Question

Four children A, B, C & D are having some chocolates each.
A gives B as many as he already has, he gives C twice of what C already has and he gives D thrice of what D already has.
Now, D gives (1/8)th of his own chocolates to B.
Then A gives 10% chocolates he now owns to C and 20% to B.
Finally, all of them have 35 chocolates each.
What is the original number of chocolates each had in the beginning?

• A. A - 110, B - 10, C - 10, D - 10
• B. A - 90, B - 20, C - 20, D - 10
• C. A - 70, B - 25, C - 25, D - 20
• D. A - 125, B - 5, C - 5, D - 5

Hint:

Work backwards from the final equal amount of 35, undoing each transaction one at a time, starting with A's last transfer of 10% and 20%.

Answer 1

The Correct Option is A

Step 1: Set up the initial amounts.
Let the original number of chocolates with A, B, C and D be A, B, C and D respectively.
Step 2: Apply the first round of giving, where A doubles B, triples C and quadruples D.
A gives B an amount equal to what B already has, so B becomes 2B and A loses B chocolates.
A gives C twice of what C already has, so C becomes 3C and A loses 2C chocolates.
A gives D thrice of what D already has, so D becomes 4D and A loses 3D chocolates.
After this round, A has (A - B - 2C - 3D), B has 2B, C has 3C, D has 4D.
Step 3: Apply D's transfer of 1/8th of his chocolates to B.
D now has 4D chocolates and gives away (1/8) of 4D, which is D/2, to B.
D is left with 4D - D/2 = 7D/2, and B now has 2B + D/2.
Step 4: Apply A's transfer of 10% to C and 20% to B.
Let A1 = A - B - 2C - 3D, the amount A holds before this step.
A gives 10% of A1 to C and 20% of A1 to B, keeping 70% of A1 for himself, so A now has 0.7 x A1.
C now has 3C + 0.1 x A1, and B now has 2B + D/2 + 0.2 x A1.
Step 5: Use the condition that everyone finally has 35 chocolates.
From A: 0.7 x A1 = 35, so A1 = 50, meaning A - B - 2C - 3D = 50.
From D: 7D/2 = 35, so D = 10.
From C: 3C + 0.1 x 50 = 35, so 3C + 5 = 35, giving C = 10.
From B: 2B + D/2 + 0.2 x 50 = 35, so 2B + 5 + 10 = 35, giving B = 10.
Step 6: Solve for A.
Substitute B = 10, C = 10, D = 10 into A - B - 2C - 3D = 50: A - 10 - 20 - 30 = 50, so A - 60 = 50, giving A = 110.
Step 7: State the final answer.
The original amounts are A = 110, B = 10, C = 10, D = 10, which matches option (1). Checking any of the other options against the same sequence of transactions does not end with everyone holding 35 chocolates, so option (1) is correct.