LCM Word Problems: Schedules, Timers, and Cycles

1) A school's fire drill bell rings every 10 minutes, while the class-change bell rings every 15 minutes. After how many minutes will both bells ring together?

20

30

40

45

We need a time that both 10 and 15 divide into. The multiples of 10 are 10, 20, 30… and for 15: 15, 30, 45… The first number they share is 30.

So both bells will ring together after 30 minutes.

Explanation

We need a time that both 10 and 15 divide into. The multiples of 10 are 10, 20, 30… and for 15: 15, 30, 45… The first number they share is 30.

So both bells will ring together after 30 minutes.

2) One light changes every 8 seconds, another changes every 12 seconds. After how many seconds will they both change at the same time?

24

36

40

48

Let’s find the LCM of 8 and 12. Multiples of 8 are 8, 16, 24… and of 12 are 12, 24… They meet at 24.

So both lights will change together every 24 seconds.

Explanation

Let’s find the LCM of 8 and 12. Multiples of 8 are 8, 16, 24… and of 12 are 12, 24… They meet at 24.

So both lights will change together every 24 seconds.

3) A gardener plants tulips in rows of 6 and daffodils in rows of 8. What is the least number of flowers needed in each row so both fit evenly?

16

24

28

32

We’re finding a number that both 6 and 8 divide into evenly — their LCM. Multiples of 6 are 6, 12, 18, 24… and of 8 are 8, 16, 24…

They meet at 24.

So, each row must have 24 flowers.

Explanation

We’re finding a number that both 6 and 8 divide into evenly — their LCM. Multiples of 6 are 6, 12, 18, 24… and of 8 are 8, 16, 24…

They meet at 24.

So, each row must have 24 flowers.

4) Two city fountains spray every 12 minutes and 18 minutes. After how many minutes will both spray together?

24

36

48

54

To find when both fountains spray at the same time again, we look for the LCM of 12 and 18.

The multiples of 12 are 12, 24, 36… and the multiples of 18 are 18, 36, 54… The first common multiple is 36.

So, both fountains will spray together after 36 minutes.

Explanation

To find when both fountains spray at the same time again, we look for the LCM of 12 and 18.

The multiples of 12 are 12, 24, 36… and the multiples of 18 are 18, 36, 54… The first common multiple is 36.

So, both fountains will spray together after 36 minutes.

5) Two runners take water breaks: one every 14 minutes, the other every 21 minutes. When will they both stop for water together?

35

42

49

56

The LCM of 14 and 21 tells us when both breaks align. Multiples of 14 are 14, 28, 42… and of 21 are 21, 42…

They’ll stop together every 42 minutes.

Explanation

The LCM of 14 and 21 tells us when both breaks align. Multiples of 14 are 14, 28, 42… and of 21 are 21, 42…

They’ll stop together every 42 minutes.

6) A digital clock beeps every 20 minutes, while a smartwatch sends reminders every 50 minutes. After how many minutes will both beep together?

50

100

150

200

We list multiples: 20, 40, 60, 80, 100… and 50, 100, 150… They match at 100.

Both will beep together every 100 minutes.

Explanation

We list multiples: 20, 40, 60, 80, 100… and 50, 100, 150… They match at 100.

Both will beep together every 100 minutes.

7) A movie projector needs a reel change every 12 minutes, while another projector changes reels every 16 minutes. When will both need to change reels at the same time?

36

48

60

72

LCM(12,16) = 48, because 12×4 = 48 and 16×3 = 48.

So both projectors will need reel changes every 48 minutes.

Explanation

LCM(12,16) = 48, because 12×4 = 48 and 16×3 = 48.

So both projectors will need reel changes every 48 minutes.

8) Two gamers log in to play: one every 9 days, the other every 15 days. In how many days will they both log in on the same day?

45

36

54

60

Multiples of 9: 9, 18, 27, 36, 45… and 15: 15, 30, 45… They meet at 45.

So, both gamers will log in again together after 45 days.

Explanation

Multiples of 9: 9, 18, 27, 36, 45… and 15: 15, 30, 45… They meet at 45.

So, both gamers will log in again together after 45 days.

9) One teacher schedules a class quiz every 6 days, another every 10 days. After how many days will they give quizzes on the same day?

30

36

40

60

LCM of 6 and 10 is 30, because 6 = 2×3 and 10 = 2×5 → 2×3×5 = 30.

So, both teachers will give quizzes every 30 days.

Explanation

LCM of 6 and 10 is 30, because 6 = 2×3 and 10 = 2×5 → 2×3×5 = 30.

So, both teachers will give quizzes every 30 days.

10) One streetlight blinks every 4 seconds, another blinks every 7 seconds. After how many seconds will they blink together?

14

21

28

35

Since 4 and 7 have no common factors, we multiply: 4×7 = 28.

They will blink together every 28 seconds.

Explanation

Since 4 and 7 have no common factors, we multiply: 4×7 = 28.

They will blink together every 28 seconds.

11) Two ferries leave the harbor every 18 minutes and 24 minutes. After how many minutes will both leave together?

72

84

90

96

LCM(18,24) = 72 because 18 = 2×3² and 24 = 2³×3 → 2³×3² = 72.

Both ferries will leave together every 72 minutes.

Explanation

LCM(18,24) = 72 because 18 = 2×3² and 24 = 2³×3 → 2³×3² = 72.

Both ferries will leave together every 72 minutes.

12) A radio station plays ads every 12 minutes, another every 15 minutes. After how many minutes will both stations play ads at the same time?

60

75

90

120

LCM(12,15) = 60, since both divide evenly into 60.

So, both stations will play ads together every 60 minutes.

Explanation

LCM(12,15) = 60, since both divide evenly into 60.

So, both stations will play ads together every 60 minutes.

13) A lighthouse flashes every 25 minutes, another every 30 minutes. After how many minutes will both flash together?

150

175

200

225

We look for a number both 25 and 30 divide into. Multiples: 25, 50, 75, 100, 125, 150… and 30, 60, 90, 120, 150…

They meet at 150.

So both lighthouses flash together every 150 minutes.

Explanation

We look for a number both 25 and 30 divide into. Multiples: 25, 50, 75, 100, 125, 150… and 30, 60, 90, 120, 150…

They meet at 150.

So both lighthouses flash together every 150 minutes.

14) A student has soccer practice every 8 days and piano lessons every 12 days. After how many days will both fall on the same day?

20

24

28

36

LCM(8,12) = 24, since 8×3 = 24 and 12×2 = 24.

They’ll both happen again after 24 days.

Explanation

LCM(8,12) = 24, since 8×3 = 24 and 12×2 = 24.

They’ll both happen again after 24 days.

15) A teacher assigns reading every 9 days and projects every 12 days. After how many days will both assignments be due on the same day?

24

36

45

48

LCM(9,12) = 36.

So both assignments will line up every 36 days.

Explanation

LCM(9,12) = 36.

So both assignments will line up every 36 days.

16) A car rental company services one car model every 15 days and another every 20 days. After how many days will both models need service together?

60

75

80

100

LCM(15,20) = 60.

So both will need servicing every 60 days.

Explanation

LCM(15,20) = 60.

So both will need servicing every 60 days.

17) A bakery bakes croissants every 6 hours and bakes pies every 9 hours. After how many hours will both be baked together?

18

24

30

36

Both items are ready every 18 hours.

Explanation

Both items are ready every 18 hours.

18) A school has art class every 10 days and science lab every 12 days. After how many days will both classes be on the same day?

60

72

80

90

Multiples of 6: 6, 12, 18… and 9: 9, 18, 27… The first match is 18.

So both will be ready every 18 hours.

Explanation

Multiples of 6: 6, 12, 18… and 9: 9, 18, 27… The first match is 18.

So both will be ready every 18 hours.

19) Two buses run on schedules of 28 minutes and 42 minutes. After how many minutes will both buses arrive together?

84

96

112

126

LCM(10,12) = 60.

So both classes will coincide every 60 days.

Explanation

LCM(10,12) = 60.

So both classes will coincide every 60 days.

20) A rocket testing facility checks one rocket every 16 days, another every 24 days. After how many days will both be tested together?

32

48

64

72

LCM(16,24) = 48 since 16 = 2⁴ and 24 = 2³×3 → 2⁴×3 = 48.

So both rockets will be tested together every 48 days.

Explanation

LCM(16,24) = 48 since 16 = 2⁴ and 24 = 2³×3 → 2⁴×3 = 48.

So both rockets will be tested together every 48 days.

Word Problems with LCM

1) A school's fire drill bell rings every 10 minutes, while the class-change bell rings every 15 minutes. After how many minutes will both bells ring together?

2) One light changes every 8 seconds, another changes every 12 seconds. After how many seconds will they both change at the same time?

3) A gardener plants tulips in rows of 6 and daffodils in rows of 8. What is the least number of flowers needed in each row so both fit evenly?

4) Two city fountains spray every 12 minutes and 18 minutes. After how many minutes will both spray together?

5) Two runners take water breaks: one every 14 minutes, the other every 21 minutes. When will they both stop for water together?

6) A digital clock beeps every 20 minutes, while a smartwatch sends reminders every 50 minutes. After how many minutes will both beep together?

7) A movie projector needs a reel change every 12 minutes, while another projector changes reels every 16 minutes. When will both need to change reels at the same time?

8) Two gamers log in to play: one every 9 days, the other every 15 days. In how many days will they both log in on the same day?

9) One teacher schedules a class quiz every 6 days, another every 10 days. After how many days will they give quizzes on the same day?

10) One streetlight blinks every 4 seconds, another blinks every 7 seconds. After how many seconds will they blink together?

11) Two ferries leave the harbor every 18 minutes and 24 minutes. After how many minutes will both leave together?

12) A radio station plays ads every 12 minutes, another every 15 minutes. After how many minutes will both stations play ads at the same time?

13) A lighthouse flashes every 25 minutes, another every 30 minutes. After how many minutes will both flash together?

14) A student has soccer practice every 8 days and piano lessons every 12 days. After how many days will both fall on the same day?

15) A teacher assigns reading every 9 days and projects every 12 days. After how many days will both assignments be due on the same day?

16) A car rental company services one car model every 15 days and another every 20 days. After how many days will both models need service together?

17) A bakery bakes croissants every 6 hours and bakes pies every 9 hours. After how many hours will both be baked together?

18) A school has art class every 10 days and science lab every 12 days. After how many days will both classes be on the same day?

19) Two buses run on schedules of 28 minutes and 42 minutes. After how many minutes will both buses arrive together?

20) A rocket testing facility checks one rocket every 16 days, another every 24 days. After how many days will both be tested together?