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Fraction Classroom · 2.6.0 · Student pages

Find when two delivery cycles meet again

List multiples of two small cycle lengths, find the first shared positive return time and convert it to a clock reading from an explicit common start.

Anonymous group code: __________   Date: __________

Every entry is elapsed minutes after the shared 8:00 start. Twenty-four is the first positive entry in both rows, so the next meeting is at 8:24.CycleReturn 1Return 2Return 3Return 46 minutes61218248 minutes8162432
Every entry is elapsed minutes after the shared 8:00 start. Twenty-four is the first positive entry in both rows, so the next meeting is at 8:24.

Start with a diagnostic

diagnostic. A cart returns every four minutes after a shared start. Enter its third positive return time in elapsed minutes, and list its first three returns.

Build the idea

model. Two supply carts are together at a classroom doorway at 8:00 a.m. The red cart returns every six minutes and the blue cart every eight minutes. After 8:00, when are they next at the doorway together? Give both elapsed minutes and the clock time.

  1. Mark the shared start as zero elapsed minutes and label it 8:00 a.m. Circle the word next: the meeting at zero is already known and is excluded. A return time must be a whole number of each cart's complete cycles.
  2. List red returns in increasing order: 6, 12, 18, 24. List blue returns up to the same point: 8, 16, 24. Continue every list without skipping a cycle so an earlier shared return cannot be hidden.
  3. Circle 24 in both lists. Confirm that 24 ÷ 6 = 4 and 24 ÷ 8 = 3, so both carts return then. None of 6, 12 or 18 occurs in the blue list, so no earlier positive red return is shared. This establishes the first meeting.
  4. Convert the elapsed result using the common start: 24 minutes after 8:00 is 8:24. Keep both labels in the answer. Continue the cycles to 48 minutes if checking the next meeting, and explain why a different starting time would require rebuilding that cart's return list.

Worked example: They next meet after 24 minutes, at 8:24 a.m. The red cart has completed four six-minute cycles and the blue cart three eight-minute cycles: 4 × 6 = 3 × 8 = 24.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. Support: two carts are together at 9:00 a.m. and return every four and six minutes. Enter the elapsed minutes until their next meeting and list both sets of returns up to that time.

p2. Core: two carts start together at 10:00 a.m. and return every six and nine minutes. Enter the elapsed minutes to the next meeting, record the clock time, and show that no earlier return is shared.

p3. Challenge: two carts start together at 11:10 a.m. and return every four and six minutes. How many meetings occur after the start and up to 11:46 a.m., including a meeting at the endpoint? Enter the count and list the meeting clock times.

e1. Independent check: two carts are together at 1:00 p.m. and return every eight and five minutes. Enter the elapsed minutes until their next meeting. List both sets of returns through that meeting, give its clock time, and justify why it is the first.

Transfer the idea

transfer. Keep the red cart's six-minute cycle starting at 8:00, but let the blue cart begin its eight-minute cycle at 8:03. Are both at the doorway at 8:24? Explain using each actual starting time.