If l : m = `2 (1)/(2) : 1(2)/(3)` and m : n = `1 (1)/(4) : 3 (1)/(2)` , find l : m : n

Answer

To solve the problem of finding the ratio \( l : m : n \) given \( l : m = 2 \frac{1}{2} : 1 \frac{2}{3} \) and \( m : n = 1 \frac{1}{4} : 3 \frac{1}{2} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers in the ratios to improper fractions.

- For \( l : m = 2 \frac{1}{2} : 1 \frac{2}{3} \):
- \( 2 \frac{1}{2} = \frac{5}{2} \)
- \( 1 \frac{2}{3} = \frac{5}{3} \)

So, \( l : m = \frac{5}{2} : \frac{5}{3} \).

- For \( m : n = 1 \frac{1}{4} : 3 \frac{1}{2} \):
- \( 1 \frac{1}{4} = \frac{5}{4} \)
- \( 3 \frac{1}{2} = \frac{7}{2} \)

So, \( m : n = \frac{5}{4} : \frac{7}{2} \).

### Step 2: Make the Denominators Equal
Next, we need to make the denominators of the ratios equal so we can combine them.

- For \( l : m = \frac{5}{2} : \frac{5}{3} \), we can multiply both parts by 3 (the denominator of the second term):
- \( l = \frac{5}{2} \times 3 = \frac{15}{2} \)
- \( m = \frac{5}{3} \times 3 = 5 \)

Thus, \( l : m = \frac{15}{2} : 5 \).

- For \( m : n = \frac{5}{4} : \frac{7}{2} \), we can multiply both parts by 4 (the denominator of the first term):
- \( m = \frac{5}{4} \times 4 = 5 \)
- \( n = \frac{7}{2} \times 4 = 14 \)

Thus, \( m : n = 5 : 14 \).

### Step 3: Combine the Ratios
Now that we have \( l : m = \frac{15}{2} : 5 \) and \( m : n = 5 : 14 \), we can express \( l : m : n \).

Since \( m \) is common in both ratios, we can write:
- \( l : m : n = \frac{15}{2} : 5 : 14 \).

### Step 4: Express All Parts with a Common Denominator
To express all parts with a common denominator, we can multiply the entire ratio by 2 to eliminate the fraction:
- \( l = \frac{15}{2} \times 2 = 15 \)
- \( m = 5 \times 2 = 10 \)
- \( n = 14 \times 2 = 28 \)

Thus, the final ratio is:
\[ l : m : n = 15 : 10 : 28 \]

### Final Answer
The ratio \( l : m : n \) is \( 15 : 10 : 28 \).

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