If `A : B = 1 : 2, B : C = 3 : 4, C : D = 6 : 9` and `D : E = 12 : 16` then `A: B : C : D : E` is equal to

Options

A. `1:3:6:12:16`
B. `2:4:6:9:16`
C. `3:4:8:12:16`
D. `3:6:8:12:16`

Answer

To solve the problem of finding the ratio \( A : B : C : D : E \) given the individual ratios, we can follow these steps:

### Step 1: Write down the given ratios
We have the following ratios:
1. \( A : B = 1 : 2 \)
2. \( B : C = 3 : 4 \)
3. \( C : D = 6 : 9 \)
4. \( D : E = 12 : 16 \)

### Step 2: Express each variable in terms of a common variable
Let's express each variable in terms of \( A \):
- From \( A : B = 1 : 2 \), we can say \( B = 2A \).
- From \( B : C = 3 : 4 \), substituting \( B \) gives \( 2A : C = 3 : 4 \). Therefore, \( C = \frac{4}{3} \times 2A = \frac{8}{3}A \).
- From \( C : D = 6 : 9 \), substituting \( C \) gives \( \frac{8}{3}A : D = 6 : 9 \). Therefore, \( D = \frac{9}{6} \times \frac{8}{3}A = \frac{12}{3}A = 4A \).
- From \( D : E = 12 : 16 \), substituting \( D \) gives \( 4A : E = 12 : 16 \). Therefore, \( E = \frac{16}{12} \times 4A = \frac{4}{3} \times 4A = \frac{16}{3}A \).

### Step 3: Write down all variables in terms of \( A \)
Now we have:
- \( A = A \)
- \( B = 2A \)
- \( C = \frac{8}{3}A \)
- \( D = 4A \)
- \( E = \frac{16}{3}A \)

### Step 4: Find a common denominator to combine the ratios
To combine these ratios, we can find a common denominator. The denominators we have are \( 1, 2, \frac{8}{3}, 4, \frac{16}{3} \). The least common multiple of the denominators \( 1, 2, 3, 4 \) is \( 12 \).

### Step 5: Convert each ratio to have the same denominator
Now we convert each variable:
- \( A = 12 \)
- \( B = 2A = 2 \times 12 = 24 \)
- \( C = \frac{8}{3}A = \frac{8}{3} \times 12 = 32 \)
- \( D = 4A = 4 \times 12 = 48 \)
- \( E = \frac{16}{3}A = \frac{16}{3} \times 12 = 64 \)

### Step 6: Write the final ratio
Thus, the final ratio \( A : B : C : D : E \) is:
\[ A : B : C : D : E = 12 : 24 : 32 : 48 : 64 \]

### Step 7: Simplify the ratio
To simplify this ratio, we can divide each term by 4:
\[ A : B : C : D : E = 3 : 6 : 8 : 12 : 16 \]

### Final Answer:
Thus, the ratio \( A : B : C : D : E \) is equal to \( 3 : 6 : 8 : 12 : 16 \).

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