Practice · Solve for the opposite-hands moment, then step back · Sensim Math

Variant 1 easy

A clock shows a time between $7$ o'clock and $8$ o'clock. Exactly $2$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 7:00 and before 8:00. 2 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $7$ and $8$ o'clock.
$2$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 7:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $2$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 7, the hour hand has moved from the 7 (which is at

210

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 210 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(210 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 30 \;\Rightarrow\; m = \frac{60}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(210 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 390

and

m = \frac{780}{11}

, which is more than

60

. That moment is after 8 o'clock, so it is not allowed.

m = \frac{780}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 2 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{60}{11}

minutes after 7, so now is

2

minutes earlier.

\frac{60}{11} - 2 = \frac{60 - 22}{11} = \frac{38}{11} = 3\frac{5}{11}

Still after 7:00, so the current time is inside the given hour.

Answer:

3\frac{5}{11}

minutes past 7 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{60}{11} \approx 5.5$ minutes, the minute hand is at about $33$ degrees and the hour hand at about $213$ degrees, $180$ degrees apart. The current time, a little after 7:03, is between 7 and 8 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 7:00 it is $210$ degrees behind, and it must close to $180$ degrees behind, a gain of $30$ degrees, taking $30 \div \frac{11}{2} = \frac{60}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 2 easy

A clock shows a time between $7$ o'clock and $8$ o'clock. Exactly $4$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 7:00 and before 8:00. 4 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $7$ and $8$ o'clock.
$4$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 7:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $4$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 7, the hour hand has moved from the 7 (which is at

210

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 210 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(210 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 30 \;\Rightarrow\; m = \frac{60}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(210 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 390

and

m = \frac{780}{11}

, which is more than

60

. That moment is after 8 o'clock, so it is not allowed.

m = \frac{780}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 4 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{60}{11}

minutes after 7, so now is

4

minutes earlier.

\frac{60}{11} - 4 = \frac{60 - 44}{11} = \frac{16}{11} = 1\frac{5}{11}

Still after 7:00, so the current time is inside the given hour.

Answer:

1\frac{5}{11}

minutes past 7 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{60}{11} \approx 5.5$ minutes, the minute hand is at about $33$ degrees and the hour hand at about $213$ degrees, $180$ degrees apart. The current time, a little after 7:01, is between 7 and 8 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 7:00 it is $210$ degrees behind, and it must close to $180$ degrees behind, a gain of $30$ degrees, taking $30 \div \frac{11}{2} = \frac{60}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 3 easy

A clock shows a time between $8$ o'clock and $9$ o'clock. Exactly $5$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 8:00 and before 9:00. 5 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $8$ and $9$ o'clock.
$5$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 8:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $5$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 8, the hour hand has moved from the 8 (which is at

240

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 240 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(240 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 60 \;\Rightarrow\; m = \frac{120}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(240 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 420

and

m = \frac{840}{11}

, which is more than

60

. That moment is after 9 o'clock, so it is not allowed.

m = \frac{840}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 5 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{120}{11}

minutes after 8, so now is

5

minutes earlier.

\frac{120}{11} - 5 = \frac{120 - 55}{11} = \frac{65}{11} = 5\frac{10}{11}

Still after 8:00, so the current time is inside the given hour.

Answer:

5\frac{10}{11}

minutes past 8 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{120}{11} \approx 10.9$ minutes, the minute hand is at about $65$ degrees and the hour hand at about $245$ degrees, $180$ degrees apart. The current time, a little after 8:05, is between 8 and 9 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 8:00 it is $240$ degrees behind, and it must close to $180$ degrees behind, a gain of $60$ degrees, taking $60 \div \frac{11}{2} = \frac{120}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 4 easy

A clock shows a time between $8$ o'clock and $9$ o'clock. Exactly $9$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 8:00 and before 9:00. 9 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $8$ and $9$ o'clock.
$9$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 8:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $9$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 8, the hour hand has moved from the 8 (which is at

240

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 240 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(240 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 60 \;\Rightarrow\; m = \frac{120}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(240 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 420

and

m = \frac{840}{11}

, which is more than

60

. That moment is after 9 o'clock, so it is not allowed.

m = \frac{840}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 9 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{120}{11}

minutes after 8, so now is

9

minutes earlier.

\frac{120}{11} - 9 = \frac{120 - 99}{11} = \frac{21}{11} = 1\frac{10}{11}

Still after 8:00, so the current time is inside the given hour.

Answer:

1\frac{10}{11}

minutes past 8 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{120}{11} \approx 10.9$ minutes, the minute hand is at about $65$ degrees and the hour hand at about $245$ degrees, $180$ degrees apart. The current time, a little after 8:01, is between 8 and 9 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 8:00 it is $240$ degrees behind, and it must close to $180$ degrees behind, a gain of $60$ degrees, taking $60 \div \frac{11}{2} = \frac{120}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 5 medium

A clock shows a time between $9$ o'clock and $10$ o'clock. Exactly $10$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 9:00 and before 10:00. 10 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $9$ and $10$ o'clock.
$10$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 9:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $10$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 9, the hour hand has moved from the 9 (which is at

270

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 270 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(270 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 90 \;\Rightarrow\; m = \frac{180}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(270 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 450

and

m = \frac{900}{11}

, which is more than

60

. That moment is after 10 o'clock, so it is not allowed.

m = \frac{900}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 10 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{180}{11}

minutes after 9, so now is

10

minutes earlier.

\frac{180}{11} - 10 = \frac{180 - 110}{11} = \frac{70}{11} = 6\frac{4}{11}

Still after 9:00, so the current time is inside the given hour.

Answer:

6\frac{4}{11}

minutes past 9 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{180}{11} \approx 16.4$ minutes, the minute hand is at about $98$ degrees and the hour hand at about $278$ degrees, $180$ degrees apart. The current time, a little after 9:06, is between 9 and 10 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 9:00 it is $270$ degrees behind, and it must close to $180$ degrees behind, a gain of $90$ degrees, taking $90 \div \frac{11}{2} = \frac{180}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 6 medium

A clock shows a time between $9$ o'clock and $10$ o'clock. Exactly $6$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 9:00 and before 10:00. 6 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $9$ and $10$ o'clock.
$6$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 9:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $6$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 9, the hour hand has moved from the 9 (which is at

270

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 270 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(270 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 90 \;\Rightarrow\; m = \frac{180}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(270 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 450

and

m = \frac{900}{11}

, which is more than

60

. That moment is after 10 o'clock, so it is not allowed.

m = \frac{900}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 6 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{180}{11}

minutes after 9, so now is

6

minutes earlier.

\frac{180}{11} - 6 = \frac{180 - 66}{11} = \frac{114}{11} = 10\frac{4}{11}

Still after 9:00, so the current time is inside the given hour.

Answer:

10\frac{4}{11}

minutes past 9 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{180}{11} \approx 16.4$ minutes, the minute hand is at about $98$ degrees and the hour hand at about $278$ degrees, $180$ degrees apart. The current time, a little after 9:10, is between 9 and 10 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 9:00 it is $270$ degrees behind, and it must close to $180$ degrees behind, a gain of $90$ degrees, taking $90 \div \frac{11}{2} = \frac{180}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 7 medium

A clock shows a time between $10$ o'clock and $11$ o'clock. Exactly $3$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 10:00 and before 11:00. 3 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $10$ and $11$ o'clock.
$3$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 10:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $3$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 10, the hour hand has moved from the 10 (which is at

300

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 300 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(300 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 120 \;\Rightarrow\; m = \frac{240}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(300 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 480

and

m = \frac{960}{11}

, which is more than

60

. That moment is after 11 o'clock, so it is not allowed.

m = \frac{960}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 3 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{240}{11}

minutes after 10, so now is

3

minutes earlier.

\frac{240}{11} - 3 = \frac{240 - 33}{11} = \frac{207}{11} = 18\frac{9}{11}

Still after 10:00, so the current time is inside the given hour.

Answer:

18\frac{9}{11}

minutes past 10 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{240}{11} \approx 21.8$ minutes, the minute hand is at about $131$ degrees and the hour hand at about $311$ degrees, $180$ degrees apart. The current time, a little after 10:18, is between 10 and 11 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 10:00 it is $300$ degrees behind, and it must close to $180$ degrees behind, a gain of $120$ degrees, taking $120 \div \frac{11}{2} = \frac{240}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 8 medium

A clock shows a time between $10$ o'clock and $11$ o'clock. Exactly $7$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 10:00 and before 11:00. 7 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $10$ and $11$ o'clock.
$7$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 10:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $7$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 10, the hour hand has moved from the 10 (which is at

300

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 300 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(300 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 120 \;\Rightarrow\; m = \frac{240}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(300 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 480

and

m = \frac{960}{11}

, which is more than

60

. That moment is after 11 o'clock, so it is not allowed.

m = \frac{960}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 7 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{240}{11}

minutes after 10, so now is

7

minutes earlier.

\frac{240}{11} - 7 = \frac{240 - 77}{11} = \frac{163}{11} = 14\frac{9}{11}

Still after 10:00, so the current time is inside the given hour.

Answer:

14\frac{9}{11}

minutes past 10 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{240}{11} \approx 21.8$ minutes, the minute hand is at about $131$ degrees and the hour hand at about $311$ degrees, $180$ degrees apart. The current time, a little after 10:14, is between 10 and 11 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 10:00 it is $300$ degrees behind, and it must close to $180$ degrees behind, a gain of $120$ degrees, taking $120 \div \frac{11}{2} = \frac{240}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 9 hard

A clock shows a time between $10$ o'clock and $11$ o'clock. Exactly $12$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 10:00 and before 11:00. 12 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $10$ and $11$ o'clock.
$12$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 10:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $12$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 10, the hour hand has moved from the 10 (which is at

300

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 300 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(300 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 120 \;\Rightarrow\; m = \frac{240}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(300 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 480

and

m = \frac{960}{11}

, which is more than

60

. That moment is after 11 o'clock, so it is not allowed.

m = \frac{960}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 12 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{240}{11}

minutes after 10, so now is

12

minutes earlier.

\frac{240}{11} - 12 = \frac{240 - 132}{11} = \frac{108}{11} = 9\frac{9}{11}

Still after 10:00, so the current time is inside the given hour.

Answer:

9\frac{9}{11}

minutes past 10 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{240}{11} \approx 21.8$ minutes, the minute hand is at about $131$ degrees and the hour hand at about $311$ degrees, $180$ degrees apart. The current time, a little after 10:09, is between 10 and 11 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 10:00 it is $300$ degrees behind, and it must close to $180$ degrees behind, a gain of $120$ degrees, taking $120 \div \frac{11}{2} = \frac{240}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 10 hard

A clock shows a time between $11$ o'clock and $12$ o'clock. Exactly $8$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 11:00 and before 12:00. 8 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $11$ and $12$ o'clock.
$8$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 11:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $8$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 11, the hour hand has moved from the 11 (which is at

330

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 330 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(330 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 150 \;\Rightarrow\; m = \frac{300}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(330 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 510

and

m = \frac{1020}{11}

, which is more than

60

. That moment is after 12 o'clock, so it is not allowed.

m = \frac{1020}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 8 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{300}{11}

minutes after 11, so now is

8

minutes earlier.

\frac{300}{11} - 8 = \frac{300 - 88}{11} = \frac{212}{11} = 19\frac{3}{11}

Still after 11:00, so the current time is inside the given hour.

Answer:

19\frac{3}{11}

minutes past 11 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{300}{11} \approx 27.3$ minutes, the minute hand is at about $164$ degrees and the hour hand at about $344$ degrees, $180$ degrees apart. The current time, a little after 11:19, is between 11 and 12 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 11:00 it is $330$ degrees behind, and it must close to $180$ degrees behind, a gain of $150$ degrees, taking $150 \div \frac{11}{2} = \frac{300}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 11 hard

A clock shows a time between $9$ o'clock and $10$ o'clock. Exactly $15$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 9:00 and before 10:00. 15 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $9$ and $10$ o'clock.
$15$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 9:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $15$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 9, the hour hand has moved from the 9 (which is at

270

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 270 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(270 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 90 \;\Rightarrow\; m = \frac{180}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(270 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 450

and

m = \frac{900}{11}

, which is more than

60

. That moment is after 10 o'clock, so it is not allowed.

m = \frac{900}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 15 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{180}{11}

minutes after 9, so now is

15

minutes earlier.

\frac{180}{11} - 15 = \frac{180 - 165}{11} = \frac{15}{11} = 1\frac{4}{11}

Still after 9:00, so the current time is inside the given hour.

Answer:

1\frac{4}{11}

minutes past 9 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{180}{11} \approx 16.4$ minutes, the minute hand is at about $98$ degrees and the hour hand at about $278$ degrees, $180$ degrees apart. The current time, a little after 9:01, is between 9 and 10 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 9:00 it is $270$ degrees behind, and it must close to $180$ degrees behind, a gain of $90$ degrees, taking $90 \div \frac{11}{2} = \frac{180}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Variant 12 hard

A clock shows a time between $11$ o'clock and $12$ o'clock. Exactly $20$ minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

It is now some time after 11:00 and before 12:00. 20 minutes later the two hands of the clock will form a straight line pointing opposite ways. Find the current time.

Givens

The time now is between $11$ and $12$ o'clock.
$20$ minutes from now the hands point in opposite directions.

Unknowns

The current time.

Constraints

The hour hand turns $\frac{1}{2}$ degree per minute; the minute hand turns $6$ degrees per minute.
Opposite directions means the angle between the hands is $180$ degrees.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#3 Eliminate Possibilities

Let the opposite-hands moment be $m$ minutes after 11:00 and write both hand positions in degrees from 12. Setting their difference to $180$ gives candidate values of $m$ ; the range of possible times eliminates all but one. Then subtract $20$ minutes.

3 · Execute4 carry out the plan

1Place the hands

#1 Draw a Diagram 7.RP.A.1

At

m

minutes after 11, the hour hand has moved from the 11 (which is at

330

degrees) by

\frac{1}{2}

degree per minute, and the minute hand has moved

6

degrees per minute from the 12.

\text{hour} = 330 + \frac{1}{2}m,\qquad \text{minute} = 6m

Each hand is just a starting angle plus a rate times the minutes.

2Set the gap to 180 degrees

#13 Convert to Algebra 7.EE.B.4

The difference of the two angles is

180

either way round.

\left(330 + \frac{1}{2}m\right) - 6m = 180 \;\Rightarrow\; \frac{11}{2}m = 150 \;\Rightarrow\; m = \frac{300}{11}

This is the case where the hour hand is still ahead of the minute hand.

3Check the other case

#3 Eliminate Possibilities 7.NS.A.3

If the minute hand were

180

degrees ahead instead, then

6m - \left(330 + \frac{1}{2}m\right) = 180

, so

\frac{11}{2}m = 510

and

m = \frac{1020}{11}

, which is more than

60

. That moment is after 12 o'clock, so it is not allowed.

m = \frac{1020}{11} > 60

The time must stay within this hour, which rules this case out.

4Step back 20 minutes

#13 Convert to Algebra 7.NS.A.3

The opposite-hands moment is

\frac{300}{11}

minutes after 11, so now is

20

minutes earlier.

\frac{300}{11} - 20 = \frac{300 - 220}{11} = \frac{80}{11} = 7\frac{3}{11}

Still after 11:00, so the current time is inside the given hour.

Answer:

7\frac{3}{11}

minutes past 11 o'clock

4 · Reviewdoes it hold up?

At $m = \frac{300}{11} \approx 27.3$ minutes, the minute hand is at about $164$ degrees and the hour hand at about $344$ degrees, $180$ degrees apart. The current time, a little after 11:07, is between 11 and 12 o'clock.

Another way: Use relative speed: the minute hand gains $\frac{11}{2}$ degrees per minute on the hour hand. At 11:00 it is $330$ degrees behind, and it must close to $180$ degrees behind, a gain of $150$ degrees, taking $150 \div \frac{11}{2} = \frac{300}{11}$ minutes.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the opposite-hands condition as a linear equation in $m$.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Using the hands' rates of $\frac{1}{2}$ and $6$ degrees per minute.
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers — Working with fractional minutes and rejecting the out-of-range case.

💡Takeaway. Each clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.

Solve for the opposite-hands moment, then step back · 12 practice problems

Generated variants — 12

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