The ratio of salary of `A:B=1:2,B:C=3:4, C:D=5:6` and `D:E=7:8` What is the ratio of salary of A and E?

Answer

To find the ratio of salaries of A and E given the ratios of salaries of A:B, B:C, C:D, and D:E, we can follow these steps:

### Step 1: Express B in terms of A
Given the ratio A:B = 1:2, we can express B as:
\[ B = 2A \]

### Step 2: Express C in terms of B
From the ratio B:C = 3:4, we can express C in terms of B:
\[ \frac{B}{C} = \frac{3}{4} \]
Cross-multiplying gives:
\[ 4B = 3C \]
Thus, we can express C as:
\[ C = \frac{4}{3}B \]
Substituting the value of B from Step 1:
\[ C = \frac{4}{3}(2A) = \frac{8}{3}A \]

### Step 3: Express D in terms of C
From the ratio C:D = 5:6, we can express D in terms of C:
\[ \frac{C}{D} = \frac{5}{6} \]
Cross-multiplying gives:
\[ 6C = 5D \]
Thus, we can express D as:
\[ D = \frac{6}{5}C \]
Substituting the value of C from Step 2:
\[ D = \frac{6}{5}\left(\frac{8}{3}A\right) = \frac{48}{15}A = \frac{16}{5}A \]

### Step 4: Express E in terms of D
From the ratio D:E = 7:8, we can express E in terms of D:
\[ \frac{D}{E} = \frac{7}{8} \]
Cross-multiplying gives:
\[ 8D = 7E \]
Thus, we can express E as:
\[ E = \frac{8}{7}D \]
Substituting the value of D from Step 3:
\[ E = \frac{8}{7}\left(\frac{16}{5}A\right) = \frac{128}{35}A \]

### Step 5: Find the ratio of A and E
Now we have both A and E in terms of A:
\[ A : E = A : \frac{128}{35}A \]
This simplifies to:
\[ A : E = 1 : \frac{128}{35} \]
To express this as a simple ratio, we can multiply both sides by 35:
\[ A : E = 35 : 128 \]

Thus, the ratio of the salaries of A and E is:
\[ \boxed{35:128} \]