If `A : B = 2 : 5, B : C = 4 : 3` and `C:D = 2:1`, what is the value of `A : C : D`?

Options

A. `6 : 5 : 2 `
B. `7 : 20 : 10`
C. `8 : 30 : 15`
D. `16 : 30 : 15 `

Answer

To find the value of \( A : C : D \) given the ratios \( A : B = 2 : 5 \), \( B : C = 4 : 3 \), and \( C : D = 2 : 1 \), we can follow these steps:

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

### Step 2: Express each variable in terms of a common variable
From \( A : B = 2 : 5 \), we can express:
- \( A = 2x \)
- \( B = 5x \)

From \( B : C = 4 : 3 \), we can express:
- \( B = 4y \)
- \( C = 3y \)

From \( C : D = 2 : 1 \), we can express:
- \( C = 2z \)
- \( D = z \)

### Step 3: Equate the values of B and C from the different ratios
Since \( B \) is expressed in two different ways, we set them equal:
\[ 5x = 4y \]
From this, we can express \( y \) in terms of \( x \):
\[ y = \frac{5x}{4} \]

Now, we can express \( C \) in terms of \( x \):
\[ C = 3y = 3 \left(\frac{5x}{4}\right) = \frac{15x}{4} \]

### Step 4: Equate the values of C from the different ratios
Now we set the two expressions for \( C \) equal:
\[ \frac{15x}{4} = 2z \]
From this, we can express \( z \) in terms of \( x \):
\[ z = \frac{15x}{8} \]

### Step 5: Substitute back to find A, C, and D
Now we can find \( A \), \( C \), and \( D \) in terms of \( x \):
- \( A = 2x \)
- \( C = \frac{15x}{4} \)
- \( D = z = \frac{15x}{8} \)

### Step 6: Find a common denominator to express A, C, and D
To express \( A : C : D \) in a simple ratio, we can express all terms with a common denominator. The common denominator for \( 4 \) and \( 8 \) is \( 8 \):
- \( A = 2x = \frac{16x}{8} \)
- \( C = \frac{15x}{4} = \frac{30x}{8} \)
- \( D = \frac{15x}{8} \)

### Step 7: Write the final ratio
Now we can write the ratio:
\[ A : C : D = \frac{16x}{8} : \frac{30x}{8} : \frac{15x}{8} \]
This simplifies to:
\[ A : C : D = 16 : 30 : 15 \]

### Step 8: Simplify the ratio
To simplify \( 16 : 30 : 15 \), we can divide each term by their greatest common divisor, which is \( 1 \) in this case:
\[ A : C : D = 16 : 30 : 15 \]

Thus, the final answer is:
\[ A : C : D = 16 : 30 : 15 \]

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