Question 47·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

Let $x$ be the number of members at the beginning. The two categories make up the whole group, so the share who aren't guides is $100\%-40\%=60\%$. Write their starting counts:

Losing $25\%$ leaves 75% of the guides. That percentage applies to the guides, not to everyone. Multiply their starting count:

Losing $10\%$ leaves 90% of the nonguides. Multiply their starting count:

None of the $32$ newcomers is a guide, so they add to the whole group, not to the guide count. Add both surviving categories and the newcomers:

The final $30\%$ is taken of the group after arrivals. The surviving guides number $0.30x$; the $0.30$ on the right represents their share of a different total. Take the final percentage of the final total. Set the guide counts equal:

Type $0.30x_{1}\sim0.30(0.30x_{1}+0.54x_{1}+32)$. Replace $x$ with $x_1$ and $=$ with $\sim$ so Desmos finds the value that makes the two counts match. Under PARAMETERS, it shows $x_1=200$. Since $x_1$ counts beginning-of-year members, the group started with $200$ members. Grid in 200.