The ratio of A and B is 2: 3 and the ratio of B and C is 4:5. Then find `A^(2):B^(2): BC.`

Options

A. `15:20:25`
B. `8:24:32`
C. `16:36:45`
D. `9:18:27`

Answer

To solve the problem where the ratio of A and B is 2:3 and the ratio of B and C is 4:5, we need to find the ratio of \( A^2 : B^2 : BC \).

### Step-by-Step Solution:

1. **Establish Ratios**:
- We know that \( A : B = 2 : 3 \).
- We also know that \( B : C = 4 : 5 \).

2. **Express A, B, and C in terms of a common variable**:
- Let’s denote \( B \) as a common variable. To do this, we need to express both ratios in terms of \( B \).
- From \( A : B = 2 : 3 \), we can express \( A \) as:
\[ A = \frac{2}{3}B \]
- From \( B : C = 4 : 5 \), we can express \( C \) as:
\[ C = \frac{5}{4}B \]

3. **Find a common value for B**:
- To make calculations easier, we can set \( B = 12k \) (since 12 is a common multiple of 3 and 4).
- Now substituting \( B = 12k \):
\[ A = \frac{2}{3} \times 12k = 8k \]
\[ C = \frac{5}{4} \times 12k = 15k \]

4. **Calculate \( A^2, B^2, \) and \( BC \)**:
- Now we can calculate \( A^2, B^2, \) and \( BC \):
\[ A^2 = (8k)^2 = 64k^2 \]
\[ B^2 = (12k)^2 = 144k^2 \]
\[ BC = (12k)(15k) = 180k^2 \]

5. **Form the ratio \( A^2 : B^2 : BC \)**:
- Now we can write the ratio:
\[ A^2 : B^2 : BC = 64k^2 : 144k^2 : 180k^2 \]
- We can simplify this by dividing each term by \( k^2 \):
\[ 64 : 144 : 180 \]

6. **Simplify the ratio**:
- To simplify \( 64 : 144 : 180 \), we can find the greatest common divisor (GCD) of these numbers.
- The GCD of 64, 144, and 180 is 4.
- Dividing each term by 4:
\[ \frac{64}{4} : \frac{144}{4} : \frac{180}{4} = 16 : 36 : 45 \]

### Final Answer:
The ratio \( A^2 : B^2 : BC \) is \( 16 : 36 : 45 \).

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