If a: b= 3 : 5 and b: c = 6 : 2 , find ` a : b : c `

Options

A. ` 4 : 9: 45`
B. ` 9 : 15: 5`
C. ` 16 : 36 : 20 `
D. ` 4 : 36 : 20`

Answer

To solve the problem of finding the ratio \( a : b : c \) given that \( a : b = 3 : 5 \) and \( b : c = 6 : 2 \), we can follow these steps:

### Step 1: Write the ratios in fraction form
We start by writing the ratios in fraction form:
- From \( a : b = 3 : 5 \), we can express this as:
\[ \frac{a}{b} = \frac{3}{5} \]
- From \( b : c = 6 : 2 \), we can express this as:
\[ \frac{b}{c} = \frac{6}{2} = 3 \]

### Step 2: Express \( b \) in terms of \( c \)
From the second ratio \( \frac{b}{c} = 3 \), we can express \( b \) in terms of \( c \):
\[ b = 3c \]

### Step 3: Find a common value for \( b \)
We need to find a common value for \( b \) from both ratios. The first ratio gives us \( b = \frac{5}{3}a \) and the second gives us \( b = 3c \). We can set these equal to each other:
\[ \frac{5}{3}a = 3c \]

### Step 4: Solve for \( a \) in terms of \( c \)
To eliminate \( b \), we can express \( a \) in terms of \( c \):
\[ a = \frac{3 \cdot 3c}{5} = \frac{9c}{5} \]

### Step 5: Write \( a \), \( b \), and \( c \) in terms of \( c \)
Now we can express all variables in terms of \( c \):
- From \( b = 3c \)
- From \( a = \frac{9c}{5} \)

### Step 6: Form the ratio \( a : b : c \)
Now we can write the ratio \( a : b : c \):
\[ a : b : c = \frac{9c}{5} : 3c : c \]

### Step 7: Eliminate \( c \) and simplify the ratio
To simplify, we can multiply through by 5 to eliminate the fraction:
\[ 9 : 15 : 5 \]

### Step 8: Simplify the ratio further
Now we can simplify the ratio \( 9 : 15 : 5 \) by dividing each term by their greatest common divisor, which is 1:
\[ 9 : 15 : 5 \]

Thus, the final ratio \( a : b : c \) is:
\[ \boxed{9 : 15 : 5} \]