If a : b=2 : 3 and b: c =4: 5,find `a^2 : b^2 : bc`

Options

A. `4: 9 : 5`
B. `16 : 36 : 45`
C. `16 : 36:20`
D. `4:36:40`

Answer

To solve the problem, we need to find the ratio \( a^2 : b^2 : bc \) given the ratios \( a : b = 2 : 3 \) and \( b : c = 4 : 5 \).

### Step-by-Step Solution:

1. **Express Ratios in Terms of a Common Variable:**
- From the ratio \( a : b = 2 : 3 \), we can express \( a \) and \( b \) in terms of a variable \( k \):
\[ a = 2k \quad \text{and} \quad b = 3k \]

2. **Express c in Terms of b:**
- From the ratio \( b : c = 4 : 5 \), we can express \( b \) and \( c \) in terms of another variable \( m \):
\[ b = 4m \quad \text{and} \quad c = 5m \]

3. **Equate the Values of b:**
- We have two expressions for \( b \):
\[ 3k = 4m \]
- From this equation, we can express \( k \) in terms of \( m \):
\[ k = \frac{4m}{3} \]

4. **Substitute k back to find a and b:**
- Substitute \( k \) into the expressions for \( a \) and \( b \):
\[ a = 2k = 2 \left(\frac{4m}{3}\right) = \frac{8m}{3} \]
\[ b = 3k = 3 \left(\frac{4m}{3}\right) = 4m \]
- We already have \( c = 5m \).

5. **Calculate \( a^2, b^2, \) and \( bc \):**
- Now we can calculate \( a^2, b^2, \) and \( bc \):
\[ a^2 = \left(\frac{8m}{3}\right)^2 = \frac{64m^2}{9} \]
\[ b^2 = (4m)^2 = 16m^2 \]
\[ bc = (4m)(5m) = 20m^2 \]

6. **Express the Ratios:**
- Now we need to express \( a^2 : b^2 : bc \):
\[ a^2 : b^2 : bc = \frac{64m^2}{9} : 16m^2 : 20m^2 \]
- To eliminate \( m^2 \), we can divide each term by \( m^2 \):
\[ = \frac{64}{9} : 16 : 20 \]

7. **Find a Common Denominator:**
- To express this in a simpler ratio, we can convert each term to have a common denominator. The least common multiple of \( 9, 1, \) and \( 1 \) is \( 9 \):
\[ = \frac{64}{9} : \frac{144}{9} : \frac{180}{9} \]
- This simplifies to:
\[ = 64 : 144 : 180 \]

8. **Simplify the Ratio:**
- Now, we can simplify this ratio by dividing each term by the greatest common divisor (GCD). The GCD of \( 64, 144, \) and \( 180 \) is \( 4 \):
\[ = \frac{64 \div 4}{4} : \frac{144 \div 4}{4} : \frac{180 \div 4}{4} = 16 : 36 : 45 \]

### Final Answer:
Thus, the final ratio is:
\[ a^2 : b^2 : bc = 16 : 36 : 45 \]