If ` a : b = 2 : 3 and b : c = 4 : 5 " find " a^(2) : b^(2) : bc`

Options

A. ` 4 : 9 : 45`
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 1: Express \( a \), \( b \), and \( c \) in terms of a common variable.

From the ratio \( a : b = 2 : 3 \), we can express \( a \) and \( b \) as:
- Let \( a = 2k \)
- Let \( b = 3k \)

For the ratio \( b : c = 4 : 5 \), we can express \( b \) and \( c \) as:
- Let \( b = 4m \)
- Let \( c = 5m \)

### Step 2: Equate the two expressions for \( b \).

Since both expressions represent \( b \), we have:
\[ 3k = 4m \]

### Step 3: Solve for \( k \) in terms of \( m \).

From \( 3k = 4m \), we can express \( k \) as:
\[ k = \frac{4m}{3} \]

### Step 4: Substitute \( k \) back into the expressions for \( a \) and \( b \).

Now substituting \( k \) back into the expressions for \( a \) and \( b \):
- \( a = 2k = 2 \left(\frac{4m}{3}\right) = \frac{8m}{3} \)
- \( b = 3k = 3 \left(\frac{4m}{3}\right) = 4m \)

### Step 5: Find \( c \) in terms of \( m \).

From the previous step, we already have:
- \( c = 5m \)

### Step 6: Calculate \( a^2 \), \( b^2 \), and \( bc \).

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

### Step 7: Write the ratios \( a^2 : b^2 : bc \).

Now we need to express the ratios:
\[ a^2 : b^2 : bc = \frac{64m^2}{9} : 16m^2 : 20m^2 \]

### Step 8: Eliminate \( m^2 \) from the ratios.

To simplify, we can divide each term by \( m^2 \):
\[ a^2 : b^2 : bc = \frac{64}{9} : 16 : 20 \]

### Step 9: Find a common denominator to express the ratios.

To express these ratios in whole numbers, we can multiply each term by 9 (the common denominator):
- \( \frac{64}{9} \times 9 = 64 \)
- \( 16 \times 9 = 144 \)
- \( 20 \times 9 = 180 \)

Thus, the ratio becomes:
\[ 64 : 144 : 180 \]

### Step 10: Simplify the ratio.

Now, we can simplify the ratio by dividing each term by the greatest common divisor (GCD). The GCD of 64, 144, and 180 is 4:
- \( 64 \div 4 = 16 \)
- \( 144 \div 4 = 36 \)
- \( 180 \div 4 = 45 \)

So, the final ratio is:
\[ 16 : 36 : 45 \]

### Final Answer:
The answer is \( 16 : 36 : 45 \).