CSAT Solved Papers/ 2021/Q16

2021 CSAT — Q16

Quant Arithmetic & numeracy 2.5 marks Medium

A boy plays with a ball and he drops it from a height of $1\cdot 5$ m. Every time the ball hits the ground, it bounces back to attain a height $\tfrac{4}{5}$ of the previous height. The ball does not bounce further if the previous height is less than $50$ cm. What is the number of times the ball hits the ground before the ball stops bouncing?

A 4
B 5
C 6 Answer
D 7

Worked rationale

Work in centimetres; the drop height is $150$ cm and each fall-height is $\tfrac45$ of the one before. List the height the ball falls from at each hit, and stop bouncing once a fall-height drops below $50$ cm.

Hit	falls from (cm)	$\ge 50$ ?	bounces to (cm)
1	$150$	yes	$120$
2	$120$	yes	$96$
3	$96$	yes	$76.8$
4	$76.8$	yes	$61.44$
5	$61.44$	yes	$49.152$
6	$49.152$	no	— (stops)

At hit $6$ the ball falls from $49.152$ cm $< 50$ cm, so it strikes the ground but does not bounce again. So the ball hits the ground $6$ times before it stops bouncing.

Answer: (c) 6.

Why the other options miss

A

stops the chain too early: cuts off when the bounce height first nears $50$ rather than tracking each fall.
B

one short: counts only the hits that are followed by a bounce, dropping the final no-rebound strike from below $50$ cm.
D

one too many: continues one fall too far, counting a hit from a height already below the $50$ cm cut-off as if it still rebounded.

Specialist insight

The whole item turns on what the threshold tests — the previous (fall) height, not the bounce-to height — and on whether the final sub- $50$ strike is counted. The clean discipline is one column (“falls from”) and a yes/no rebound test: $150,120,96,76.8,61.44$ all rebound; $49.152$ does not, but the ball still hits once more. Count strikes, not rebounds: $6$ . Multiplying $\times\tfrac45$ five times takes $61.44\to 49.152$ , which is the pivot value to nail.

The trap, in one line

Five rebounds $(150\to49.152)$ then one final no-rebound strike $=6$ hits $\Rightarrow$ (c).

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