Least Common Multiple Word Problems (A) - Solve Word Problems

Sep 23, 2026

Uploaded by Jacob

A set of eight least common multiple word problems set in music contexts, asking when different rhythms or schedules will coincide again. Learners solve timing questions involving drum hits, stage performances, a piano chord and bell, rotating performers, overlapping spotlight colors, lesson days, recording schedules for two bands, and rotating concert events across days. An answer key provides final times/days and the LCM reasoning steps for each problem.

LanguageEnglish

CategoryExam

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Directions

Problems 1-8

Answer Key

Name: Least Common Multiple Word Problems (A) MUSIC & PERFORMANCES Directions: Solve each word problem. Show all of your work and circle your answer. 1. A drummer hits a bass drum every 4 beats and a snare drum every 6 beats. If both drums are hit at beat 1, after how many beats will they both be hit at the same time again? Show your work: 2. At a music festival, Stage A has a new performance every 15 minutes and Stage B has a new performance every 20 minutes. If both stages start at the same time, how many minutes until they both start a new performance simultaneously? Show your work: 3. A music teacher plays a chord on the piano every 8 seconds and rings a bell every 12 seconds. Starting at the same moment, how many seconds until both happen at the same time again? How many times will she play the chord in that time? Show your work: Mashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com) 4. Three performers rotate on stage: a singer every 6 minutes, a dancer every 9 minutes, and a magician every 4 minutes. If all three start at the same time, after how many minutes will all three perform at the same time again? Show your work: 5. A spotlight cycles red every 10 seconds, blue every 15 seconds, and green every 6 seconds. If all three flash at time zero, after how many seconds will all three flash together again? Show your work: 6. A music school holds guitar lessons every 8 days and piano lessons every 12 days. Both are held on Day 1. On what day will both next be held on the same day? How many of each lesson will have occurred by then? Show your work: Mashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com) 7. Two bands are recording. Band A records a new song every 18 days and Band B every 24 days. Both record on Day 1. What is the next day both will record on the same day? How many songs will each band have recorded by then (including Day 1)? Show your work: Challenge Problem 8. A concert hall hosts three events on rotation: jazz concerts every 14 days, classical concerts every 21 days, and pop concerts every 35 days. All three are held on Day 1. What is the next day all three will be held on the same day? How many of each event will have occurred by then (including Day 1)? Show your work: Mashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com) ANSWER KEY 1.) 1. ) Answer: Beat 12 Steps: Multiples of 4: 4, 8, 12, ... → Multiples of 6: 6, 12, ... → LCM(4,6) = 12 2.) 2. ) Answer: 60 minutes Steps: 15 = 3×5 20 = 2² ×5 → LCM(15,20) = 2² ×3×5 = 60 minutes 3.) 3. ) Answer: 24 seconds; chord plays 3 times Steps: 8 = 2³ 12 = 2² ×3 → LCM(8,12) = 2³ ×3 = 24 seconds → Chord plays at 8, 16, 24 → 3 times in 24 seconds 4.) 4. ) Answer: 36 minutes Steps: LCM(6,9): 6=2×3, 9=3² → LCM=2×3²=18 → LCM(18,4): 4=2² → LCM=2² ×3²=36 minutes 5.) 5. ) Answer: 30 seconds Steps: LCM(10,15): 10=2×5, 15=3×5 → LCM=2×3×5=30 → LCM(30,6): 6=2×3 → LCM=30 seconds 6.) 6. ) Answer: Day 24; guitar 3 times, piano 2 times Steps: LCM(8,12) = 2³ ×3 = 24 → Both meet again on Day 24 → Guitar: 24÷8 = 3 lessons Piano: 24÷12 = 2 lessons 7.) 7. ) Answer: Day 72; Band A: 4 songs, Band B: 3 songs Steps: 18 = 2×3² 24 = 2³ ×3 → LCM(18,24) = 2³ ×3² = 72 → Next shared day: Day 72 → Band A: 72÷18 = 4 songs Band B: 72÷24 = 3 songs 8.) 8. ) Answer: Day 210; Jazz: 15, Classical: 10, Pop: 6 Steps: LCM(14,21): 14=2×7, 21=3×7 → LCM=2×3×7=42 → LCM(42,35): 35=5×7 → LCM=2×3×5×7=210 → Jazz: 210÷14=15 Classical: 210÷21=10 Pop: 210÷35=6 Mashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mas

FAQs

How do you find when two events happen again at the same time?

Compute the LCM of the two intervals (beats, minutes, seconds, or days). The LCM gives the next time both events coincide.

Why does the answer key use prime factorization in some problems?

Prime factorization helps calculate the LCM accurately by combining the highest powers of each prime factor from the given intervals.

In problem 3, how many times does the chord play within the shared time?

The LCM of 8 seconds and 12 seconds is 24 seconds. The chord plays every 8 seconds, so it occurs at 8, 16, and 24—3 times.

Sep 23, 2026

Uploaded by Jacob