If average of `23, 15, 19, x, 12, 7` is `16` while average of `37, 28, 53, y, 42` is `41` then find `(x+y)/(y-x)` = ?

Options

A. `2.4`
B. `2.6`
C. `2.9`
D. `2.5`

Answer

To solve the problem step-by-step, we need to find the values of \(x\) and \(y\) based on the given averages and then calculate \(\frac{x+y}{y-x}\).

### Step 1: Find the value of \(x\)

We know that the average of the numbers \(23, 15, 19, x, 12, 7\) is \(16\). The formula for average is given by:

\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

In this case, the sum of observations is:

\[ 23 + 15 + 19 + x + 12 + 7 = 76 + x \]

The number of observations is \(6\). Thus, we can set up the equation:

\[ 16 = \frac{76 + x}{6} \]

Now, we can cross-multiply to solve for \(x\):

\[ 16 \times 6 = 76 + x \]

\[ 96 = 76 + x \]

Subtract \(76\) from both sides:

\[ x = 96 - 76 = 20 \]

### Step 2: Find the value of \(y\)

Next, we find the average of the numbers \(37, 28, 53, y, 42\), which is \(41\). Again, using the average formula, we have:

\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

The sum of observations is:

\[ 37 + 28 + 53 + y + 42 = 160 + y \]

The number of observations is \(5\). Thus, we can set up the equation:

\[ 41 = \frac{160 + y}{5} \]

Cross-multiplying gives us:

\[ 41 \times 5 = 160 + y \]

\[ 205 = 160 + y \]

Subtract \(160\) from both sides:

\[ y = 205 - 160 = 45 \]

### Step 3: Calculate \(\frac{x+y}{y-x}\)

Now that we have \(x = 20\) and \(y = 45\), we can substitute these values into the expression \(\frac{x+y}{y-x}\):

\[ \frac{x+y}{y-x} = \frac{20 + 45}{45 - 20} \]

Calculating the numerator and denominator:

\[ = \frac{65}{25} \]

Now simplify:

\[ = \frac{65 \div 5}{25 \div 5} = \frac{13}{5} \]

### Final Answer

Thus, the final answer is:

\[ \frac{x+y}{y-x} = \frac{13}{5} = 2.6 \]