Why does multiplying a number on a clock face by 10 and then halving, give the minutes? ${}{}$

Question

My daughter in grade 3 is learning about telling time at her school. She eagerly showed me this method she has discovered on her own to tell the minutes part of the time on an analogue clock. I wasn't sure at first because I have never heard about it before but it works really well.

Here is her method:

• Look at the minute hand and see what number it is pointing to, let's say it's $3$.

• Add a zero at the end to make it $30$.

• Halve it to get the minutes, so $3$ becomes $30$ and halving it gives $15$, $6$ becomes $60$ and halving it gives $30$ and so on.

It works well but I am not sure why. What mathematically justifies the method used?

Answer 1

Each number on the clock face is worth five minutes. One good way to multiply by $5$ is to first multiply by $10$, and then divide by $2$. This works because $5=10\div 2$.

Answer 2

The minute hand passes over $12$ numbers in $60$ minutes.

That is $5$ minutes for each number.

Note that multiplying by $5$ is the same as multiplying by $10$ and dividing by $2$

Thus $3$ translates into $15$ which is $3(10)/2$.

Similarly $6$ translates into $30$ which is $6(10)/2$.

Answer 3

The minutes are reckoned as

(Minutes pointing numeral N)(number of minutes in one hour =60)/(maximum minutes digits available on dial= 12) $= 5 N $

The procedure your daughter gave has effect of multiplying pointed figure $N$ by $5$; .. so it works.

Answer 4

From the perspective of the minute hand, the numbers mark the 1/12ths of an hour. But we are usually more interested in the 1/60ths of an hour: the minutes.

Since 12 and 60 differ by a factor of 5 (i.e. $12*5=60$), converting between the two is quite easy. We just multiply by 5.

Or as your daughter prefers, you could multiply by 10 (by adding a 0) and then divide by 2. Since $10/2=5$, it amounts to the same thing.