Boys to girls

The ratio of boys to girls at a party is 3:5. If six more boys arrived and four girls left the party, the ratio of boys to girls would be 5:6. How many people were at the party initially?

Correct answer:

t =  64

Step-by-step explanation:

$$\begin{gathered} t=a+b \\ a=3/5\cdot b \\ (a+6)=5/6\cdot (b-4) \\ a+b-t=0 \\ 5a-3b=0 \\ 6a-5b=-56 \\ Pivot:Row1\leftrightarrow Row3 \\ 6a-5b=-56 \\ 5a-3b=0 \\ a+b-t=0 \\ Row2-\frac{5}{6}\cdot Row1\to Row2 \\ 6a-5b=-56 \\ 1.167b=46.667 \\ a+b-t=0 \\ Row3-\frac{1}{6}\cdot Row1\to Row3 \\ 6a-5b=-56 \\ 1.167b=46.667 \\ 1.833b-t=9.333 \\ Pivot:Row2\leftrightarrow Row3 \\ 6a-5b=-56 \\ 1.833b-t=9.333 \\ 1.167b=46.667 \\ Row3-\frac{1.16666667}{1.83333333}\cdot Row2\to Row3 \\ 6a-5b=-56 \\ 1.833b-t=9.333 \\ 0.636t=40.727 \\ t=\frac{40.72727273}{0.63636364}=64 \\ b=\frac{9.33333333+t}{1.83333333}=\frac{9.33333333+64}{1.83333333}=40 \\ a=\frac{-56+5b}{6}=\frac{-56+5\cdot 40}{6}=24 \\ a=24 \\ b=40 \\ t=64 \end{gathered}$$

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