Question

A, B, C and D purchase a gift worth ₹60. A pays $\tfrac{1}{2}$ of what others are paying, B pays $\tfrac{1}{3}$ of what others are paying, and C pays $\tfrac{1}{4}$ of what others are paying. What is the amount paid by D? 

• A. ₹14
• B. ₹15
• C. ₹16
• D. ₹13

Hint:

Phrases like "pays $\frac{1}{k}$ of what others are paying" translate to $x=\frac{1}{k}(S-x)$ where $S$ is the total. Solve each to get $x$ directly.

Answer 1

The Correct Option is D

Let payments be $a,b,c,d$ with total $S=60$. A's condition: $a=\tfrac{1}{2}(S-a)$ $\Rightarrow$ $2a=S-a$ $\Rightarrow$ $3a=S$ $\Rightarrow$ $a=20$.
B's condition: $b=\tfrac{1}{3}(S-b)$ $\Rightarrow$ $3b=S-b$ $\Rightarrow$ $4b=S$ $\Rightarrow$ $b=15$.
C's condition: $c=\tfrac{1}{4}(S-c)$ $\Rightarrow$ $4c=S-c$ $\Rightarrow$ $5c=S$ $\Rightarrow$ $c=12$.
Then $d=S-(a+b+c)=60-(20+15+12)=\boxed{13}$. 

Answer 2

Instead of finding each of \(A\), \(B\), \(C\)'s payments as absolute amounts and subtracting them from the total, we can express each person's payment as a fraction of the total directly, then check each option for \(D\)'s payment.

1. Option A (₹14): Since \(A\) pays \(\tfrac12\) of what the others (\(B+C+D\)) pay, \(A=\tfrac12(S-A)\Rightarrow 3A=S\Rightarrow A=\tfrac{S}{3}\). Similarly \(B=\tfrac{S}{4}\) and \(C=\tfrac{S}{5}\). So \(D\)'s share of the total is \[ \frac{D}{S}=1-\left(\frac13+\frac14+\frac15\right)=1-\frac{20+15+12}{60}=1-\frac{47}{60}=\frac{13}{60}. \] With \(S=60\), \(D=60\times\tfrac{13}{60}=13\), not \(14\), so this option is incorrect.
2. Option B (₹15): This is actually the value of \(B\)'s payment (\(S/4=15\)), not \(D\)'s, so it is incorrect.
3. Option C (₹16): This does not match the computed fraction \(\tfrac{13}{60}\) of \(60\), so it is incorrect.
4. Option D (₹13): As computed, \(D=60\times\dfrac{13}{60}=13\), matching this option exactly.

Expressing each person's share as a fraction of the total directly confirms \(D\) pays ₹13.

Hence, the correct answer is option D: ₹13.