The average of the test scores of a group of x students is 76 and the average of the test scores of a group of y students is 90. When the scores of the two groups of students are combined, the average test score is 85. What is the value of of `(x)/(y)`?

Options

A. `(4)/(7)`
B. `(5)/(9)`
C. `(2)/(3)`
D. `(7)/(4)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Define the sums of scores
Let the sum of the test scores of the group of \( x \) students be \( A \) and the sum of the test scores of the group of \( y \) students be \( B \).

### Step 2: Set up equations for the averages
From the information given:
- The average score of \( x \) students is 76, so we can write:
\[ A = 76x \quad \text{(Equation 1)} \]
- The average score of \( y \) students is 90, so we can write:
\[ B = 90y \quad \text{(Equation 2)} \]

### Step 3: Set up the equation for the combined average
When the scores of both groups are combined, the average score is 85. Therefore, we can express this as:
\[ \frac{A + B}{x + y} = 85 \]

### Step 4: Substitute \( A \) and \( B \) into the combined average equation
Substituting the values of \( A \) and \( B \) from Equations 1 and 2 into the combined average equation gives:
\[ \frac{76x + 90y}{x + y} = 85 \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying results in:
\[ 76x + 90y = 85(x + y) \]

### Step 6: Expand the right-hand side
Expanding the right-hand side gives:
\[ 76x + 90y = 85x + 85y \]

### Step 7: Rearrange the equation
Rearranging the equation to isolate terms involving \( x \) and \( y \):
\[ 76x + 90y - 85x - 85y = 0 \]
This simplifies to:
\[ -9x + 5y = 0 \]
or
\[ 9x = 5y \]

### Step 8: Solve for the ratio \( \frac{x}{y} \)
To find the ratio \( \frac{x}{y} \), we can rearrange the equation:
\[ \frac{x}{y} = \frac{5}{9} \]

### Final Answer
Thus, the value of \( \frac{x}{y} \) is:
\[ \frac{x}{y} = \frac{5}{9} \]

---