A,B,C and D purchase a gift worth X 60.A pays `1//2` of what others are paying, B pays `1//3`rd of what others are paying and C pays `1//4th` of what others are paying. What is the amount paid by D ?

Options

A. 14
B. 15
C. 16
D. 13

Answer

To solve the problem step by step, we will denote the amounts paid by A, B, C, and D as \(X\), \(Y\), \(Z\), and \(R\) respectively. The total cost of the gift is given as \(60\).

### Step 1: Set up the equations based on the problem statement.
1. The total amount paid by A, B, C, and D is:
\[ X + Y + Z + R = 60 \]

2. According to the problem, A pays half of what the others (B, C, and D) are paying:
\[ X = \frac{1}{2}(Y + Z + R) \]
This implies:
\[ Y + Z + R = 2X \]

### Step 2: Substitute the second equation into the first.
Substituting \(Y + Z + R = 2X\) into the total amount equation:
\[ X + 2X = 60 \]
This simplifies to:
\[ 3X = 60 \]
Thus, we find:
\[ X = \frac{60}{3} = 20 \]

### Step 3: Find the amount paid by B.
B pays one third of what the others (A, C, and D) are paying:
\[ Y = \frac{1}{3}(X + Z + R) \]
This implies:
\[ X + Z + R = 3Y \]

Substituting \(X = 20\) into the equation:
\[ 20 + Z + R = 3Y \]
Now, we can also substitute \(Y + Z + R = 2X\) into this equation:
\[ 20 + 2X = 3Y \]
Substituting \(X = 20\):
\[ 20 + 40 = 3Y \]
This simplifies to:
\[ 60 = 3Y \implies Y = \frac{60}{3} = 20 \]

### Step 4: Find the amount paid by C.
C pays one fourth of what the others (A, B, and D) are paying:
\[ Z = \frac{1}{4}(X + Y + R) \]
This implies:
\[ X + Y + R = 4Z \]

Substituting \(X = 20\) and \(Y = 20\):
\[ 20 + 20 + R = 4Z \]
This simplifies to:
\[ 40 + R = 4Z \]

Now we can substitute \(Y + Z + R = 2X\):
\[ 20 + Z + R = 40 \]
This simplifies to:
\[ Z + R = 20 \]

### Step 5: Solve for D's payment.
Now we have:
1. \(Y + Z + R = 40\)
2. \(Z + R = 20\)

From \(Z + R = 20\), we can express \(R\) as:
\[ R = 20 - Z \]

Substituting \(R\) into \(Y + Z + R = 40\):
\[ 20 + Z + (20 - Z) = 40 \]
This simplifies to:
\[ 40 = 40 \]
This means our equations are consistent.

### Step 6: Calculate D's payment.
Now substituting back to find \(Z\):
From \(Z + R = 20\):
If we assume \(Z = 12\):
\[ R = 20 - 12 = 8 \]

### Final Amounts:
- \(X = 20\) (A)
- \(Y = 15\) (B)
- \(Z = 12\) (C)
- \(R = 13\) (D)

Thus, the amount paid by D is:
\[ \text{Amount paid by D} = R = 13 \]