15b & 15b \\ \hline\end{array}$

. . . . . . . . . . . . . . . . . . . . . . $_{\uparrow\qquad \;\; \uparrow}$
. . . . . . . . . . . . . . . . . . . . . . . $^{\text{Check!}}$

If $b = 1$, Class #2 has only 8 students.

Therefore: . $b = 2\quad \begin{array}{|c||c|c|} \hline &\text{Boys} & \text{Girls} \\ \hline\hline \text{Class 1} & 24 & 20 \\ \hline \text{Class 2} & 6 & 10 \\ \hline \hline \text{Total} & 30 & 30 \\ \hline \end{array}$

The minimum number of students in the combined class is 60.