If l:m=1 1/2 :  1 3/4 and m:n = 2 1/3 :   4 1/5, find l:n and l:m:n

We are given the ratios \( l:m = 1\frac{1}{2} : 1\frac{3}{4} \) and \( m:n = 2\frac{1}{3} : 4\frac{1}{5} \).

### Step 1: Express the ratios as improper fractions.
Convert the mixed fractions to improper fractions:

- \( l:m = 1\frac{1}{2} : 1\frac{3}{4} = \frac{3}{2} : \frac{7}{4} \)
- \( m:n = 2\frac{1}{3} : 4\frac{1}{5} = \frac{7}{3} : \frac{21}{5} \)

### Step 2: Simplify each ratio.

#### For \( l:m = \frac{3}{2} : \frac{7}{4} \):
To simplify this, multiply both sides by 4 to eliminate the denominators:

\[
l:m = \frac{3}{2} \cdot 4 : \frac{7}{4} \cdot 4 = 6 : 7
\]

Thus, \( l:m = 6:7 \).

#### For \( m:n = \frac{7}{3} : \frac{21}{5} \):
To simplify this, multiply both sides by 15 (the LCM of 3 and 5) to eliminate the denominators:

\[
m:n = \frac{7}{3} \cdot 15 : \frac{21}{5} \cdot 15 = 35 : 63
\]

Simplify \( 35 : 63 \) by dividing both sides by 7:

\[
m:n = 5 : 9
\]

### Step 3: Find \( l:n \).
Now that we have \( l:m = 6:7 \) and \( m:n = 5:9 \), we need to find the combined ratio \( l:m:n \).

1. Multiply the first ratio \( l:m = 6:7 \) by 5 (the value of \( m \) from the second ratio):
   \[
   l:m = 6 \times 5 : 7 \times 5 = 30 : 35
   \]

2. Multiply the second ratio \( m:n = 5:9 \) by 7 (the value of \( m \) from the first ratio):
   \[
   m:n = 5 \times 7 : 9 \times 7 = 35 : 63
   \]

Thus, the combined ratio \( l:m:n \) is:

\[
l:m:n = 30:35:63
\]

### Final Answer:
- \( l:n = 30:63 \), which simplifies to \( l:n = 10:21 \).
- \( l:m:n = 30:35:63 \).

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What are the steps to convert a mixed fraction to an improper fraction?
2. How can you find the least common multiple (LCM) of two numbers?
3. Can the ratios \( l:n \) and \( m:n \) be represented on a number line?
4. How can the concept of proportions be applied to real-life problems?
5. What are some common mistakes when simplifying ratios involving fractions?

**Tip:** Always simplify ratios by dividing through the greatest common divisor (GCD) to make them as simple as possible!

If l:m = 1 1/2 : 1 3/4 and m:n = 2 1/3 : 4 1/5, find l:n and l:m:n

Learn how to find the combined ratio l:m:n from the given ratios l:m = 1 1/2 : 1 3/4 and m:n = 2 1/3 : 4 1/5. This solution involves converting mixed fractions to improper fractions and simplifying the ratios step-by-step. Suitable for students in Grades 8-10.

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