Question 1·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

Take the 2024 total as $100$ because the problem gives only percentages. Marketing's $35\%$ share is $35$, leaving $100-35=65$ for manufacturing. These are the starting amounts for the two changes.

A multiplier turns an old amount into a new one. Marketing is $20\%$ greater than its 2024 allocation, so it keeps the original $100\%$ and adds $20\%$. Its 2025 amount is $35(1.20)$. Using $0.20$ alone would count only the increase.

A base is the starting amount a percent acts on. Manufacturing's base is its own starting amount, $65$, not the entire $100$ budget. A $p\%$ decrease leaves $1-\frac{p}{100}$ of that $65$, so its 2025 amount is $65\left(1-\frac{p}{100}\right)$.

The total budget is $2\%$ greater than the starting $100$, so the 2025 total is $102$. Both allocations together make up that total. Add their 2025 amounts and set the sum equal to $102$:

Type $35(1.20)+65\left(1-\frac{x_1}{100}\right)\sim102$ in Desmos. Here $x_1$ stands for $p$; the $\sim$ tells Desmos to find a value that makes the sides equal rather than create a slider. Under PARAMETERS, it shows $x_1\approx7.692307692$. Type $x_1$ on the next line and use its fraction button to read $\frac{100}{13}$. Manufacturing's allocation decreased by $\frac{100}{13}\%$. Grid in 100/13.

Instead of finding both new allocations, compare changes on a starting budget of $100$. Marketing starts at $35$, so its $20\%$ increase adds $35(0.20)=7$.

The whole $100$ budget grows by $2\%$, a gain of $2$. Marketing adds $7$, so manufacturing must lose the difference to leave a net gain of $2$: $7-2=5$.

Manufacturing started at $65$, so divide its loss by its own starting amount and multiply by $100$ to get the percent decrease. Type $\frac{5}{65}\cdot100$ in Desmos. It shows $7.692307\ldots$; its fraction button gives $\frac{100}{13}$. So manufacturing decreased by $\frac{100}{13}\%$. Grid in 100/13.