Question

A man distributed 43 chocolates to his children. How many of his children are more than five years old?

Statements:
I. A child older than five years gets 5 chocolates. 
II. A child 5 years or younger in age gets 6 chocolates. 

• A. The question can be answered with the help of statement I alone.
• B. The question can be answered with the help of statement II alone.
• C. Both statements I and II are needed to answer the question.
• D. The question cannot be answered even with the help of both the statements.

Hint:

In DS problems with distributions, define variables for each group and use the total. Modular checks often confirm uniqueness of integer solutions.

Answer 1

The Correct Option is C

Step 1: Set variables and equation (using both statements). 
Let $x =$ number of children older than $5$ years, $y =$ number of children $\le 5$ years. 
Using I and II with total $43$:   $5x + 6y = 43$. 

Step 2: Show each statement alone is insufficient. 
I alone: only tells older children get $5$ each—no info for younger $\Rightarrow$ cannot form a total. 
II alone: only tells younger get $6$ each—no info for older $\Rightarrow$ cannot form a total. 
Thus neither alone suffices. 

Step 3: Solve with both. 
$5x + 6y = 43 \Rightarrow 6y \equiv 43 \pmod{5} \Rightarrow y \equiv 3 \pmod{5}$. 
Try $y=3 \Rightarrow 5x = 43 - 18 = 25 \Rightarrow x=5$. 
Next $y=8$ makes $5x=43-48$ impossible. 
So the unique solution is $(x,y)=(5,3)$. \[ \boxed{\text{Number older than five} = x = 5} \]

Answer 2

Instead of solving the combined equation through modular arithmetic, the same conclusion follows from a direct search over the possible number of older children.

Why neither statement alone is enough. Statement I alone only tells us that each older child gets 5 chocolates; without knowing anything about the younger children's share, the total of 43 could be split in infinitely many ways. Statement II alone only tells us about the younger children's rate of 6 each, with the same problem for older children. Neither pins down a number by itself.

Combining both and searching. With \(x\)=older children and \(y\)=younger children, \(5x+6y=43\). Trying \(x=0,1,2,\dots,8\) (since \(5x\le43\)) and checking whether \(43-5x\) is divisible by 6: \[ x=0:43,\ x=1:38,\ x=2:33,\ x=3:28,\ x=4:23,\ x=5:18\ (\div6=3),\ x=6:13,\ x=7:8,\ x=8:3. \] Only \(x=5\) gives a value divisible by 6, yielding \(y=3\).

1. Option "Statement I alone": Insufficient — gives no way to isolate the number of older children without knowing the younger children's contribution.
2. Option "Statement II alone": Insufficient — likewise gives no way to isolate the older-children count alone.
3. Option "Both statements needed": Correct — only combining both rates and searching over integer solutions pins down \(x=5\) uniquely.
4. Option "Cannot be answered": Incorrect — the search above does find a unique, valid solution \(x=5,\ y=3\).

So the correct answer is both statements I and II are needed to answer the question.