CSAT Solved Papers/ 2025/Q26

2025 CSAT — Q26

Quant Arithmetic & numeracy 2.5 marks Medium

If $4 \le x \le 8$ and $2 \le y \le 7$ , then what is the ratio of maximum value of $(x + y)$ to minimum value of $(x - y)$ ?

A 6
B $\dfrac{15}{2}$
C $-\dfrac{15}{2}$
D None of the above Answer

Worked rationale

Optimise each part at the interval endpoints:

$(x+y)$ is largest at the largest $x$ and largest $y$ : $8 + 7 = 15$ .
$(x-y)$ is smallest at the smallest $x$ and largest $y$ : $4 - 7 = -3$ .

The required ratio is

\frac{\max(x+y)}{\min(x-y)} = \frac{15}{-3} = -5.

Now check the options: $-5$ is not $6$ , not $\tfrac{15}{2}$ , not $-\tfrac{15}{2}$ .

Answer: (d) None of the above.

Why the other options miss

A

took the wrong endpoints: minimises $(x-y)$ as $|{\cdot}|$ or uses $\min(x-y) = 4 - 2 = 2$ (smallest $x$ , smallest $y$ ), giving $\tfrac{15}{2.5}$ -type slips toward $6$ .
B

dropped the sign: takes $\min(x-y)$ as the magnitude $x_{\min} - y_{\max}$ $= 2$ (e.g. $4-2$ ) and divides $15/2$ , losing the sign and mis-choosing $y$ .
C

used the wrong $y$ -endpoint: correctly signs the difference but uses $\min(x-y) = -2$ (taking $y_{\max}=6$ or $x=4,y=6$ ), missing that $y$ ranges up to $7$ .

Specialist insight

Two precision habits decide this item. First, minimise a difference by pushing the subtrahend up: $\min(x-y)$ wants $x$ as small as allowed and $y$ as large as allowed — $4 - 7 = -3$ , a negative number. Second, keep the sign — the ratio $15/(-3) = -5$ is negative, and the examiner has deliberately omitted $-5$ from (a)–(c) so that every student who mishandles the endpoints or the sign finds a “matching” option. The correct exam reflex on “ratio of extremes” items is to compute the honest value first and only then scan the options; finding it absent is itself the answer. The presence of a clean negative distractor $-\tfrac{15}{2}$ is the tell that sign-handling is the trap.

The trap, in one line

$\min(x-y) = 4 - 7 = -3$ (smallest $x$ , largest $y$ ), so the ratio is $15/(-3) = -5$ — which is *not* listed, making (d) correct.

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