Clocks and Calendar Problems

What do clock problems ask?

They ask for the angle between the two hands at a given time. They may also ask when the hands meet or make a right angle.

Some ask about a clock that runs fast or slow.

What do calendar problems ask?

They give one date and its weekday. You find the weekday of another date.

Leap years and month lengths are the only tricky parts.

Why group them together?

Both are about cycles. A clock face cycles every 12 hours. A week cycles every 7 days.

Both are solved with a simple rule and a little arithmetic.

How fast does the minute hand move?

The minute hand covers 360 degrees in 60 minutes. That is 6 degrees every minute.

It covers 30 degrees every five minutes.

How fast does the hour hand move?

The hour hand covers 360 degrees in 12 hours. That is 30 degrees every hour.

In one minute it moves half a degree.

Why this works: 30 degrees in 60 minutes is half a degree a minute.

What does this give us?

The minute hand gains 5.5 degrees on the hour hand every minute.

This one number solves most clock problems.

What is the formula?

Angle = |30 x H – 5.5 x M|. Here H is the hour and M is the minutes.

If the answer is above 180, subtract it from 360.

Can you see it solved?

Find the angle at 5:00. Here H is 5 and M is 0.

30 x 5 = 150. So the angle is 150 degrees.

Why this works: at 5:00 the minute hand is at 12 and the hour hand is at 5. That is five marks of 30 degrees.

Can you try one with minutes?

Find the angle at 8:30. Here 30 x 8 = 240 and 5.5 x 30 = 165.

The gap is 240 – 165 = 75. The angle is 75 degrees.

Why this works: the hour hand has moved 15 degrees past the 8 mark, so it sits at 255 degrees. The minute hand is at 180. The gap is 75.

Can you try one that needs 360?

Find the angle at 10:20. 30 x 10 = 300 and 5.5 x 20 = 110.

The gap is 300 – 110 = 190. That is above 180, so the angle is 360 – 190 = 170 degrees.

Why this works: the smaller angle is the one that is at most 180.

How do you find a meeting time?

At H o’clock the hour hand is 5 x H minute-spaces ahead of the minute hand.

The minute hand gains 5.5 degrees each minute, so divide the gap by that.

Can you see it solved?

When do the hands coincide between 5 and 6? At 5 the gap is 150 degrees.

150 divided by 5.5 is 27 3/11. So the time is 5:27 3/11.

Why this works: the minute hand gains 5.5 degrees each minute, so it needs 150 divided by 5.5 minutes to catch up.

Can you find the next right angle?

At 3:00 the hands are 90 degrees apart. When are they next at a right angle?

The minute hand must gain 180 degrees on the hour hand. 180 divided by 5.5 is 32 8/11.

So the next right angle is at 3:32 8/11.

Why this works: after the hands were 90 apart, the minute hand must run through 90 and out to the other side.

What about opposite hands or right angles?

For opposite hands, the gap must be 180 degrees. For a right angle, it must be 90.

Use 30 x H plus or minus the target angle, then divide by 5.5.

What is an odd day?

Divide the number of days by 7. The remainder is the number of odd days.

Move that many weekdays ahead.

How do months count?

A normal year has 365 days, which is 1 odd day. A leap year has 366 days, which is 2 odd days.

Months with 31 days give 3 odd days. Those with 30 days give 2.

Can you see it solved?

Suppose 1 January 2025 is a Wednesday. What is 1 March 2025?

January has 31 days and February has 28. That is 59 days, and 59 divided by 7 leaves 3. Wednesday plus 3 is Saturday.

Why this works: only the leftover days change the weekday. Whole weeks bring it back to the same day.

Can you move across a year?

15 August 2026 is a Saturday. What day is 15 August 2027?

A year from August to August holds 365 days, because February 2027 has 28 days.

365 divided by 7 leaves 1. Saturday plus 1 is Sunday.

Why this works: a normal year has 1 odd day, so the weekday moves forward by one.

Can you move across a century?

1 January 1900 was a Monday. What day was 1 January 2000?

From 1900 to 1999 there are 100 years. Of these, 24 are leap years, since 1900 is not.

That is 76 x 1 + 24 x 2 = 124 odd days. 124 divided by 7 leaves 5, so Monday plus 5 is Saturday.

Why this works: 100 years has 5 odd days, which is the number to memorise.

How do you spot a leap year?

A year divisible by 4 is a leap year. A century year must be divisible by 400.

So 2000 is a leap year, but 1900 is not.

How do you handle a fast or slow clock?

Find how much it gains or loses per hour. Multiply by the real hours passed.

Add the gain, or subtract the loss, from the real time.

Can you see it solved?

A clock gains 3 minutes every hour. It is set right at 8:00 AM.

At real 12:00 noon, 4 hours have passed. It gained 12 minutes, so it shows 12:12 PM.

Why this works: the clock gains the same 3 minutes every real hour, so we multiply.

Can you try a slow clock?

A clock loses 5 minutes every hour. It is set right at 6:00 AM.

At real 2:00 PM, 8 hours have passed. It lost 8 x 5 = 40 minutes.

So it shows 2:00 PM minus 40 minutes, which is 1:20 PM.

Why this works: a slow clock falls behind the real time, so we subtract.

What mistakes cost marks?

Forgetting to take the smaller angle when the formula gives more than 180.

Counting the leap day in a year that has none.

How do you stay quick?

Remember the two speeds: 6 degrees a minute and 0.5 degrees a minute.

Memorise that 100 years has 5 odd days and 400 years has 0.

How should you practise?

Do five angle problems and five weekday problems a day.

Then try the quiz below and read every explanation.