Question 6·Hard·Percentages

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At the beginning of a year, a community center's membership consisted only of student members and adult members. During the year, the number of student members increased by $50\%$ , and the number of adult members decreased by $10\%$ . The total membership increased by $8\%$ .

Which choice gives the percentage of the membership that consisted of student members at the end of the year?

In a two-group percent-change problem, the two starting shares add to the original total. Multiply each share by its own change factor, then add the results to match the given overall change. Desmos can solve for the unknown starting share. The key trap is finding that share and stopping: a question about the end of the year uses the ending membership as its whole.

Hints

Hint 1
A share is a group's part of a total. Call the students' starting share $s$ . Since students and adults make up everyone, what must the adults' starting share be?
Hint 2
An increase of $50\%$ leaves the students with $150\%$ of their starting number. A decrease of $10\%$ leaves the adults with $90\%$ of theirs. Multiply each starting share by its own factor.
Hint 3
The ending total is $108\%$ of the starting total. After you find the students' new amount, which total should go beneath it to find their share at the end?

Step-by-step

Track both groups from start to finish
Step 1
Write each group's ending amount
Count the starting membership as $1$ , or $100\%$ , and let $s$ be the students' starting share. Adults start with $1-s$ because these are the only two groups. The student increase gives a factor of $1.50$ ; the adult decrease leaves a factor of $0.90$ . So, measured against the starting total:
$\text{students at end}=1.50s,\qquad \text{adults at end}=0.90(1-s).$
Step 2
Match the sum to the ending total
An $8\%$ increase makes the whole membership $1.08$ times its starting size. The two ending amounts must add to that whole, so:
$1.50s+0.90(1-s)=1.08.$
Step 3
Find the starting student share
Type $1.50x_1+0.90(1-x_1)\sim1.08$ in Desmos, using $x_1$ for $s$ . The $\sim$ tells Desmos to find the value that makes both sides equal. Under PARAMETERS, it shows $x_1=0.30$ , so students were $30\%$ of the membership at the start.
Step 4
Carry the students through their increase
Add $1.50x_1$ in Desmos. It prints $0.45$ : the students at year's end amount to $45\%$ of the starting total. That's not yet their percentage of the membership at year's end.
Step 5
Divide by the new total
For an ending share, use the ending total as the whole. Add $100\frac{1.50x_1}{1.08}$ in Desmos. It prints $41.666\ldots$ ; the fraction button shows $\frac{125}{3}$ . Convert that to a mixed-number percent: $\frac{125}{3}\%=41\frac{2}{3}\%$ . So $41\frac{2}{3}\%$ of the members at the end of the year were students. Choice C.

Question 6·Hard·Percentages

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