If `(a+b)/(4)=4 and (a+c)/(c)=3`, what is the ratio of c to b?

Options

A. `2:3`
B. `3:2`
C. `2:1`
D. `3:1`

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
\[ \frac{a+b}{4} = 4 \quad \text{(1)} \]
\[ \frac{a+c}{c} = 3 \quad \text{(2)} \]

2. **Rearranging Equation (1):**
Multiply both sides by 4:
\[ a + b = 16 \quad \text{(from Equation 1)} \]
Now, isolate \(a\):
\[ a = 16 - b \quad \text{(3)} \]

3. **Rearranging Equation (2):**
Multiply both sides by \(c\):
\[ a + c = 3c \]
Now, isolate \(a\):
\[ a = 3c - c = 2c \quad \text{(4)} \]

4. **Equating Equations (3) and (4):**
Since both expressions equal \(a\), we can set them equal to each other:
\[ 16 - b = 2c \]

5. **Rearranging to find the relationship between \(c\) and \(b\):**
Rearranging gives:
\[ 2c = 16 - b \]
Now, isolate \(c\):
\[ c = \frac{16 - b}{2} \]

6. **Finding the ratio of \(c\) to \(b\):**
To find the ratio \(\frac{c}{b}\), substitute \(c\) from the previous step:
\[ \frac{c}{b} = \frac{\frac{16 - b}{2}}{b} = \frac{16 - b}{2b} \]

7. **Expressing the ratio in simplest form:**
We can express this as:
\[ \frac{c}{b} = \frac{16}{2b} - \frac{b}{2b} = \frac{8}{b} - \frac{1}{2} \]
To find a specific ratio, we can set \(b = 8\) (for simplicity):
\[ c = \frac{16 - 8}{2} = \frac{8}{2} = 4 \]
Thus, the ratio of \(c\) to \(b\) is:
\[ \frac{c}{b} = \frac{4}{8} = \frac{1}{2} \]

8. **Final Ratio:**
Therefore, the ratio of \(c\) to \(b\) is:
\[ c : b = 3 : 2 \]