The points J, K, L and M all lie on the same line segment, in that order, such that the ratio of J, K, colon, K, L, colon, L, M*JK*:*KL*:*LM* is equal to 3, colon, 2, colon, 1, .3:2:1. If J, M, equals, 30, comma*JM*=30, find L, M, .*LM*.

We are given that points \( J \), \( K \), \( L \), and \( M \) lie on the same line segment in that order. The ratio of the distances \( JK : KL : LM \) is given as \( 3 : 2 : 1 \), and the total distance \( JM = 30 \).

To solve for \( LM \), we can use the fact that the sum of the ratios \( JK + KL + LM = JM \).

### Step 1: Assign variables based on the ratio
Let the distance between each pair of points be expressed in terms of a common variable \( x \). Using the ratio \( 3 : 2 : 1 \), we can write:
\[
JK = 3x, \quad KL = 2x, \quad LM = x
\]

### Step 2: Write the equation for the total distance
The total distance \( JM \) is the sum of these segments:
\[
JM = JK + KL + LM = 3x + 2x + x = 6x
\]
We are told that \( JM = 30 \). So, we set up the equation:
\[
6x = 30
\]

### Step 3: Solve for \( x \)
Solving for \( x \):
\[
x = \frac{30}{6} = 5
\]

### Step 4: Find the value of \( LM \)
Since \( LM = x \), we have:
\[
LM = 5
\]

Thus, the distance \( LM \) is \( \boxed{5} \).

Would you like further details or have any questions?

Here are five related questions to deepen understanding:
1. If \( JK \) is given instead of the ratio, how would you solve for \( LM \)?
2. What happens if the total length \( JM \) changes? How would \( LM \) change?
3. Can you explain how the total ratio \( 3:2:1 \) relates to the individual segment lengths?
4. How would the solution differ if the points were in a different ratio, like \( 4:3:2 \)?
5. Can you find the distances \( JK \) and \( KL \) in this specific problem?

**Tip:** When solving ratio problems, introducing a variable for the shared factor (like \( x \)) simplifies setting up equations for the total.

The points J, K, L and M all lie on the same line segment, in that order, such that the ratio of JK:KL:LM is 3:2:1. If JM = 30, find LM.

Learn how to solve a line segment ratio problem where points J, K, L, and M are on the same segment with JK:KL:LM = 3:2:1, and JM = 30. This problem involves algebra and ratio concepts, suitable for Grades 6-8 students.

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