Question
If a rubber ball consistently bounces back $$\frac{{2}}{{3}}$$ of the height from which it is dropped, what fraction of its original height will the ball bounce after being dropped and bounced four times without being stopped?
Answer: Option A
Each time the ball is dropped and it bounces back, it reaches $$\frac{{2}}{{3}}$$ of the height it was dropped from.
After the first bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height from which it was dropped - let us call it the original height.
After the second bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height it would have reached after the first bounce.
So, at the end of the second bounce, the ball would have reached $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ of the original height = $$\frac{{4}}{{9}}$$ th of the original height.
After the third bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height it would have reached after the second bounce.
So, at the end of the third bounce, the ball would have reached $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ = $$\frac{{8}}{{27}}$$ th of the original height.
After the fourth and last bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height it would have reached after the third bounce.
So, at the end of the last bounce, the ball would have reached $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ of the original height = $$\frac{{16}}{{81}}$$ of the original height.
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Each time the ball is dropped and it bounces back, it reaches $$\frac{{2}}{{3}}$$ of the height it was dropped from.
After the first bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height from which it was dropped - let us call it the original height.
After the second bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height it would have reached after the first bounce.
So, at the end of the second bounce, the ball would have reached $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ of the original height = $$\frac{{4}}{{9}}$$ th of the original height.
After the third bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height it would have reached after the second bounce.
So, at the end of the third bounce, the ball would have reached $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ = $$\frac{{8}}{{27}}$$ th of the original height.
After the fourth and last bounce, the ball will reach $$\frac{{2}}{{3}}$$ of the height it would have reached after the third bounce.
So, at the end of the last bounce, the ball would have reached $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ × $$\frac{{2}}{{3}}$$ of the original height = $$\frac{{16}}{{81}}$$ of the original height.
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