LCM word problems (ACT). Game on.

LCM word problems (ACT) is a grade 9 math skill aligned to Common Core standard HSN.Q.A.2. Below are 8 practice questions with answers and step-by-step explanations, drawn from the 10 lcm word problems (act) problems our math games drill.

CCSS HSN.Q.A.210 questions in the bank
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Warm-upeasy

The school band practices every 66 school days, and the drama club meets every 99 school days. Both groups met today. In how many school days will they next meet on the same day?

The next shared meeting day is the least common multiple of 66 and 9.9\textsf{.} Since the GCF of 66 and 99 is 3,3\textsf{,} divide their product by the GCF: 6×93=18.\dfrac{6 \times 9}{3}=18\textsf{.} So they meet together again in 1818 school days.

Mid-gameeasy

A recycling truck visits a neighborhood every 1212 days, and a compost truck visits the same neighborhood every 1818 days. Both trucks came today. In how many days will both trucks next visit on the same day?

The next day both trucks visit is the least common multiple of 1212 and 18.18\textsf{.} Since the GCF of 1212 and 1818 is 6,6\textsf{,} divide their product by the GCF: 12×186=36.\dfrac{12 \times 18}{6}=36\textsf{.} So they next coincide in 3636 days.

Mid-gamemedium

A community garden waters the vegetable beds every 1010 days and fertilizes the same beds every 1414 days. Both jobs were done on the same day. In how many days will watering and fertilizing next fall on the same day?

The next day both jobs occur is the least common multiple of 1010 and 14.14\textsf{.} Since the GCF of 1010 and 1414 is 2,2\textsf{,} divide their product by the GCF: 10×142=70.\dfrac{10 \times 14}{2}=70\textsf{.} So they coincide again in 7070 days.

Mid-gamemedium

A campus security cart leaves the main gate every 1818 minutes, and a campus shuttle leaves the same gate every 2424 minutes. Both left together at 9:00 a.m. After how many minutes will they next leave together?

The cart and shuttle next leave together at the least common multiple of 1818 and 24.24\textsf{.} Since the GCF of 1818 and 2424 is 6,6\textsf{,} divide their product by the GCF: 18×246=72.\dfrac{18 \times 24}{6}=72\textsf{.} So they next leave together after 7272 minutes.

Mid-gamemedium

An electronic billboard refreshes every 1616 seconds, and a traffic signal cycle restarts every 2020 seconds. Both update at the same moment at t=0.t=0\textsf{.} After how many seconds do they next update at the same time?

The billboard and signal next update together at the least common multiple of 1616 and 20.20\textsf{.} Since the GCF of 1616 and 2020 is 4,4\textsf{,} divide their product by the GCF: 16×204=80.\dfrac{16 \times 20}{4}=80\textsf{.} So the next shared update is after 8080 seconds.

Mid-gamemedium

A cafe sells muffins in packs of 88 and bagels in packs of 14.14\textsf{.} What is the least number of muffins the cafe can buy so that it also buys exactly that many bagels?

The cafe needs the least common multiple of 88 and 1414 so the muffin and bagel totals match. Since the GCF of 88 and 1414 is 2,2\textsf{,} divide their product by the GCF: 8×142=56.\dfrac{8 \times 14}{2}=56\textsf{.} So the least matching total is 56.56\textsf{.}

Mid-gamehard

Bus route A leaves the station every 2020 minutes, and bus route B leaves every 3030 minutes. Both buses leave together at noon. Over the next 33 hours, not counting the noon departure, how many times do both buses leave the station at the same time?

Both buses leave together every least common multiple of 2020 and 3030 minutes. Since the GCF of 2020 and 3030 is 10,10\textsf{,} divide their product by the GCF to get that interval: 20×3010=60\dfrac{20 \times 30}{10}=60 minutes. In 180180 minutes after noon, the shared departures are at 60,60\textsf{,} 120,120\textsf{,} and 180180 minutes, so there are 33 additional times.

Buzzer beaterhard

A parking-garage gate opens every 99 minutes, and a ferry docks every 1515 minutes. Both events occur together at noon. Over the next 9090 minutes, not counting noon, how many times do both events occur at the same time?

Both events coincide every least common multiple of 99 and 1515 minutes. Since the GCF of 99 and 1515 is 3,3\textsf{,} divide their product by the GCF to get that interval: 9×153=45\dfrac{9 \times 15}{3}=45 minutes. In 9090 minutes after noon, the shared times are at 4545 and 9090 minutes, so there are 22 additional times.

Tests that cover lcm word problems (act):
ACTSBACVTCAPMCAP
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