What we see: a Two-Part Analysis question with two limits met at the same time, weight and space. We need the one pair of crate counts that uses up both limits exactly.
Step 1. Turn each limit into an equation.
What we see: 25 kg per Type L crate and 15 kg per Type S crate, 600 kg in all; 0.4 cubic meter per Type L and 0.3 per Type S, 10.8 in all.
What it means: with \(L\) Type L crates and \(S\) Type S crates, \(25L + 15S = 600\) and \(0.4L + 0.3S = 10.8\).
What we do and why: two unknowns and two equations pin down one pair. Either equation alone allows several pairs, and the answer rows are built to catch that.
Step 2. Make the numbers easy.
What we see: every number in the weight equation divides by 5, and the space equation has decimals.
What it means: weight becomes \(5L + 3S = 120\). Space times 10 becomes \(4L + 3S = 108\).
What we do and why: both equations now have \(3S\), so subtracting one from the other removes \(S\).
Step 3. Subtract to find L, then S.
What we see: \((5L + 3S) - (4L + 3S) = 120 - 108\).
What it means: \(L = 12\). Put it back in: \(4(12) + 3S = 108\), so \(3S = 60\) and \(S = 20\).
What we do and why: both numbers are in the table, so we check them before selecting.
Step 4. Check against both limits.
What we see: weight \(25 \times 12 + 15 \times 20 = 300 + 300 = 600\) kg. Space \(0.4 \times 12 + 0.3 \times 20 = 4.8 + 6 = 10.8\) cubic meters.
What it means: both limits are reached exactly, as the question says.
The trap: using one limit only. Weight alone also allows 18 Type L with 10 Type S (\(450 + 150 = 600\) kg), but those take only 10.2 cubic meters. Space alone also allows 18 Type L with 12 Type S (\(7.2 + 3.6 = 10.8\)), but those weigh 630 kg. Another trap is swapping the columns: 20 Type L with 12 Type S weighs 680 kg.
Faster way: scale the two equations so one letter has the same number in front of it (here \(3S\)), then subtract. One line of arithmetic gives \(L\).
Takeaway: When a question says two limits are both met exactly, write one equation per limit and solve them together. A pair that fits only one limit is a wrong answer placed there on purpose.
Answer: Type L 12, Type S 20
