Least Common Multiple (LCM) Word Problem - Free Printable

Let's solve both LCM (Least Common Multiple) word problems step by step.

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Problem 1:

Find the least length of a rope which can be cut into whole number of pieces of lengths 45 cm, 75 cm and 81 cm.

We are to find the least length that is divisible by 45, 75, and 81 — meaning we need the LCM of these three numbers.

#### Step 1: Prime Factorization

- $ 45 = 3^2 \times 5 $
- $ 75 = 3 \times 5^2 $
- $ 81 = 3^4 $

#### Step 2: Take the highest powers of all primes

- $ 3^4 $ (from 81)
- $ 5^2 $ (from 75)

So,
$$
\text{LCM} = 3^4 \times 5^2 = 81 \times 25 = 2025
$$

✔ So, the least length of the rope is 2025 cm.

> Answer: d. 2025 cm

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Problem 2:

The local bus service has 2 lines of buses that start together at 8 a.m. Buses on line A leave every 15 minutes, while buses on line B leave every 20 minutes. In a day, how many times do buses on both line A and B leave together between 8 a.m. and 11 a.m.?

We need to find how many times both buses leave together between 8 a.m. and 11 a.m.

This happens when the time elapsed is a common multiple of 15 and 20 minutes.

So, find the LCM of 15 and 20.

#### Step 1: Prime Factorization

- $ 15 = 3 \times 5 $
- $ 20 = 2^2 \times 5 $

Take highest powers:
- $ 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 $

So, LCM = 60 minutes

That means buses leave together every 60 minutes.

They start together at 8:00 a.m.

Next together at:
- 9:00 a.m.
- 10:00 a.m.
- 11:00 a.m.

But the question says: between 8 a.m. and 11 a.m.

Now, check what "between" means.

Usually, in such problems:

- "Between 8 a.m. and 11 a.m." includes 8 a.m. but excludes 11 a.m. unless stated otherwise.

But let’s see the exact timing:

- 8:00 → included (start time)
- 9:00 → included
- 10:00 → included
- 11:00 → is it included?

Since the interval is between 8 a.m. and 11 a.m., and 11 a.m. is the end point, if it says "between", it usually does not include 11 a.m.

But sometimes, especially in real-life contexts, "between 8 a.m. and 11 a.m." may include both endpoints.

Let’s clarify:

- From 8:00 to 11:00 is 3 hours.
- They meet every 60 minutes → at:
- 8:00
- 9:00
- 10:00
- 11:00

But if 11:00 is excluded, then only three times: 8, 9, 10.

If 11:00 is included, then four times.

But option a is 3, b is 5, c is 6, d is 9 → so 3 is an option.

Let’s re-read: "between 8 a.m. and 11 a.m."

In mathematics, “between” often excludes endpoints, but since they start at 8 a.m., and the problem says "between 8 a.m. and 11 a.m.", it likely includes 8 a.m. but not 11 a.m.

So:
- 8:00 → yes
- 9:00 → yes
- 10:00 → yes
- 11:00 → no (since it's the end)

So 3 times.

But wait — is 11:00 considered within the interval?

Let’s suppose the buses leave at 11:00 — does that count?

The phrase is: "between 8 a.m. and 11 a.m."

This typically means from 8 a.m. up to but not including 11 a.m.

So 11:00 is not included.

Thus, together departures:
- 8:00
- 9:00
- 10:00

✔ That’s 3 times.

> Answer: a. 3

But wait — let’s double-check.

Alternatively, some might interpret "between 8 a.m. and 11 a.m." as including both ends, but 11 a.m. is the end of the period, and if buses leave at 11 a.m., is that before or after?

But the problem says: "between 8 a.m. and 11 a.m." — this usually implies up to but not including 11 a.m.

Moreover, if it were inclusive, then:
- 8:00, 9:00, 10:00, 11:00 → 4 times

But 4 is not an option.

Options are: a. 3, b. 5, c. 6, d. 9

So 3 is the closest possible.

But maybe we're missing something.

Wait — is 11 a.m. part of the interval?

Let’s calculate total time:

From 8:00 to 11:00 is 3 hours = 180 minutes

Buses leave together every 60 minutes, starting at 8:00.

So the times are:
- 8:00 (0 min)
- 9:00 (60 min)
- 10:00 (120 min)
- 11:00 (180 min)

Now, is 180 minutes included?

If the interval is from 8 a.m. to 11 a.m., and 11 a.m. is the end, then 11:00 is included if the departure happens at 11 a.m.

But the wording is "between 8 a.m. and 11 a.m.", which in most math contexts excludes endpoints.

But in real-world terms, if a bus leaves at 11 a.m., and you're considering events from 8 to 11, it might be included.

But again, 11:00 is not listed in options if we include it — 4 times isn't an option.

But wait! Let's think differently.

Maybe the buses start at 8 a.m., and we want how many times they leave together during the time from 8 a.m. to 11 a.m.

So:

- At 8:00 → first time → counts
- At 9:00 → second
- At 10:00 → third
- At 11:00 → fourth

But 11:00 is the end of the interval.

If the question says "between 8 a.m. and 11 a.m.", it might exclude 11 a.m., so only 3 times.

But if it includes 11 a.m., then 4 times — but 4 is not an option.

So this suggests our interpretation must be wrong.

Wait — perhaps we made a mistake in LCM?

No: LCM(15,20) = 60 → correct.

But maybe the time window is 3 hours, and buses leave together every hour.

So at:
- 8:00
- 9:00
- 10:00
- 11:00

But if 11:00 is not included, then 3 times.

But option a is 3, so that’s plausible.

But why are there options like 5, 6, 9?

Wait — maybe I misread the problem.

Let me read again:

> "how many times do buses on both line A and B leave together between 8 a.m. and 11 a.m."

Another possibility: do they leave together at 8:00?

Yes — they start together at 8 a.m.

Then next at 9:00, 10:00, 11:00.

Now, if the time frame is from 8 a.m. to 11 a.m., and 11 a.m. is included, then 4 times.

But 4 is not an option.

Unless... maybe the question means how many times they leave together *after* 8 a.m.?

But it says "between 8 a.m. and 11 a.m.", which typically includes 8 a.m.

Alternatively, maybe the buses leave at intervals, and we need to count how many joint departures occur in the span of 3 hours.

Let’s list the departure times:

Line A: every 15 min → 8:00, 8:15, 8:30, 8:45, 9:00, 9:15, ..., 10:30, 10:45, 11:00

Line B: every 20 min → 8:00, 8:20, 8:40, 9:00, 9:20, 9:40, 10:00, 10:20, 10:40, 11:00

Common times: when both have a bus.

We already know they coincide every 60 minutes.

So common times:
- 8:00
- 9:00
- 10:00
- 11:00

So 4 times.

But 4 is not an option.

Options: a. 3, b. 5, c. 6, d. 9

Hmm.

Wait — maybe 11:00 is not included because the period ends at 11 a.m., and the bus leaving at 11 a.m. is not within the interval.

So only:
- 8:00
- 9:00
- 10:00

→ 3 times.

So answer a. 3

But why is 4 not an option?

Wait — maybe I miscalculated.

Let’s compute how many multiples of 60 minutes are there from t=0 to t=180 minutes (8 a.m. to 11 a.m.)

At t = 0, 60, 120, 180

So:
- 0 min → 8:00
- 60 min → 9:00
- 120 min → 10:00
- 180 min → 11:00

Now, if 180 min is included, then 4 times.

But 180 min is exactly 11:00, which is the end.

But the phrase "between 8 a.m. and 11 a.m." usually means excluding 11 a.m.

For example, "between 8 and 11" in math means (8,11), open interval.

So 11:00 is not included.

Therefore, only:
- 8:00
- 9:00
- 10:00

→ 3 times

✔ So answer is a. 3

But let’s check if there are more common times.

Is it possible that buses leave together at other times?

For example, is there a time less than 60 minutes where both leave?

Let’s check:

Do any other common departure times exist?

Suppose at t = 0 → yes

Next: LCM(15,20) = 60 → so only every 60 minutes.

So no other times.

So only three times in the interval [8:00, 11:00) → 8:00, 9:00, 10:00

So 3 times

✔ Answer: a. 3

But let’s consider another interpretation: "between 8 a.m. and 11 a.m." might mean from 8:00 to 11:00 inclusive.

Then 4 times.

But 4 is not an option.

So must be 3

Perhaps the problem considers only after 8 a.m., so excluding 8:00?

But it says "start together at 8 a.m." — so that counts.

And "between 8 a.m. and 11 a.m." includes 8 a.m.

But again, if 11 a.m. is excluded, then 3 times.

So I think a. 3 is correct.

But let’s see — maybe I made a mistake in LCM?

No: LCM(15,20) = 60 → correct.

So every 60 minutes.

Total time: 3 hours = 180 minutes.

Number of 60-minute intervals from 0 to 180: 0, 60, 120, 180 → 4 times.

But if 180 minutes (11:00) is not included, then 3 times.

Given that 4 is not an option, and 3 is, it must be that 11:00 is not counted.

Hence, answer is a. 3

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✔ Final Answers:

1. d. 2025 cm
2. a. 3

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✔ Explanation Summary:

- Problem 1: Find LCM of 45, 75, 81 → LCM = 2025 cm → d
- Problem 2: LCM of 15 and 20 = 60 minutes → buses meet every hour → at 8, 9, 10 a.m. → 3 times (11 a.m. not included) → a

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Final Answers:
1. d. 2025 cm
2. a. 3 ✔