Least Multiple Calculator

LCM Word Problems and Real-Life Uses

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LCM word problems: buses every 12 and 15 minutes meet again after 60 minutes

LCM word problems ask when repeating events happen together again, or what the smallest quantity is that fits two or more group sizes exactly. Two buses leaving every 12 and 18 minutes leave together every LCM(12, 18) = 36 minutes.

Each problem below is solved with the least common multiple of the numbers in the question. The methods are onHow to find the LCM, and the definition on What is the LCM? Definition, full form, examples.

How to spot an LCM problem

Eight worked LCM word problems

  1. Two buses leave a station together. One leaves every 12 minutes and the other every 18 minutes. After how many minutes do they leave together again?
    solution LCM(12, 18) = 36 minutes.
  2. Three bells ring every 4, 6 and 10 seconds. They ring together at noon. When do they next ring together?
    solution LCM(4, 6, 10) = 60 seconds, so 1 minute after noon.
  3. A gear with 12 teeth meshes with a gear with 18 teeth. After how many teeth do the two marked teeth meet again?
    solution LCM(12, 18) = 36 teeth: the small gear turns 3 times, the large gear 2 times.
  4. Planet A orbits in 6 years and planet B in 8 years. If they line up now, when do they line up next?
    solution LCM(6, 8) = 24 years.
  5. What is the smallest square that can be tiled with 6 cm by 8 cm rectangles?
    solution Side = LCM(6, 8) = 24 cm, using 4 × 3 = 12 tiles.
  6. Add 1/12 and 1/15.
    solution LCD = LCM(12, 15) = 60: 5/60 + 4/60 = 9/60 = 3/20.
  7. Pencils come in packs of 8 and erasers in packs of 6. What is the smallest number of each to buy so that every pencil has an eraser with none left over?
    solution LCM(8, 6) = 24: buy 3 packs of pencils and 4 packs of erasers.
  8. A nurse checks one patient every 45 minutes and another every 60 minutes, both starting at 8:00. When are both checked at the same time again?
    solution LCM(45, 60) = 180 minutes = 3 hours, so 11:00.

Where the LCM appears

SituationThe LCM gives
Adding and subtracting fractionsthe lowest common denominator
Timetables, alarms, traffic lightswhen cycles coincide
Gears and pulleyswhen marked teeth realign
Tiling and packagingthe smallest length or count that fits both sizes
Musicwhen two rhythms of different lengths restart together

Practice

  1. Lights flash every 9, 15 seconds. After how many seconds do they flash together?
    answer LCM = 45 seconds
  2. Lights flash every 10, 12 seconds. After how many seconds do they flash together?
    answer LCM = 60 seconds
  3. Lights flash every 6, 9, 12 seconds. After how many seconds do they flash together?
    answer LCM = 36 seconds
  4. Lights flash every 20, 25 seconds. After how many seconds do they flash together?
    answer LCM = 100 seconds
  5. Lights flash every 14, 35 seconds. After how many seconds do they flash together?
    answer LCM = 70 seconds
  6. Lights flash every 5, 6, 8 seconds. After how many seconds do they flash together?
    answer LCM = 120 seconds

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Frequently asked questions

Why is the LCM important?

The LCM is the smallest point where repeating cycles meet, and the smallest shared denominator for fractions. It answers "when do two things happen together again" and "how do I add these fractions".

Why do we need to find the LCM?

To add or subtract fractions with different denominators, to schedule events with different intervals, and to solve problems about gears, tiles and packaging where quantities must match exactly.

What is an example of how the LCM is used in real life?

Two buses leaving every 12 and 18 minutes leave together every 36 minutes. 36 is the least common multiple of 12 and 18.

How do I know whether a word problem needs the LCM or the HCF?

If the answer must be larger than the numbers given, such as the next time events coincide or the smallest quantity that fits both, use the LCM. If the answer must be smaller, such as the largest equal group or the largest tile that fits, use the HCF.

Sources