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Reason about a fast or slow clock · 8 practice problems

3.MD.A.1

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Generated variants — 8

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

Variant 1 easy

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $1{:}00$ and then looked at it $4$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $1{:}00$ , and the clock after the arrow labeled " $4$ hours later" reads $5{:}08$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 1:00. After 4 real hours it read 5:08 instead of 5:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 1:00.
After 4 actual hours the clock displays 5:08.
The figure shows a digital display 01:00 with an arrow labeled '4 hours later' pointing to a display reading 05:08.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

4 hours after 1:00, a correct clock would read 5:00.

1{:}00 + 4 \text{ hours} = 5{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 5:08 but should show 5:00, so it has gained 8 minutes over the 4 hours.

5{:}08 - 5{:}00 = 8 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 8 extra minutes were gained evenly across 4 hours, so divide 8 by 4.

8 \div 4 = 2

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 2 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 2 minutes fast each hour for 4 hours is 8 minutes, so the clock shows 5:00 + 8 min = 5:08, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

Variant 2 easy

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $2{:}00$ and then looked at it $3$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $2{:}00$ , and the clock after the arrow labeled " $3$ hours later" reads $5{:}09$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 2:00. After 3 real hours it read 5:09 instead of 5:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 2:00.
After 3 actual hours the clock displays 5:09.
The figure shows a digital display 02:00 with an arrow labeled '3 hours later' pointing to a display reading 05:09.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

3 hours after 2:00, a correct clock would read 5:00.

2{:}00 + 3 \text{ hours} = 5{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 5:09 but should show 5:00, so it has gained 9 minutes over the 3 hours.

5{:}09 - 5{:}00 = 9 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 9 extra minutes were gained evenly across 3 hours, so divide 9 by 3.

9 \div 3 = 3

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 3 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 3 minutes fast each hour for 3 hours is 9 minutes, so the clock shows 5:00 + 9 min = 5:09, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

Variant 3 medium

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $10{:}00$ and then looked at it $3$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $10{:}00$ , and the clock after the arrow labeled " $3$ hours later" reads $1{:}03$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 10:00. After 3 real hours it read 1:03 instead of 1:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 10:00.
After 3 actual hours the clock displays 1:03.
The figure shows a digital display 10:00 with an arrow labeled '3 hours later' pointing to a display reading 01:03.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

3 hours after 10:00, a correct clock would read 1:00.

10{:}00 + 3 \text{ hours} = 1{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 1:03 but should show 1:00, so it has gained 3 minutes over the 3 hours.

1{:}03 - 1{:}00 = 3 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 3 extra minutes were gained evenly across 3 hours, so divide 3 by 3.

3 \div 3 = 1

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 1 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 1 minutes fast each hour for 3 hours is 3 minutes, so the clock shows 1:00 + 3 min = 1:03, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

Variant 4 medium

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $3{:}00$ and then looked at it $2$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $3{:}00$ , and the clock after the arrow labeled " $2$ hours later" reads $5{:}10$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 3:00. After 2 real hours it read 5:10 instead of 5:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 3:00.
After 2 actual hours the clock displays 5:10.
The figure shows a digital display 03:00 with an arrow labeled '2 hours later' pointing to a display reading 05:10.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

2 hours after 3:00, a correct clock would read 5:00.

3{:}00 + 2 \text{ hours} = 5{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 5:10 but should show 5:00, so it has gained 10 minutes over the 2 hours.

5{:}10 - 5{:}00 = 10 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 10 extra minutes were gained evenly across 2 hours, so divide 10 by 2.

10 \div 2 = 5

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 5 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 5 minutes fast each hour for 2 hours is 10 minutes, so the clock shows 5:00 + 10 min = 5:10, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

Variant 5 medium

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $4{:}00$ and then looked at it $2$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $4{:}00$ , and the clock after the arrow labeled " $2$ hours later" reads $6{:}12$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 4:00. After 2 real hours it read 6:12 instead of 6:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 4:00.
After 2 actual hours the clock displays 6:12.
The figure shows a digital display 04:00 with an arrow labeled '2 hours later' pointing to a display reading 06:12.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

2 hours after 4:00, a correct clock would read 6:00.

4{:}00 + 2 \text{ hours} = 6{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 6:12 but should show 6:00, so it has gained 12 minutes over the 2 hours.

6{:}12 - 6{:}00 = 12 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 12 extra minutes were gained evenly across 2 hours, so divide 12 by 2.

12 \div 2 = 6

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 6 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 6 minutes fast each hour for 2 hours is 12 minutes, so the clock shows 6:00 + 12 min = 6:12, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

Variant 6 medium

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $9{:}00$ and then looked at it $3$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $9{:}00$ , and the clock after the arrow labeled " $3$ hours later" reads $12{:}12$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 9:00. After 3 real hours it read 12:12 instead of 12:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 9:00.
After 3 actual hours the clock displays 12:12.
The figure shows a digital display 09:00 with an arrow labeled '3 hours later' pointing to a display reading 12:12.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

3 hours after 9:00, a correct clock would read 12:00.

9{:}00 + 3 \text{ hours} = 12{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 12:12 but should show 12:00, so it has gained 12 minutes over the 3 hours.

12{:}12 - 12{:}00 = 12 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 12 extra minutes were gained evenly across 3 hours, so divide 12 by 3.

12 \div 3 = 4

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 4 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 4 minutes fast each hour for 3 hours is 12 minutes, so the clock shows 12:00 + 12 min = 12:12, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

Variant 7 hard

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $6{:}00$ and then looked at it $5$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $6{:}00$ , and the clock after the arrow labeled " $5$ hours later" reads $11{:}15$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 6:00. After 5 real hours it read 11:15 instead of 11:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 6:00.
After 5 actual hours the clock displays 11:15.
The figure shows a digital display 06:00 with an arrow labeled '5 hours later' pointing to a display reading 11:15.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

5 hours after 6:00, a correct clock would read 11:00.

6{:}00 + 5 \text{ hours} = 11{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 11:15 but should show 11:00, so it has gained 15 minutes over the 5 hours.

11{:}15 - 11{:}00 = 15 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 15 extra minutes were gained evenly across 5 hours, so divide 15 by 5.

15 \div 5 = 3

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 3 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 3 minutes fast each hour for 5 hours is 15 minutes, so the clock shows 11:00 + 15 min = 11:15, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

Variant 8 hard

Hana accidentally dropped her clock, and after that the clock suddenly began running fast. To find out how fast it was running, Hana set it to exactly $12{:}00$ and then looked at it $4$ hours later, and it looked like this. By how many minutes per hour does Hana's clock run fast?

There are two digital clock displays. The clock that was set correctly at the start reads $12{:}00$ , and the clock after the arrow labeled " $4$ hours later" reads $4{:}20$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Hana set her clock to exactly 12:00. After 4 real hours it read 4:20 instead of 4:00, because it runs fast. Find how many minutes per hour the clock gains.

Givens

The clock was set correctly to 12:00.
After 4 actual hours the clock displays 4:20.
The figure shows a digital display 12:00 with an arrow labeled '4 hours later' pointing to a display reading 04:20.
The clock runs fast at a steady rate.

Unknowns

How many minutes per hour the clock runs fast.

Constraints

The fast-running rate is constant each hour.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

First find the total extra time the clock gained over the whole span (subproblem 1), then share that extra time equally across the hours to get the gain per hour (subproblem 2).

3 · Execute3 carry out the plan

1Find what a correct clock would show

#7 Identify Subproblems 3.MD.A.1

4 hours after 12:00, a correct clock would read 4:00.

12{:}00 + 4 \text{ hours} = 4{:}00

Adding the hours to the start shows where a good clock should be.

2Find the total extra minutes

#7 Identify Subproblems 3.MD.A.1

Hana's clock shows 4:20 but should show 4:00, so it has gained 20 minutes over the 4 hours.

4{:}20 - 4{:}00 = 20 \text{ minutes}

The difference between the displayed time and the true time is the total time the clock ran ahead.

3Share the extra over each hour

#8 Analyze the Units 3.MD.A.1

The 20 extra minutes were gained evenly across 4 hours, so divide 20 by 4.

20 \div 4 = 5

Splitting the total gain equally among the hours gives the gain in each single hour.

Answer: 5 minutes per hour

4 · Reviewdoes it hold up?

Check by building up: 5 minutes fast each hour for 4 hours is 20 minutes, so the clock shows 4:00 + 20 min = 4:20, matching the figure.

Another way: Look for a pattern (tool 5): add the per-hour gain one hour at a time and check that after all the hours the displayed time matches the figure.

Standardsmin grade 3

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems — Comparing the fast clock's displayed time with the true time and dividing the gained minutes across the hours.

💡Takeaway. Find the total minutes the clock got ahead, then split it across the hours: just Grade 3 time-and-dividing!

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