If a:b = 5:6 and b:c = 3:4, then find a:b:c.

Answer

To solve the problem where \( a:b = 5:6 \) and \( b:c = 3:4 \), we need to find the combined ratio \( a:b:c \). Here's a step-by-step solution:

### Step 1: Write the given ratios
We start with the ratios provided:
- \( a:b = 5:6 \)
- \( b:c = 3:4 \)

### Step 2: Express the ratios in terms of a common variable
From the first ratio \( a:b = 5:6 \), we can express \( a \) and \( b \) as:
- \( a = 5k \)
- \( b = 6k \)
(where \( k \) is a common multiplier).

From the second ratio \( b:c = 3:4 \), we can express \( b \) and \( c \) as:
- \( b = 3m \)
- \( c = 4m \)
(where \( m \) is another common multiplier).

### Step 3: Equate the values of \( b \)
Since both expressions represent \( b \), we can set them equal to each other:
\[ 6k = 3m \]

### Step 4: Solve for one variable in terms of the other
To find a relationship between \( k \) and \( m \), we can rearrange the equation:
\[ m = \frac{6k}{3} = 2k \]

### Step 5: Substitute back to find \( a \) and \( c \)
Now that we have \( m \) in terms of \( k \), we can find \( c \):
\[ c = 4m = 4(2k) = 8k \]

### Step 6: Write the combined ratio
Now we have all the variables in terms of \( k \):
- \( a = 5k \)
- \( b = 6k \)
- \( c = 8k \)

Thus, the combined ratio \( a:b:c \) can be written as:
\[ a:b:c = 5k:6k:8k \]

### Step 7: Simplify the ratio
Since \( k \) is a common factor, we can simplify the ratio:
\[ a:b:c = 5:6:8 \]

### Final Answer
Therefore, the final answer is:
\[ a:b:c = 5:6:8 \]

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