Question 40·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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A nonprofit distributes food boxes from Site A and Site B. In 2023, the two sites distributed a combined total number of boxes. In 2024, Site A distributed 18% more boxes than it had in 2023, and Site B distributed 7% fewer boxes than it had in 2023. The combined number of boxes distributed increased by 3% from 2023 to 2024.

In 2025, Site A distributed 15% fewer boxes than it had in 2024, and Site B distributed 8% more boxes than it had in 2024. Relative to the combined number of boxes distributed in 2023, what was the percentage increase in the combined number of boxes distributed in 2025?

Your answer

An unknown starting split and different changes for two sites call for percent-change multipliers. Treat the first year's total as $1$ , then use the known combined change to find each site's starting share with a Desmos regression. Keep the sites separate through both years: applying either site's second-year rate to the combined total gives the wrong result.

Hints

Hint 1
A starting share is one site's portion of the 2023 total. Call Site A's share $s$ , so Site B's is $1-s$ . What multiplier takes each site from 2023 to 2024?
Hint 2
The combined $3\%$ increase means the 2024 total is $1.03$ times the 2023 total. Add the sites' 2024 amounts, each measured against the 2023 total, and set their sum equal to $1.03$ .
Hint 3
A successive percent change acts on the amount from the year before. Site A keeps $85\%$ of its 2024 boxes, while Site B has $108\%$ of its 2024 boxes. Multiply each site by both yearly factors before adding.

Step-by-step

Track each site's share
Step 1
Write the 2024 total in terms of the 2023 shares
Treat the 2023 combined total as $1$ . Let $s$ be Site A's share, or portion, of that total; Site B's share is $1-s$ . An $18\%$ increase multiplies A by $1.18$ , and a $7\%$ decrease multiplies B by $0.93$ . The combined $3\%$ increase makes the 2024 total $1.03$ , so add the two site amounts:
$1.18s+0.93(1-s)=1.03.$
Step 2
Find the starting split
Type $1.18x_1+0.93(1-x_1)\sim1.03$ in Desmos, using $x_1$ for $s$ . The $\sim$ tells Desmos to fit the unknown value; under PARAMETERS it shows $x_1=0.4$ . So Site A had $0.4$ of the 2023 total, and Site B had the remaining $0.6$ .
Step 3
Carry each site through 2025
A factor is the number that multiplies an amount after a percent change. A's $15\%$ decrease gives $0.85$ of its 2024 amount; B's $8\%$ increase gives $1.08$ of its 2024 amount. Type the two lines below, naming their 2025 amounts $a$ and $b$ . Desmos shows $a=0.4012$ and $b=0.60264$ , each measured against the 2023 total.
Step 4
Add the 2025 site amounts
Type $a+b$ . Desmos prints $1.00384$ , the overall factor comparing the 2025 combined total with the 2023 combined total. Don't apply one site's 2025 factor to the 2024 combined total: the sites changed at different rates.
Step 5
Convert the factor to a percent increase
The original total is $1$ , so subtract $1$ to get only the increase. Then multiply by $100$ to express it as a percent. Type $(a+b-1)100$ ; Desmos prints $0.384$ . The combined number of boxes increased by $0.384\%$ from 2023 to 2025. Grid in 0.384.

Question 40·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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