5 on the Clock
3 out of 3
On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?
Problem
On a digital clock showing 24-hour time, over a whole day, how many times does a 5 appear?
Is it the same number for a 12-hour clock over a whole day?
Getting Started
When does $5$ appear in the minutes display?
When does $5$ appear in the hours display?
How will you know that you have got all the different times?
Student Solutions
We had a very impressive number of responses to this tricky problem. Thank you everyone who sent in their solutions. There were a couple of different ways of looking at the problem and because you explained your solutions so clearly, we can see that both are equally good.
Cassandra from Impington Village College (Cambridge) says that the 5 appears 170 times on a 24 hour clock and has clearly explained how she worked that out:
I counted how many times the 5 was in the units of the minutes (eg 01:25) each hour (6 times) then multiplied it by 24 (144).
Then I counted how many times the 5 appeared in the tens of minutes (eg 12:50) in 24 hours (24).
Finally I counted how many times it appears in the hours section (eg 05:00) (2).
I added it all up and got 170. The answer is the same for the 12 hour clock, because there's just two 5 o'clocks instead of 05:00 and 15:00.
Those of you who agreed with this interpretation were:
Laurie from Rhodes Avenue, George (who didn't tell us where he goes to school), Miriam from Archbishop Temple High School, Edward from WMMS, Xavier from Thomas Deacon Academy, Elliot (who didn't tell us where he goes to school), Alex and Grant from St Benedict', Mike (who didn't tell us where he goes to school) and Shasvat from GIIS. Well done.
Another solution came in from Jonathan from St. Andrews Primary (Devon). This is quite a different solution to Cassandra's because Jonathan looked at the problem in a different way. When Jonathan worked out his total, he counted the 5's that would be on the clock if he looked at it every minute. So, for example, at 05:53 there are two, then at 05:54 there are two, and at 05:55 there are three and so on. Here is his explanation:
My answer is 504. This is how I worked it out. I worked out that 5 appears 16 times in 22 hours (05, 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59), but I had to add 60 more 5s when the hour was 5 or 15 to make 76.
I multiplied 16 x 22 to make 352, added 2 x 76 (152) to make my total.
Here is the sum to show what I did: (16 x22) + (76 x2) = 504.
Yes I think it is the same with a twelve hour clock, it just has 2 5s instead of 15.
Michael from Henry Park Primary School in Singapore, Laura and Matthew from Radstock School, Amy from Talycopa Primary, Emily, Tishtrya, Michelle and Fionnuala from Ursuline High School, Oliver from Aycliffe, Calum, Fergus and Nancy from St Anne's Primary, Class 5AH from Aston Fields, Khalid, Piotr and Antranig from Dubai International Academy, Julieta from The Grange School, Jozie and Molly from Cumnor Primary, Jacob from St Edburg's, Caitlin, Molly and Oliver from Great Torrington Junior School, Katie, Theona and Abigail from St Joseph's RC High School, Christopher and Zoe from Randlay Primary School, Isabella from Parkland Junior School, KS2 Class from Ysgol Aberdyfi, Jessica, Isobel, Emma, Lily, Alice, Lotty, Helena, Molly, Martha, Isabella, Lucy, Matilda and Nathalie from St Ives, The Extenstion Maths Group at St Nicolas C of E Junior School and Alex from Waddinton Redwood Primary School all sent in good solutions which agree with Jonathan.
Finally, a special mention to the Junior Class at Ysgol Bryncrug who sent in this Word document which explained their thinking very, very clearly. They also thought the solution was 504 and in the document, they talk us through how they solved the problem. Thank you!
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem will help reinforce telling the time with the help of a digital clock. It also requires children to be systematic in their working.
Possible approach
It would be good to have an interactive digital clock on the whiteboard for the duration of this lesson so that you and the class can refer to it whenever necessary. You may want to begin by asking a few oral questions based on the clock before moving on to the problem as it stands.
Once you have introduced the problem, you could ask for some estimates of the total number of times 5 appears. The problem can be interpreted in slightly different ways, so you may need to reach a consensus as to whether, for example, 05:55 counts as three times, because there are three fives, or just once, because the three fives are all present at just one moment. Give pairs of learners time to work on the problem - they are likely to need paper or mini-whiteboards for recording.
After a short time, share a few insights they have gained and invite some pairs to explain what they have done so far. Some may have started by looking at the minutes digits, some with the hours, some may have looked at certain times of day. Talk about the different approaches and in particular ask the class how they will make sure they don't miss out any times. If no-one suggests a system, then you could offer some ideas such as concentrating on the minutes in ascending order, or perhaps beginning with 00:00 and working minute by minute through to 23:59. (You could also discuss which way might be most efficient.)
Leave them to work more on the problem before bringing them together to talk about their final solutions. You could use the 12-hour display question for discussion in the plenary.
This problem could be presented as a challenge to parents and children while waiting to see teachers at parents' evening! To avoid confusion, specify that, for example, 15.55 counts as three times, and ask families to write their name and answer on a card and post it in a suitably-labelled box. A small prize could go to the family whose correct answer is the first to be drawn by the head at the next school assembly.
Key questions
When does 5 appear in the minutes display?
When does 5 appear in the hours display?
How will you know that you have got all the different times?
Possible extension
You could ask learners whether each digit will appear the same number of times as the 5 and invite them to explore this.
Possible support
It might be useful for some children to have access to an interactive version of a digital clock themselves, perhaps at a shared computer.