If `a/b=b/c=c/d=d/e=e/f=1/3`, find the value of `((a+b+c+d+e)/(b+c+d+e+f)):`

Options

A. `1/81`
B. `1/27`
C. `1/3`
D. 1

Answer

To solve the problem, we need to find the value of \(\frac{a+b+c+d+e}{b+c+d+e+f}\) given that \(\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \frac{d}{e} = \frac{e}{f} = \frac{1}{3}\).

### Step-by-Step Solution:

1. **Express each variable in terms of \(a\)**:
- From \(\frac{a}{b} = \frac{1}{3}\), we can write:
\[ b = 3a \]
- From \(\frac{b}{c} = \frac{1}{3}\):
\[ c = 3b = 3 \times 3a = 9a \]
- From \(\frac{c}{d} = \frac{1}{3}\):
\[ d = 3c = 3 \times 9a = 27a \]
- From \(\frac{d}{e} = \frac{1}{3}\):
\[ e = 3d = 3 \times 27a = 81a \]
- From \(\frac{e}{f} = \frac{1}{3}\):
\[ f = 3e = 3 \times 81a = 243a \]

2. **Substitute the values into the expression**:
- Now we substitute \(b\), \(c\), \(d\), \(e\), and \(f\) in terms of \(a\):
\[ a + b + c + d + e = a + 3a + 9a + 27a + 81a = 121a \]
\[ b + c + d + e + f = 3a + 9a + 27a + 81a + 243a = 363a \]

3. **Form the ratio**:
- Now we can form the ratio:
\[ \frac{a+b+c+d+e}{b+c+d+e+f} = \frac{121a}{363a} \]

4. **Simplify the ratio**:
- The \(a\) in the numerator and denominator cancels out:
\[ = \frac{121}{363} \]
- Now simplify \(\frac{121}{363}\):
\[ = \frac{121 \div 121}{363 \div 121} = \frac{1}{3} \]

### Final Answer:
\[ \frac{a+b+c+d+e}{b+c+d+e+f} = \frac{1}{3} \]