At a collage basketball game, the ratio of the number of freshmen who attended to the number of juniors who attended is 3:4. The ratio of the number of juniors who attended to the number of seniors who attended is 7:6. What is the ratio of the number of freshmen to the number of seniors who attended the basketball game?

Options

A. `7:8`
B. `3:4`
C. `2:3`
D. `1:2`

Answer

To solve the problem step by step, we will denote the number of freshmen as \( F \), the number of juniors as \( J \), and the number of seniors as \( S \).

### Step 1: Establish the first ratio
According to the problem, the ratio of the number of freshmen to the number of juniors is given as \( 3:4 \). This can be expressed mathematically as:
\[ \frac{F}{J} = \frac{3}{4} \]
From this, we can express the number of freshmen in terms of juniors:
\[ F = \frac{3}{4} J \quad \text{(Equation 1)} \]

### Step 2: Establish the second ratio
The problem also states that the ratio of the number of juniors to the number of seniors is \( 7:6 \). This can be expressed as:
\[ \frac{J}{S} = \frac{7}{6} \]
From this, we can express the number of juniors in terms of seniors:
\[ J = \frac{7}{6} S \quad \text{(Equation 2)} \]

### Step 3: Substitute the expression for juniors into the expression for freshmen
Now we have expressions for both freshmen and juniors. We can substitute Equation 2 into Equation 1 to find the number of freshmen in terms of seniors:
\[ F = \frac{3}{4} J = \frac{3}{4} \left(\frac{7}{6} S\right) \]

### Step 4: Simplify the expression
Now we simplify the expression for \( F \):
\[ F = \frac{3 \cdot 7}{4 \cdot 6} S = \frac{21}{24} S \]
This can be further simplified to:
\[ F = \frac{7}{8} S \quad \text{(Equation 3)} \]

### Step 5: Find the ratio of freshmen to seniors
Now we can find the ratio of the number of freshmen to the number of seniors:
\[ \frac{F}{S} = \frac{7}{8} \]

### Conclusion
Thus, the ratio of the number of freshmen to the number of seniors who attended the basketball game is:
\[ \boxed{\frac{7}{8}} \]