← Events coincide at common multiples of intervals · Repeating Cycle Patterns

Events coincide at common multiples of intervals · 12 practice problems

4.OA.B.46.NS.B.44.OA.A.3

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Variant 1 easy

At a station, the train to Denver departs every $15$ minutes and the train to Seattle departs every $10$ minutes. If the Denver train and the Seattle train both leave together at $8\!:\!15$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 8:15. From now on the Denver train leaves every 15 minutes and the Seattle train every 10 minutes. We need the next clock time when both leave at once.

Givens

The Denver train departs every 15 minutes.
The Seattle train departs every 10 minutes.
Both departed together at 8:15 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 8:15 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Denver departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 8:15, the Denver train leaves at every multiple of 15.

15, 30, \cdots

These are simply the multiples of 15.

2List the Seattle departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Seattle train leaves at every multiple of 10 minutes after 8:15.

10, 20, 30, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 15 and 10. The earliest one is their least common multiple.

\operatorname{lcm}(15, 10) = 30

Before minute 30, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 30 minutes after 8:15, so count forward on the clock.

8:15 + 30\ \text{min} = 8:45

The next joint departure is 8:45 a.m.

Answer: 8:45 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 30 / 15 = 2 and 30 / 10 = 3, both whole numbers, so at minute 30 each train is exactly at a departure.

Another way: Use gcd instead: 15 x 10 / gcd(15, 10) = 150 / 5 = 30 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 15 and of 10 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 30 as the least common multiple of 15 and 10.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 30 minutes to 8:15 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 2 easy

At a station, the train to Boston departs every $18$ minutes and the train to Chicago departs every $12$ minutes. If the Boston train and the Chicago train both leave together at $8\!:\!05$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 8:05. From now on the Boston train leaves every 18 minutes and the Chicago train every 12 minutes. We need the next clock time when both leave at once.

Givens

The Boston train departs every 18 minutes.
The Chicago train departs every 12 minutes.
Both departed together at 8:05 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 8:05 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Boston departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 8:05, the Boston train leaves at every multiple of 18.

18, 36, \cdots

These are simply the multiples of 18.

2List the Chicago departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Chicago train leaves at every multiple of 12 minutes after 8:05.

12, 24, 36, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 18 and 12. The earliest one is their least common multiple.

\operatorname{lcm}(18, 12) = 36

Before minute 36, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 36 minutes after 8:05, so count forward on the clock.

8:05 + 36\ \text{min} = 8:41

The next joint departure is 8:41 a.m.

Answer: 8:41 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 36 / 18 = 2 and 36 / 12 = 3, both whole numbers, so at minute 36 each train is exactly at a departure.

Another way: Use gcd instead: 18 x 12 / gcd(18, 12) = 216 / 6 = 36 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 18 and of 12 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 36 as the least common multiple of 18 and 12.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 36 minutes to 8:05 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 3 easy

At a station, the train to Austin departs every $14$ minutes and the train to Phoenix departs every $6$ minutes. If the Austin train and the Phoenix train both leave together at $7\!:\!20$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 7:20. From now on the Austin train leaves every 14 minutes and the Phoenix train every 6 minutes. We need the next clock time when both leave at once.

Givens

The Austin train departs every 14 minutes.
The Phoenix train departs every 6 minutes.
Both departed together at 7:20 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 7:20 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Austin departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 7:20, the Austin train leaves at every multiple of 14.

14, 28, 42, \cdots

These are simply the multiples of 14.

2List the Phoenix departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Phoenix train leaves at every multiple of 6 minutes after 7:20.

6, 12, 18, 24, 30, 36, 42, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 14 and 6. The earliest one is their least common multiple.

\operatorname{lcm}(14, 6) = 42

Before minute 42, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 42 minutes after 7:20, so count forward on the clock.

7:20 + 42\ \text{min} = 8:02

The next joint departure is 8:02 a.m.

Answer: 8:02 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 42 / 14 = 3 and 42 / 6 = 7, both whole numbers, so at minute 42 each train is exactly at a departure.

Another way: Use gcd instead: 14 x 6 / gcd(14, 6) = 84 / 2 = 42 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 14 and of 6 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 42 as the least common multiple of 14 and 6.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 42 minutes to 7:20 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 4 easy

At a station, the train to Miami departs every $21$ minutes and the train to Omaha departs every $14$ minutes. If the Miami train and the Omaha train both leave together at $7\!:\!10$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 7:10. From now on the Miami train leaves every 21 minutes and the Omaha train every 14 minutes. We need the next clock time when both leave at once.

Givens

The Miami train departs every 21 minutes.
The Omaha train departs every 14 minutes.
Both departed together at 7:10 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 7:10 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Miami departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 7:10, the Miami train leaves at every multiple of 21.

21, 42, \cdots

These are simply the multiples of 21.

2List the Omaha departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Omaha train leaves at every multiple of 14 minutes after 7:10.

14, 28, 42, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 21 and 14. The earliest one is their least common multiple.

\operatorname{lcm}(21, 14) = 42

Before minute 42, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 42 minutes after 7:10, so count forward on the clock.

7:10 + 42\ \text{min} = 7:52

The next joint departure is 7:52 a.m.

Answer: 7:52 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 42 / 21 = 2 and 42 / 14 = 3, both whole numbers, so at minute 42 each train is exactly at a departure.

Another way: Use gcd instead: 21 x 14 / gcd(21, 14) = 294 / 7 = 42 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 21 and of 14 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 42 as the least common multiple of 21 and 14.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 42 minutes to 7:10 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 5 medium

At a station, the train to Austin departs every $12$ minutes and the train to Phoenix departs every $8$ minutes. If the Austin train and the Phoenix train both leave together at $9\!:\!25$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 9:25. From now on the Austin train leaves every 12 minutes and the Phoenix train every 8 minutes. We need the next clock time when both leave at once.

Givens

The Austin train departs every 12 minutes.
The Phoenix train departs every 8 minutes.
Both departed together at 9:25 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 9:25 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Austin departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 9:25, the Austin train leaves at every multiple of 12.

12, 24, \cdots

These are simply the multiples of 12.

2List the Phoenix departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Phoenix train leaves at every multiple of 8 minutes after 9:25.

8, 16, 24, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 12 and 8. The earliest one is their least common multiple.

\operatorname{lcm}(12, 8) = 24

Before minute 24, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 24 minutes after 9:25, so count forward on the clock.

9:25 + 24\ \text{min} = 9:49

The next joint departure is 9:49 a.m.

Answer: 9:49 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 24 / 12 = 2 and 24 / 8 = 3, both whole numbers, so at minute 24 each train is exactly at a departure.

Another way: Use gcd instead: 12 x 8 / gcd(12, 8) = 96 / 4 = 24 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 12 and of 8 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 24 as the least common multiple of 12 and 8.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 24 minutes to 9:25 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 6 medium

At a station, the train to Miami departs every $20$ minutes and the train to Omaha departs every $15$ minutes. If the Miami train and the Omaha train both leave together at $9\!:\!30$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 9:30. From now on the Miami train leaves every 20 minutes and the Omaha train every 15 minutes. We need the next clock time when both leave at once.

Givens

The Miami train departs every 20 minutes.
The Omaha train departs every 15 minutes.
Both departed together at 9:30 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 9:30 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Miami departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 9:30, the Miami train leaves at every multiple of 20.

20, 40, 60, \cdots

These are simply the multiples of 20.

2List the Omaha departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Omaha train leaves at every multiple of 15 minutes after 9:30.

15, 30, 45, 60, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 20 and 15. The earliest one is their least common multiple.

\operatorname{lcm}(20, 15) = 60

Before minute 60, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 60 minutes after 9:30, so count forward on the clock.

9:30 + 60\ \text{min} = 10:30

The next joint departure is 10:30 a.m.

Answer: 10:30 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 60 / 20 = 3 and 60 / 15 = 4, both whole numbers, so at minute 60 each train is exactly at a departure.

Another way: Use gcd instead: 20 x 15 / gcd(20, 15) = 300 / 5 = 60 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 20 and of 15 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 60 as the least common multiple of 20 and 15.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 60 minutes to 9:30 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 7 medium

At a station, the train to Austin departs every $10$ minutes and the train to Phoenix departs every $4$ minutes. If the Austin train and the Phoenix train both leave together at $6\!:\!35$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 6:35. From now on the Austin train leaves every 10 minutes and the Phoenix train every 4 minutes. We need the next clock time when both leave at once.

Givens

The Austin train departs every 10 minutes.
The Phoenix train departs every 4 minutes.
Both departed together at 6:35 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 6:35 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Austin departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 6:35, the Austin train leaves at every multiple of 10.

10, 20, \cdots

These are simply the multiples of 10.

2List the Phoenix departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Phoenix train leaves at every multiple of 4 minutes after 6:35.

4, 8, 12, 16, 20, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 10 and 4. The earliest one is their least common multiple.

\operatorname{lcm}(10, 4) = 20

Before minute 20, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 20 minutes after 6:35, so count forward on the clock.

6:35 + 20\ \text{min} = 6:55

The next joint departure is 6:55 a.m.

Answer: 6:55 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 20 / 10 = 2 and 20 / 4 = 5, both whole numbers, so at minute 20 each train is exactly at a departure.

Another way: Use gcd instead: 10 x 4 / gcd(10, 4) = 40 / 2 = 20 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 10 and of 4 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 20 as the least common multiple of 10 and 4.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 20 minutes to 6:35 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 8 hard

At a station, the train to Dallas departs every $10$ minutes and the train to Detroit departs every $6$ minutes. If the Dallas train and the Detroit train both leave together at $11\!:\!40$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 11:40. From now on the Dallas train leaves every 10 minutes and the Detroit train every 6 minutes. We need the next clock time when both leave at once.

Givens

The Dallas train departs every 10 minutes.
The Detroit train departs every 6 minutes.
Both departed together at 11:40 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 11:40 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Dallas departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 11:40, the Dallas train leaves at every multiple of 10.

10, 20, 30, \cdots

These are simply the multiples of 10.

2List the Detroit departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Detroit train leaves at every multiple of 6 minutes after 11:40.

6, 12, 18, 24, 30, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 10 and 6. The earliest one is their least common multiple.

\operatorname{lcm}(10, 6) = 30

Before minute 30, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 30 minutes after 11:40, so count forward on the clock.

11:40 + 30\ \text{min} = 12:10

The next joint departure is 12:10 a.m.

Answer: 12:10 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 30 / 10 = 3 and 30 / 6 = 5, both whole numbers, so at minute 30 each train is exactly at a departure.

Another way: Use gcd instead: 10 x 6 / gcd(10, 6) = 60 / 2 = 30 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 10 and of 6 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 30 as the least common multiple of 10 and 6.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 30 minutes to 11:40 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 9 medium

At a station, the train to Austin departs every $8$ minutes and the train to Phoenix departs every $6$ minutes. If the Austin train and the Phoenix train both leave together at $8\!:\!40$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 8:40. From now on the Austin train leaves every 8 minutes and the Phoenix train every 6 minutes. We need the next clock time when both leave at once.

Givens

The Austin train departs every 8 minutes.
The Phoenix train departs every 6 minutes.
Both departed together at 8:40 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 8:40 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Austin departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 8:40, the Austin train leaves at every multiple of 8.

8, 16, 24, \cdots

These are simply the multiples of 8.

2List the Phoenix departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Phoenix train leaves at every multiple of 6 minutes after 8:40.

6, 12, 18, 24, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 8 and 6. The earliest one is their least common multiple.

\operatorname{lcm}(8, 6) = 24

Before minute 24, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 24 minutes after 8:40, so count forward on the clock.

8:40 + 24\ \text{min} = 9:04

The next joint departure is 9:04 a.m.

Answer: 9:04 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 24 / 8 = 3 and 24 / 6 = 4, both whole numbers, so at minute 24 each train is exactly at a departure.

Another way: Use gcd instead: 8 x 6 / gcd(8, 6) = 48 / 2 = 24 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 8 and of 6 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 24 as the least common multiple of 8 and 6.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 24 minutes to 8:40 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 10 hard

At a station, the train to Portland departs every $9$ minutes and the train to Atlanta departs every $12$ minutes. If the Portland train and the Atlanta train both leave together at $10\!:\!45$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 10:45. From now on the Portland train leaves every 9 minutes and the Atlanta train every 12 minutes. We need the next clock time when both leave at once.

Givens

The Portland train departs every 9 minutes.
The Atlanta train departs every 12 minutes.
Both departed together at 10:45 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 10:45 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Portland departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 10:45, the Portland train leaves at every multiple of 9.

9, 18, 27, 36, \cdots

These are simply the multiples of 9.

2List the Atlanta departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Atlanta train leaves at every multiple of 12 minutes after 10:45.

12, 24, 36, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 9 and 12. The earliest one is their least common multiple.

\operatorname{lcm}(9, 12) = 36

Before minute 36, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 36 minutes after 10:45, so count forward on the clock.

10:45 + 36\ \text{min} = 11:21

The next joint departure is 11:21 a.m.

Answer: 11:21 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 36 / 9 = 4 and 36 / 12 = 3, both whole numbers, so at minute 36 each train is exactly at a departure.

Another way: Use gcd instead: 9 x 12 / gcd(9, 12) = 108 / 3 = 36 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 9 and of 12 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 36 as the least common multiple of 9 and 12.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 36 minutes to 10:45 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 11 hard

At a station, the train to Portland departs every $9$ minutes and the train to Atlanta departs every $6$ minutes. If the Portland train and the Atlanta train both leave together at $7\!:\!50$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 7:50. From now on the Portland train leaves every 9 minutes and the Atlanta train every 6 minutes. We need the next clock time when both leave at once.

Givens

The Portland train departs every 9 minutes.
The Atlanta train departs every 6 minutes.
Both departed together at 7:50 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 7:50 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Portland departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 7:50, the Portland train leaves at every multiple of 9.

9, 18, \cdots

These are simply the multiples of 9.

2List the Atlanta departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Atlanta train leaves at every multiple of 6 minutes after 7:50.

6, 12, 18, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 9 and 6. The earliest one is their least common multiple.

\operatorname{lcm}(9, 6) = 18

Before minute 18, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 18 minutes after 7:50, so count forward on the clock.

7:50 + 18\ \text{min} = 8:08

The next joint departure is 8:08 a.m.

Answer: 8:08 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 18 / 9 = 2 and 18 / 6 = 3, both whole numbers, so at minute 18 each train is exactly at a departure.

Another way: Use gcd instead: 9 x 6 / gcd(9, 6) = 54 / 3 = 18 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 9 and of 6 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 18 as the least common multiple of 9 and 6.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 18 minutes to 7:50 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Variant 12 hard

At a station, the train to Dallas departs every $16$ minutes and the train to Detroit departs every $12$ minutes. If the Dallas train and the Detroit train both leave together at $6\!:\!55$ a.m., at what time will the two trains next depart at the same moment?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two trains just left together at 6:55. From now on the Dallas train leaves every 16 minutes and the Detroit train every 12 minutes. We need the next clock time when both leave at once.

Givens

The Dallas train departs every 16 minutes.
The Detroit train departs every 12 minutes.
Both departed together at 6:55 a.m.

Unknowns

The next time both trains depart together.

Constraints

Both trains keep their intervals exactly, with no delays.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List

List the minutes after 6:55 when each train leaves, then look for the first minute that appears on both lists. That first shared minute is the next time they coincide.

3 · Execute4 carry out the plan

1List the Dallas departures

#2 Make a Systematic List 4.OA.B.4

Counting minutes after 6:55, the Dallas train leaves at every multiple of 16.

16, 32, 48, \cdots

These are simply the multiples of 16.

2List the Detroit departures

#2 Make a Systematic List 4.OA.B.4

In the same way, the Detroit train leaves at every multiple of 12 minutes after 6:55.

12, 24, 36, 48, \cdots

Two separate lists, one per train.

3Find the first shared minute

#5 Look for a Pattern 6.NS.B.4

A minute on both lists is a common multiple of 16 and 12. The earliest one is their least common multiple.

\operatorname{lcm}(16, 12) = 48

Before minute 48, one train has always just missed the other.

4Add that gap to the start time

#5 Look for a Pattern 4.OA.A.3

The two trains coincide 48 minutes after 6:55, so count forward on the clock.

6:55 + 48\ \text{min} = 7:43

The next joint departure is 7:43 a.m.

Answer: 7:43 a.m.

4 · Reviewdoes it hold up?

Check the gap divides evenly both ways: 48 / 16 = 3 and 48 / 12 = 4, both whole numbers, so at minute 48 each train is exactly at a departure.

Another way: Use gcd instead: 16 x 12 / gcd(16, 12) = 192 / 4 = 48 minutes, the same gap.

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples of whole numbers — Listing the multiples of 16 and of 12 as the departure minutes.
6.NS.B.4 Find the greatest common factor and least common multiple of whole numbers — Identifying 48 as the least common multiple of 16 and 12.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Adding 48 minutes to 6:55 to land on the clock time.

💡Takeaway. Two repeating schedules meet again at their least common multiple, so find that number instead of listing every departure.

Where next?

← Back to the problem: Events coincide at common multiples of intervals