If `A : B = 5 : 6` and `B : C = 8 : 9`, find `A : B : C`

Options

A. `20 : 24 : 27`
B. `30 : 48 : 54`
C. `20 : 48 : 54`
D. `40 : 24 : 27`

Answer

To solve the problem of finding the ratio \( A : B : C \) given the ratios \( A : B = 5 : 6 \) and \( B : C = 8 : 9 \), we can follow these steps:

### Step 1: Write the given ratios
We have:
- \( A : B = 5 : 6 \)
- \( B : C = 8 : 9 \)

### Step 2: Express the ratios in terms of a common variable
Let's express \( A \), \( B \), and \( C \) in terms of a common variable.

From \( A : B = 5 : 6 \), we can write:
- \( A = 5x \)
- \( B = 6x \)

From \( B : C = 8 : 9 \), we can write:
- \( B = 8y \)
- \( C = 9y \)

### Step 3: Set the values of \( B \) equal to each other
Since both expressions represent \( B \), we can set them equal:
\[ 6x = 8y \]

### Step 4: Solve for one variable in terms of the other
To find a relationship between \( x \) and \( y \), we can rearrange the equation:
\[ \frac{x}{y} = \frac{8}{6} = \frac{4}{3} \]
This implies:
\[ x = \frac{4}{3}y \]

### Step 5: Substitute back to find \( A \), \( B \), and \( C \)
Now, we can substitute \( x \) back into the expressions for \( A \), \( B \), and \( C \):
- \( A = 5x = 5 \left(\frac{4}{3}y\right) = \frac{20}{3}y \)
- \( B = 6x = 6 \left(\frac{4}{3}y\right) = 8y \)
- \( C = 9y \)

### Step 6: Write the ratios in terms of \( y \)
Now we can express the ratios \( A : B : C \):
\[ A : B : C = \frac{20}{3}y : 8y : 9y \]

### Step 7: Eliminate \( y \) and find a common denominator
To eliminate \( y \), we can multiply each term by 3 to get rid of the fraction:
\[ A : B : C = 20 : 24 : 27 \]

### Final Answer
Thus, the ratio \( A : B : C = 20 : 24 : 27 \).

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