CAT 2002 | Question: 78

Question

A thief was stealing diamonds from a jewellery store. On his way out, he encountered three guards, each was given half of the existing diamonds and two, to cover it by the thief. In the end, he was left with one diamond. How many did the thief steal?

• A. $40$
• B. $36$
• C. $42$
• D. $38$

Answer 1

Answer: Option B

The thief stole 36 diamonds. Solution:-

Let’s assume that the thief stole $x$ diamonds from the store.

When he encountered the first guard, he gave him half of his diamonds and two more, so he was left with $(x/2) - 2$ diamonds.

When he encountered the second guard, he gave him half of his remaining diamonds and two more, so he was left with $[(x/2) - 2]/2 - 2 = (x-12)/4$ diamonds.

When he encountered the third guard, he gave him half of his remaining diamonds and two more, so he was left with $[(x-12)/2] - 2 = (x-28)/8$ diamonds.

Since he was left with one diamond in the end, we can set up an equation:

$(x-28)/8 = 1$

Solving for $x$ gives us:

$x = 36$

Therefore, the thief stole $36$ diamonds.

Another Way (Short Trick):-

To solve this problem, we can start with one diamond left in the end and add $2$ to get $3$. Double it to get $6$ and add $2$ to get $8$. Double it again to get $16$ and add $2$ to get $18$. Double it again to get $36$. Therefore, the thief stole $36$ diamonds.

Comments

Meaning of “and two cover it by the thief”?

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it means it gives half of the existing diamonds + 2 to the guard all the three times

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Error: The terms that show what he is left with should have minus sign(not plus).

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i have told +2 goes to the guard that meant same that -2 from thief

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You are right. I meant that there are typos in the answer.

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@okntk thanks for that clarification, I was losing my sanity over that xD