Solve Word Problems With Lcm Worksheets [PDF] (6.NS.B.4): 6th ... - Free Printable

Let's solve each of these word problems using the Least Common Multiple (LCM). The LCM is the smallest number that is a multiple of two or more numbers. It’s useful for finding when events that repeat at different intervals will coincide.

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Problem 1

Ramesh and Ciara love to jog on a track at the sports center. Ramesh can complete a lap in 4 minutes and Ciara can complete a lap in 3 minutes. If they both start their first lap at the same time, in how many minutes will they both be at the starting point together at the same time again?

#### Solution:
We need to find the LCM of 4 and 3.

- Multiples of 4: 4, 8, 12, 16, 20, ...
- Multiples of 3: 3, 6, 9, 12, 15, ...

The least common multiple is 12.

✔ Answer: 12 minutes

They will both be at the starting point together again after 12 minutes.

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Problem 2

Manny attends violin practice every 6 days and swimming practice every 2 days. If he does both violin and swimming practice today, in how many days will he next do both practices on the same day?

#### Solution:
Find the LCM of 6 and 2.

- Multiples of 6: 6, 12, 18, ...
- Multiples of 2: 2, 4, 6, 8, 10, 12, ...

The LCM is 6.

✔ Answer: 6 days

He will do both practices on the same day again in 6 days.

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Problem 3

In Kimberly's music class, they are learning about rhythm. You hear the sound of a shaker every 10 seconds and the beat of a drum every 12 seconds. If you hear both sounds at the same time right now, what is the shortest time in which you will hear both the sounds again?

#### Solution:
Find the LCM of 10 and 12.

First, factor the numbers:
- 10 = 2 × 5
- 12 = 2² × 3

Take the highest powers of all primes:
- 2² × 3 × 5 = 4 × 3 × 5 = 60

✔ Answer: 60 seconds

You will hear both sounds together again in 60 seconds.

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Problem 4

Teesta and Eric work at a widget factory. Their job is to inspect the widgets for defects. Teesta inspects every 15th widget and Eric inspects every 12th widget. What is the number of the first widget that they will both inspect?

#### Solution:
This is asking for the LCM of 15 and 12, because we want the first widget number divisible by both 15 and 12.

Factor:
- 15 = 3 × 5
- 12 = 2² × 3

Highest powers:
- 2² × 3 × 5 = 4 × 3 × 5 = 60

✔ Answer: 60

The first widget they both inspect is the 60th widget.

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✔ Final Answers:

1. 12 minutes
2. 6 days
3. 60 seconds
4. 60th widget

These problems all use LCM to find when repeated events align — whether it's people finishing laps, practicing, hearing sounds, or inspecting widgets.

Let me know if you'd like a visual explanation or step-by-step LCM calculation!