Reasoning · Grade 6-1 Clocks and Angles

Problem

When the hour and minute hands overlap

Sometimes the hour hand and the minute hand lie exactly on top of each other. Look at the stretch from midnight to noon. Midnight itself is not counted. Count the overlaps and find the first one after noon.
Your answer
How to solve
Strategy Look for a Pattern — Chasing every overlap one at a time would be slow and easy to miscount. Instead I look at the whole 12 hours at once and count laps: in 12 hours the minute hand goes round 12 times while the hour hand goes round once, so the minute hand gains 11 laps, and every gained lap is exactly one overlap. That single count answers the first two parts. Then a systematic hour-by-hour list confirms where the missing twelfth overlap went. For the third part I switch to units — degrees gained per minute — because the overlaps are evenly spaced and the spacing is what the question really asks for.
1STEP 1

Say exactly what an overlap is

An overlap is the minute hand gaining a full lap.

one overlap ⇔ the minute hand gains exactly 360° on the hour hand
2STEP 2

Count the laps in 12 hours

In twelve hours it gains 11 laps.

12 - 1 = 11 laps gained → 11 overlaps
3STEP 3

Check the count hour by hour and find the missing one

Hour by hour, only the eleven o'clock hour has none.

0 + 1 + 1 + … + 1₁₀ hours, 1--2 up to 10--11 + 0 + 1 (at 12:00) = 11
4STEP 4

Double the count for a whole day

A whole day doubles it to 22.

11 + 11 = 22
5STEP 5

Find how fast the minute hand gains on the hour hand

The minute hand gains 5.5 degrees a minute.

6° - 0.5° = 5.5° gained per minute
6STEP 6

Work out how long one whole gained lap takes

A full gained lap takes 720/11 minutes.

360/5.5 = 720/11 = 65 5/11 minutes
7STEP 7

Turn that into a clock time

As a time that is 1:05 and 5/11 p.m.

65 5/11 = 60 + 5 5/11 → 1 : 05 5/11 p.m.
8STEP 8

Check that the eleven gaps fill the twelve hours exactly

Eleven gaps fill the twelve hours exactly.

11 × 720/11 = 720 minutes = 12 hours
Answer
11, 22, 1:05 and 5/11 p.m.
360 ÷ 5.5 = 720/11
The units are right: parts (1) and (2) are counts and part (3) is a time. The size is right too. Overlaps come a little more than an hour apart, so in 12 hours you expect a bit fewer than 12 of them, and 11 is exactly that; 12 would be too many and 10 too few. Doubling gives 22 for the day, which likewise sits just below 24. For part (3), 720 over 11 minutes is about 65.45 minutes, a little over an hour, which matches the pictures: the hands meet just past the 1, then a little further past the 2, then further still past the 3, drifting later and later by about 5 minutes each time. Finally, the parts agree with each other: 11 gaps of 720 over 11 minutes come to exactly 720 minutes, the full 12 hours.
Takeaway

The hands meet once for every whole lap the minute hand gains — 11 laps in half a day, so 11 meetings, spaced exactly 720/11 minutes apart.

  • Say exactly what an overlap is
  • Count the laps in 12 hours
  • Check the count hour by hour and find the missing one
  • Double the count for a whole day
  • Find how fast the minute hand gains on the hour hand
  • Work out how long one whole gained lap takes
  • Turn that into a clock time
  • Check that the eleven gaps fill the twelve hours exactly