`a:b::2:3,b :c::4:5,` find `a:b:c.`

Options

A. `8:12:15`
B. `4:3:5`
C. `4:6:9`
D. `2:4:7`

Answer

To find the ratio \( a:b:c \) given the ratios \( a:b::2:3 \) and \( b:c::4:5 \), we can follow these steps:

### Step 1: Write down the given ratios
We have:
- \( a:b = 2:3 \)
- \( b:c = 4:5 \)

### Step 2: Express the ratios in terms of a common variable
From the first ratio \( a:b = 2:3 \), we can express \( a \) and \( b \) as:
- \( a = 2k \)
- \( b = 3k \)

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

### Step 3: Equate the two expressions for \( b \)
Since both expressions represent \( b \), we can set them equal to each other:
\[ 3k = 4m \]

### Step 4: Solve for one variable in terms of the other
From \( 3k = 4m \), we can express \( k \) in terms of \( m \):
\[ k = \frac{4m}{3} \]

### Step 5: Substitute back to find \( a \) and \( c \)
Now, substitute \( k \) back into the expressions for \( a \) and \( c \):
- For \( a \):
\[ a = 2k = 2 \left(\frac{4m}{3}\right) = \frac{8m}{3} \]

- For \( c \):
\[ c = 5m \]

### Step 6: Write the ratios \( a:b:c \)
Now we have:
- \( a = \frac{8m}{3} \)
- \( b = 4m \)
- \( c = 5m \)

To express \( a:b:c \) in a simpler form, we can eliminate \( m \) by multiplying through by 3 (the denominator of \( a \)):
- \( a = 8m \)
- \( b = 12m \)
- \( c = 15m \)

Thus, the ratio \( a:b:c \) is:
\[ a:b:c = 8:12:15 \]

### Final Answer
The final answer for the ratio \( a:b:c \) is:
\[ 8:12:15 \]

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