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A graduating class has 236 students. Among them, 142 enrolled in algebra and 121 enrolled in chemistry. What is the greatest possible number of students who could have taken both algebra and chemistry?
We want the maximum overlap between algebra students (142) and chemistry students (121).
The overlap is maximized when every chemistry student also took algebra (since 121 < 142, this is possible).
Maximum overlap $= \min(142, 121) = \mathbf{121}$.
Check: if 121 students took both, then students taking at least one subject $= 142 + 121 - 121 = 142 \leq 236$. ✓
So the greatest possible number is $121$.
Two capacitors $C_1$ and $C_2$ are charged to 120 V and 200 V respectively. When connected together, the potential on each becomes zero. Which relation holds?
For the potential to become zero after connection, the total charge must be zero (charges cancel). Taking signs into account:
$Q_1 - Q_2 = 0 \Rightarrow C_1 \times 120 = C_2 \times 200$
$\frac{C_1}{C_2} = \frac{200}{120} = \frac{5}{3}$
$\Rightarrow 3C_1 = 5C_2$
The capacitors must have been charged with opposite polarities so that when connected, charges cancel. Conservation of charge gives $120C_1 = 200C_2$, yielding $3C_1 = 5C_2$.
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