If `A : B = 3 : 5` and `B : C = 10 : 13` find `A: B : C`

Answer

To solve the problem of finding the ratio \( A : B : C \) given \( A : B = 3 : 5 \) and \( B : C = 10 : 13 \), we will follow these steps:

### Step 1: Write the given ratios in fractional form
We start with the ratios provided:
- \( A : B = 3 : 5 \) can be written as \( \frac{A}{B} = \frac{3}{5} \).
- \( B : C = 10 : 13 \) can be written as \( \frac{B}{C} = \frac{10}{13} \).

### Step 2: Express \( A \) and \( C \) in terms of \( B \)
From the first ratio \( \frac{A}{B} = \frac{3}{5} \), we can express \( A \) in terms of \( B \):
\[ A = \frac{3}{5}B \]

From the second ratio \( \frac{B}{C} = \frac{10}{13} \), we can express \( C \) in terms of \( B \):
\[ C = \frac{13}{10}B \]

### Step 3: Find a common value for \( B \)
To combine these ratios, we need to express them in a way that allows us to eliminate \( B \). The value of \( B \) in both ratios needs to be the same. The simplest way to do this is to find a common multiple of the denominators of \( B \) in both ratios.

The current values of \( B \) are \( 5 \) (from \( A : B \)) and \( 10 \) (from \( B : C \)). The least common multiple (LCM) of \( 5 \) and \( 10 \) is \( 10 \).

### Step 4: Scale the ratios
To make \( B = 10 \):
- For \( A : B = 3 : 5 \), we multiply both parts by \( 2 \) (since \( 5 \times 2 = 10 \)):
\[ A : B = 3 \times 2 : 5 \times 2 = 6 : 10 \]

- For \( B : C = 10 : 13 \), \( B \) is already \( 10 \):
\[ B : C = 10 : 13 \]

### Step 5: Combine the ratios
Now we can combine the ratios:
\[ A : B : C = 6 : 10 : 13 \]

### Final Answer
Thus, the ratio \( A : B : C \) is \( 6 : 10 : 13 \).

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