If A, B, C and D are four numbers such that A : B = 2 : 3, B : C = 4 : 5, C : D = 5 : 8, then A : D is equal to

Options

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

Answer

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

### Step 1: Write down the given ratios
We have:
- A : B = 2 : 3
- B : C = 4 : 5
- C : D = 5 : 8

### Step 2: Express each ratio in fraction form
From the ratios, we can express them as:
- A/B = 2/3
- B/C = 4/5
- C/D = 5/8

### Step 3: Find the value of A in terms of C
To find A in terms of C, we can rearrange the first ratio:
\[ A = \frac{2}{3} B \]

### Step 4: Find the value of B in terms of C
From the second ratio, we can express B in terms of C:
\[ B = \frac{4}{5} C \]

### Step 5: Substitute B in the equation for A
Now, substitute the value of B into the equation for A:
\[ A = \frac{2}{3} \left(\frac{4}{5} C\right) \]
\[ A = \frac{2 \times 4}{3 \times 5} C \]
\[ A = \frac{8}{15} C \]

### Step 6: Find the value of C in terms of D
From the third ratio, we can express C in terms of D:
\[ C = \frac{5}{8} D \]

### Step 7: Substitute C in the equation for A
Now, substitute the value of C into the equation for A:
\[ A = \frac{8}{15} \left(\frac{5}{8} D\right) \]
\[ A = \frac{8 \times 5}{15 \times 8} D \]
\[ A = \frac{5}{15} D \]
\[ A = \frac{1}{3} D \]

### Step 8: Write the final ratio A : D
Thus, we can express the ratio A : D as:
\[ A : D = 1 : 3 \]

### Final Answer
A : D = 1 : 3

---