← Blinking lights coincide at the LCM · Repeating Cycle Patterns

Blinking lights coincide at the LCM · 12 practice problems

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

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 60 seconds

A white light stays on for 88 seconds and then off for 44 seconds, while a yellow light stays on for 55 seconds and then off for 55 seconds. If both lights are switched on at the same moment, for how many seconds during the first 33 minute are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 8 seconds then off 4; the yellow one is on 5 seconds then off 5. Both start on together. We need the total time in the first 180 seconds when both are on at once.

Givens
  • White: on 8 s, off 4 s, repeating.
  • Yellow: on 5 s, off 5 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 180 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 180 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 60-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
8+4=12,5+5=108 + 4 = 12,\quad 5 + 5 = 10
White repeats every 12 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 12 and of 10.
lcm⁡(12,10)=60\operatorname{lcm}(12, 10) = 60
After 60 s the same picture starts over.

3List the ON stretches inside the first 60 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−8,12−20,24−32,36−44,48−56;yellow:0−5,10−15,20−25,30−35,40−45,50−55\text{white}: 0-8, 12-20, 24-32, 36-44, 48-56;\quad \text{yellow}: 0-5, 10-15, 20-25, 30-35, 40-45, 50-55
Two rows of intervals, ready to compare.

4Find the overlaps in one 60-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
20 s per 60-second cycle20\ \text{s per 60-second cycle}
The short problem is solved: 20 s.

5Scale up to the full 180 seconds

#5 Look for a Pattern 6.NS.B.4
180 seconds is 3 whole cycles of 60 seconds, and each cycle looks the same.
20×3=6020 \times 3 = 60
Both lights are on for 60 seconds in all.
Answer: 60 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 8 of every 12 seconds and yellow 5 of every 10, so an overlap of 60 out of 180 seconds is in the range those two fractions suggest.

Another way: Mark all 180 seconds on two timelines and count the seconds shaded twice; it gives 60, but takes 3 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 12 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 60 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 2 easy answer: 100 seconds

A white light stays on for 55 seconds and then off for 77 seconds, while a yellow light stays on for 88 seconds and then off for 22 seconds. If both lights are switched on at the same moment, for how many seconds during the first 55 minute are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 5 seconds then off 7; the yellow one is on 8 seconds then off 2. Both start on together. We need the total time in the first 300 seconds when both are on at once.

Givens
  • White: on 5 s, off 7 s, repeating.
  • Yellow: on 8 s, off 2 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 300 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 300 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 60-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
5+7=12,8+2=105 + 7 = 12,\quad 8 + 2 = 10
White repeats every 12 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 12 and of 10.
lcm⁡(12,10)=60\operatorname{lcm}(12, 10) = 60
After 60 s the same picture starts over.

3List the ON stretches inside the first 60 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−5,12−17,24−29,36−41,48−53;yellow:0−8,10−18,20−28,30−38,40−48,50−58\text{white}: 0-5, 12-17, 24-29, 36-41, 48-53;\quad \text{yellow}: 0-8, 10-18, 20-28, 30-38, 40-48, 50-58
Two rows of intervals, ready to compare.

4Find the overlaps in one 60-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
20 s per 60-second cycle20\ \text{s per 60-second cycle}
The short problem is solved: 20 s.

5Scale up to the full 300 seconds

#5 Look for a Pattern 6.NS.B.4
300 seconds is 5 whole cycles of 60 seconds, and each cycle looks the same.
20×5=10020 \times 5 = 100
Both lights are on for 100 seconds in all.
Answer: 100 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 5 of every 12 seconds and yellow 8 of every 10, so an overlap of 100 out of 300 seconds is in the range those two fractions suggest.

Another way: Mark all 300 seconds on two timelines and count the seconds shaded twice; it gives 100, but takes 5 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 12 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 60 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 3 medium answer: 60 seconds

A white light stays on for 88 seconds and then off for 77 seconds, while a yellow light stays on for 99 seconds and then off for 11 seconds. If both lights are switched on at the same moment, for how many seconds during the first 22 minute are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 8 seconds then off 7; the yellow one is on 9 seconds then off 1. Both start on together. We need the total time in the first 120 seconds when both are on at once.

Givens
  • White: on 8 s, off 7 s, repeating.
  • Yellow: on 9 s, off 1 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 120 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 120 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 30-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
8+7=15,9+1=108 + 7 = 15,\quad 9 + 1 = 10
White repeats every 15 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 15 and of 10.
lcm⁡(15,10)=30\operatorname{lcm}(15, 10) = 30
After 30 s the same picture starts over.

3List the ON stretches inside the first 30 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−8,15−23;yellow:0−9,10−19,20−29\text{white}: 0-8, 15-23;\quad \text{yellow}: 0-9, 10-19, 20-29
Two rows of intervals, ready to compare.

4Find the overlaps in one 30-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
15 s per 30-second cycle15\ \text{s per 30-second cycle}
The short problem is solved: 15 s.

5Scale up to the full 120 seconds

#5 Look for a Pattern 6.NS.B.4
120 seconds is 4 whole cycles of 30 seconds, and each cycle looks the same.
15×4=6015 \times 4 = 60
Both lights are on for 60 seconds in all.
Answer: 60 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 8 of every 15 seconds and yellow 9 of every 10, so an overlap of 60 out of 120 seconds is in the range those two fractions suggest.

Another way: Mark all 120 seconds on two timelines and count the seconds shaded twice; it gives 60, but takes 4 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 15 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 30 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 4 easy answer: 81 seconds

A white light stays on for 99 seconds and then off for 33 seconds, while a yellow light stays on for 66 seconds and then off for 44 seconds. If both lights are switched on at the same moment, for how many seconds during the first 33 minute are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 9 seconds then off 3; the yellow one is on 6 seconds then off 4. Both start on together. We need the total time in the first 180 seconds when both are on at once.

Givens
  • White: on 9 s, off 3 s, repeating.
  • Yellow: on 6 s, off 4 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 180 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 180 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 60-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
9+3=12,6+4=109 + 3 = 12,\quad 6 + 4 = 10
White repeats every 12 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 12 and of 10.
lcm⁡(12,10)=60\operatorname{lcm}(12, 10) = 60
After 60 s the same picture starts over.

3List the ON stretches inside the first 60 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−9,12−21,24−33,36−45,48−57;yellow:0−6,10−16,20−26,30−36,40−46,50−56\text{white}: 0-9, 12-21, 24-33, 36-45, 48-57;\quad \text{yellow}: 0-6, 10-16, 20-26, 30-36, 40-46, 50-56
Two rows of intervals, ready to compare.

4Find the overlaps in one 60-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
27 s per 60-second cycle27\ \text{s per 60-second cycle}
The short problem is solved: 27 s.

5Scale up to the full 180 seconds

#5 Look for a Pattern 6.NS.B.4
180 seconds is 3 whole cycles of 60 seconds, and each cycle looks the same.
27×3=8127 \times 3 = 81
Both lights are on for 81 seconds in all.
Answer: 81 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 9 of every 12 seconds and yellow 6 of every 10, so an overlap of 81 out of 180 seconds is in the range those two fractions suggest.

Another way: Mark all 180 seconds on two timelines and count the seconds shaded twice; it gives 81, but takes 3 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 12 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 60 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 5 easy answer: 26 seconds

A white light stays on for 99 seconds and then off for 66 seconds, while a yellow light stays on for 77 seconds and then off for 33 seconds. If both lights are switched on at the same moment, for how many seconds during the first 11 minute are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 9 seconds then off 6; the yellow one is on 7 seconds then off 3. Both start on together. We need the total time in the first 60 seconds when both are on at once.

Givens
  • White: on 9 s, off 6 s, repeating.
  • Yellow: on 7 s, off 3 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 60 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 60 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 30-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
9+6=15,7+3=109 + 6 = 15,\quad 7 + 3 = 10
White repeats every 15 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 15 and of 10.
lcm⁡(15,10)=30\operatorname{lcm}(15, 10) = 30
After 30 s the same picture starts over.

3List the ON stretches inside the first 30 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−9,15−24;yellow:0−7,10−17,20−27\text{white}: 0-9, 15-24;\quad \text{yellow}: 0-7, 10-17, 20-27
Two rows of intervals, ready to compare.

4Find the overlaps in one 30-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
13 s per 30-second cycle13\ \text{s per 30-second cycle}
The short problem is solved: 13 s.

5Scale up to the full 60 seconds

#5 Look for a Pattern 6.NS.B.4
60 seconds is 2 whole cycles of 30 seconds, and each cycle looks the same.
13×2=2613 \times 2 = 26
Both lights are on for 26 seconds in all.
Answer: 26 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 9 of every 15 seconds and yellow 7 of every 10, so an overlap of 26 out of 60 seconds is in the range those two fractions suggest.

Another way: Mark all 60 seconds on two timelines and count the seconds shaded twice; it gives 26, but takes 2 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 15 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 30 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 6 medium answer: 120 seconds

A white light stays on for 1010 seconds and then off for 55 seconds, while a yellow light stays on for 44 seconds and then off for 44 seconds. If both lights are switched on at the same moment, for how many seconds during the first 66 minute are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 10 seconds then off 5; the yellow one is on 4 seconds then off 4. Both start on together. We need the total time in the first 360 seconds when both are on at once.

Givens
  • White: on 10 s, off 5 s, repeating.
  • Yellow: on 4 s, off 4 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 360 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 360 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 120-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
10+5=15,4+4=810 + 5 = 15,\quad 4 + 4 = 8
White repeats every 15 s, yellow every 8 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 15 and of 8.
lcm⁡(15,8)=120\operatorname{lcm}(15, 8) = 120
After 120 s the same picture starts over.

3List the ON stretches inside the first 120 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−10,15−25,30−40,45−55,60−70,75−85,90−100,105−115;yellow:0−4,8−12,16−20,24−28,32−36,40−44,48−52,56−60,64−68,72−76,80−84,88−92,96−100,104−108,112−116\text{white}: 0-10, 15-25, 30-40, 45-55, 60-70, 75-85, 90-100, 105-115;\quad \text{yellow}: 0-4, 8-12, 16-20, 24-28, 32-36, 40-44, 48-52, 56-60, 64-68, 72-76, 80-84, 88-92, 96-100, 104-108, 112-116
Two rows of intervals, ready to compare.

4Find the overlaps in one 120-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
40 s per 120-second cycle40\ \text{s per 120-second cycle}
The short problem is solved: 40 s.

5Scale up to the full 360 seconds

#5 Look for a Pattern 6.NS.B.4
360 seconds is 3 whole cycles of 120 seconds, and each cycle looks the same.
40×3=12040 \times 3 = 120
Both lights are on for 120 seconds in all.
Answer: 120 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 10 of every 15 seconds and yellow 4 of every 8, so an overlap of 120 out of 360 seconds is in the range those two fractions suggest.

Another way: Mark all 360 seconds on two timelines and count the seconds shaded twice; it gives 120, but takes 3 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 15 and 8.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 120 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 7 medium answer: 60 seconds

A white light stays on for 1212 seconds and then off for 66 seconds, while a yellow light stays on for 55 seconds and then off for 55 seconds. If both lights are switched on at the same moment, for how many seconds during the first 33 minute are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 12 seconds then off 6; the yellow one is on 5 seconds then off 5. Both start on together. We need the total time in the first 180 seconds when both are on at once.

Givens
  • White: on 12 s, off 6 s, repeating.
  • Yellow: on 5 s, off 5 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 180 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 180 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 90-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
12+6=18,5+5=1012 + 6 = 18,\quad 5 + 5 = 10
White repeats every 18 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 18 and of 10.
lcm⁡(18,10)=90\operatorname{lcm}(18, 10) = 90
After 90 s the same picture starts over.

3List the ON stretches inside the first 90 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−12,18−30,36−48,54−66,72−84;yellow:0−5,10−15,20−25,30−35,40−45,50−55,60−65,70−75,80−85\text{white}: 0-12, 18-30, 36-48, 54-66, 72-84;\quad \text{yellow}: 0-5, 10-15, 20-25, 30-35, 40-45, 50-55, 60-65, 70-75, 80-85
Two rows of intervals, ready to compare.

4Find the overlaps in one 90-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
30 s per 90-second cycle30\ \text{s per 90-second cycle}
The short problem is solved: 30 s.

5Scale up to the full 180 seconds

#5 Look for a Pattern 6.NS.B.4
180 seconds is 2 whole cycles of 90 seconds, and each cycle looks the same.
30×2=6030 \times 2 = 60
Both lights are on for 60 seconds in all.
Answer: 60 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 12 of every 18 seconds and yellow 5 of every 10, so an overlap of 60 out of 180 seconds is in the range those two fractions suggest.

Another way: Mark all 180 seconds on two timelines and count the seconds shaded twice; it gives 60, but takes 2 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 18 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 90 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 8 medium answer: 24 seconds

A white light stays on for 66 seconds and then off for 66 seconds, while a yellow light stays on for 99 seconds and then off for 33 seconds. If both lights are switched on at the same moment, for how many seconds during the first 4848 seconds are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 6 seconds then off 6; the yellow one is on 9 seconds then off 3. Both start on together. We need the total time in the first 48 seconds when both are on at once.

Givens
  • White: on 6 s, off 6 s, repeating.
  • Yellow: on 9 s, off 3 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 48 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 48 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 12-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
6+6=12,9+3=126 + 6 = 12,\quad 9 + 3 = 12
White repeats every 12 s, yellow every 12 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 12 and of 12.
lcm⁡(12,12)=12\operatorname{lcm}(12, 12) = 12
After 12 s the same picture starts over.

3List the ON stretches inside the first 12 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−6;yellow:0−9\text{white}: 0-6;\quad \text{yellow}: 0-9
Two rows of intervals, ready to compare.

4Find the overlaps in one 12-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
6 s per 12-second cycle6\ \text{s per 12-second cycle}
The short problem is solved: 6 s.

5Scale up to the full 48 seconds

#5 Look for a Pattern 6.NS.B.4
48 seconds is 4 whole cycles of 12 seconds, and each cycle looks the same.
6×4=246 \times 4 = 24
Both lights are on for 24 seconds in all.
Answer: 24 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 6 of every 12 seconds and yellow 9 of every 12, so an overlap of 24 out of 48 seconds is in the range those two fractions suggest.

Another way: Mark all 48 seconds on two timelines and count the seconds shaded twice; it gives 24, but takes 4 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 12 and 12.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 12 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 9 hard answer: 38 seconds

A white light stays on for 1010 seconds and then off for 88 seconds, while a yellow light stays on for 1111 seconds and then off for 11 seconds. If both lights are switched on at the same moment, for how many seconds during the first 7272 seconds are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 10 seconds then off 8; the yellow one is on 11 seconds then off 1. Both start on together. We need the total time in the first 72 seconds when both are on at once.

Givens
  • White: on 10 s, off 8 s, repeating.
  • Yellow: on 11 s, off 1 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 72 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 72 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 36-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
10+8=18,11+1=1210 + 8 = 18,\quad 11 + 1 = 12
White repeats every 18 s, yellow every 12 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 18 and of 12.
lcm⁡(18,12)=36\operatorname{lcm}(18, 12) = 36
After 36 s the same picture starts over.

3List the ON stretches inside the first 36 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−10,18−28;yellow:0−11,12−23,24−35\text{white}: 0-10, 18-28;\quad \text{yellow}: 0-11, 12-23, 24-35
Two rows of intervals, ready to compare.

4Find the overlaps in one 36-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
19 s per 36-second cycle19\ \text{s per 36-second cycle}
The short problem is solved: 19 s.

5Scale up to the full 72 seconds

#5 Look for a Pattern 6.NS.B.4
72 seconds is 2 whole cycles of 36 seconds, and each cycle looks the same.
19×2=3819 \times 2 = 38
Both lights are on for 38 seconds in all.
Answer: 38 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 10 of every 18 seconds and yellow 11 of every 12, so an overlap of 38 out of 72 seconds is in the range those two fractions suggest.

Another way: Mark all 72 seconds on two timelines and count the seconds shaded twice; it gives 38, but takes 2 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 18 and 12.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 36 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 10 hard answer: 18 seconds

A white light stays on for 77 seconds and then off for 88 seconds, while a yellow light stays on for 44 seconds and then off for 66 seconds. If both lights are switched on at the same moment, for how many seconds during the first 9090 seconds are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 7 seconds then off 8; the yellow one is on 4 seconds then off 6. Both start on together. We need the total time in the first 90 seconds when both are on at once.

Givens
  • White: on 7 s, off 8 s, repeating.
  • Yellow: on 4 s, off 6 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 90 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 90 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 30-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
7+8=15,4+6=107 + 8 = 15,\quad 4 + 6 = 10
White repeats every 15 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 15 and of 10.
lcm⁡(15,10)=30\operatorname{lcm}(15, 10) = 30
After 30 s the same picture starts over.

3List the ON stretches inside the first 30 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−7,15−22;yellow:0−4,10−14,20−24\text{white}: 0-7, 15-22;\quad \text{yellow}: 0-4, 10-14, 20-24
Two rows of intervals, ready to compare.

4Find the overlaps in one 30-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
6 s per 30-second cycle6\ \text{s per 30-second cycle}
The short problem is solved: 6 s.

5Scale up to the full 90 seconds

#5 Look for a Pattern 6.NS.B.4
90 seconds is 3 whole cycles of 30 seconds, and each cycle looks the same.
6×3=186 \times 3 = 18
Both lights are on for 18 seconds in all.
Answer: 18 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 7 of every 15 seconds and yellow 4 of every 10, so an overlap of 18 out of 90 seconds is in the range those two fractions suggest.

Another way: Mark all 90 seconds on two timelines and count the seconds shaded twice; it gives 18, but takes 3 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 15 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 30 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 11 hard answer: 48 seconds

A white light stays on for 1111 seconds and then off for 44 seconds, while a yellow light stays on for 77 seconds and then off for 33 seconds. If both lights are switched on at the same moment, for how many seconds during the first 9090 seconds are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 11 seconds then off 4; the yellow one is on 7 seconds then off 3. Both start on together. We need the total time in the first 90 seconds when both are on at once.

Givens
  • White: on 11 s, off 4 s, repeating.
  • Yellow: on 7 s, off 3 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 90 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 90 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 30-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
11+4=15,7+3=1011 + 4 = 15,\quad 7 + 3 = 10
White repeats every 15 s, yellow every 10 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 15 and of 10.
lcm⁡(15,10)=30\operatorname{lcm}(15, 10) = 30
After 30 s the same picture starts over.

3List the ON stretches inside the first 30 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−11,15−26;yellow:0−7,10−17,20−27\text{white}: 0-11, 15-26;\quad \text{yellow}: 0-7, 10-17, 20-27
Two rows of intervals, ready to compare.

4Find the overlaps in one 30-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
16 s per 30-second cycle16\ \text{s per 30-second cycle}
The short problem is solved: 16 s.

5Scale up to the full 90 seconds

#5 Look for a Pattern 6.NS.B.4
90 seconds is 3 whole cycles of 30 seconds, and each cycle looks the same.
16×3=4816 \times 3 = 48
Both lights are on for 48 seconds in all.
Answer: 48 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 11 of every 15 seconds and yellow 7 of every 10, so an overlap of 48 out of 90 seconds is in the range those two fractions suggest.

Another way: Mark all 90 seconds on two timelines and count the seconds shaded twice; it gives 48, but takes 3 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 15 and 10.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 30 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.
Variant 12 hard answer: 44 seconds

A white light stays on for 77 seconds and then off for 55 seconds, while a yellow light stays on for 66 seconds and then off for 22 seconds. If both lights are switched on at the same moment, for how many seconds during the first 9696 seconds are both lights on at the same time?

Show solution
1 · Understandwhat's really being asked

Two lights blink forever. The white one is on 7 seconds then off 5; the yellow one is on 6 seconds then off 2. Both start on together. We need the total time in the first 96 seconds when both are on at once.

Givens
  • White: on 7 s, off 5 s, repeating.
  • Yellow: on 6 s, off 2 s, repeating.
  • Both switch on at the same moment.
Unknowns
  • How many of the first 96 seconds have both lights on.
Constraints
  • Each light keeps its own rhythm exactly; nothing resets them.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#9 Solve an Easier Related Problem

Checking all 96 seconds one by one is slow. The pair of rhythms repeats after their least common multiple, so solve the shorter 24-second problem and multiply.

3 · Execute5 carry out the plan

1Find each light's full cycle length

#5 Look for a Pattern 4.OA.B.4
A cycle is one on-stretch plus one off-stretch.
7+5=12,6+2=87 + 5 = 12,\quad 6 + 2 = 8
White repeats every 12 s, yellow every 8 s.

2Find when the whole pattern repeats (the LCM)

#5 Look for a Pattern 6.NS.B.4
Both lights are back at the start together at the first time that is a multiple of 12 and of 8.
lcm⁡(12,8)=24\operatorname{lcm}(12, 8) = 24
After 24 s the same picture starts over.

3List the ON stretches inside the first 24 seconds

#2 Make a Systematic List 4.OA.B.4
Write each light's lit intervals, counting seconds from 0.
white:0−7,12−19;yellow:0−6,8−14,16−22\text{white}: 0-7, 12-19;\quad \text{yellow}: 0-6, 8-14, 16-22
Two rows of intervals, ready to compare.

4Find the overlaps in one 24-second cycle

#9 Solve an Easier Related Problem 4.OA.A.3
Where a white stretch and a yellow stretch cover the same second, both lights are on. Adding those shared seconds gives the total for one cycle.
11 s per 24-second cycle11\ \text{s per 24-second cycle}
The short problem is solved: 11 s.

5Scale up to the full 96 seconds

#5 Look for a Pattern 6.NS.B.4
96 seconds is 4 whole cycles of 24 seconds, and each cycle looks the same.
11×4=4411 \times 4 = 44
Both lights are on for 44 seconds in all.
Answer: 44 seconds
4 · Reviewdoes it hold up?

Sanity-check the size: white is on 7 of every 12 seconds and yellow 6 of every 8, so an overlap of 44 out of 96 seconds is in the range those two fractions suggest.

Another way: Mark all 96 seconds on two timelines and count the seconds shaded twice; it gives 44, but takes 4 times as long as using the cycle.

Standardsmin grade 6
  • 4.OA.B.4 Find all factor pairs for a whole number in the range 1-100 and recognize multiples — Treating each light's lit seconds as a repeating pattern of period 12 and 8.
  • 6.NS.B.4 Find the greatest common factor and least common multiple of two whole numbers — Finding 24 as the least common multiple, the length after which the combined pattern repeats.
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations — Counting the overlapping seconds in one cycle and scaling to the full window.
💡Takeaway. Two repeating rhythms make one bigger rhythm that repeats at their least common multiple -- solve one of those and multiply.