MATHS WEEK is under way and, as is our annual tradition, we’re setting our readers some puzzles. Give them a go! You can check out other events being held over the next week here. 

Have You Time for Maths?

Regardless of whether you have time for maths, it is very important to have maths for time. As usual, there were some differing opinions in the comments on The Journal‘s poll on Daylight Savings Time this week. 

While there are real and practical implications, many people just feel it is an outdated and complicated procedure. Many find it confusing and can never remember if you spring forward and fall back or fall forward and spring back. Some see it as tradition and would miss it. Others like to be reminded of the passage of time and the journey of our planet around our Sun.

Perhaps if we had two new public holidays marking this change in the calendar, it might become very popular. 

The desire and need to measure the seasons and time for agricultural, religious or other reasons spurred the development of mathematics in many ancient cultures. We have 60 seconds in a minute and 60 minutes in an hour thanks to the ancient Babylonians. They liked 60 because it was divisible by so many smaller numbers making it useful for fractions and calculation (decimals hadn’t been invented then).   

Many readers are probably now thinking: “This is old fashioned, why don’t we have decimal time?”

Well in fact it has been tried. 

After the French revolution, Republican time divided the day into 10 hours (instead of 24) and each hour into 100 minutes and each minute into a hundred seconds. This is easier to follow and has advantages in calculating. 

For instance, if we want to divide an hour by 16, we might divide 60 minutes by 16 which gives 3.75 minutes. Now as a minute is 60 seconds, 0.75 minutes (three quarters of 60) is 45 seconds, so the answer is 3 minutes 45 seconds.

However, if an hour is 100 minutes and we divide by 16 we get 6.25 minutes, which in the decimal system is 6 minutes 25 seconds. No need to convert the seconds. 

Try the following questions to see how well you would adapt to decimal time.  

1. What whole numbers divide into 60 exactly?

2. The decimal time divides the day (our 24 hours) into 10 hours. What time would our 12 noon be in the decimal clock?

3. The Maths Week Puzzle is published at 7.30pm each evening. What time would this be on a decimal clock? 

Don’t worry about decimal time being introduced despite its advantages. It was introduced in 1794 and lasted until 1800. Today changing all our systems would be too much to even consider.

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Wednesday’s puzzle: The answers

1. 9 sparrows, 10 turtle-doves and 11 doves is 30 birds. £1/3 per sparrow, £1/2 per turtle
dove, £2 per dove. That is £3 + £5 + £22 = £30 (discussion below)

2. 4 oranges, 1 banana and 5 apples

3. 20c and 50c. 1x 5c + 9x 20c and 7x 5c + 3x 50c

Fibonacci’s explanation:

“I first suppose that the man buys 30 sparrows for 10 pounds, and keeps back 20 pounds,
which is the difference between 30 pounds and 10 pounds.

Now suppose I changed one of the sparrows into a turtle-dove which would increase the money spent by this change by 1/6 of a pound since the sparrow was worth 1/3 of a pound and the turtle-dove 1/2 of a pound, that is 1/6 more than the price of a sparrow.

Suppose now that I changed one of the sparrows into a dove and by that change increased my spending by 1 and 2/3 of a pound, that is the difference between 2 pounds and 1/3 of a pound.

If I made six of these 1 and 2/3 changes then these six would make in total an increase of 10 pounds.

Now I must change sparrows into turtle-doves and into doves until I have used up the 20 pounds that I kept back earlier.

So I multiplied by six and so obtained 120 which I split into two parts, one of which could be divided exactly by 10 and the other by 1. The total of these two divisions was not to be as large as 30. So the first part is 110 and the other 10; and I divided the first part, that is 110, by 10, and the second by 1.

This gives 11 doves and 10 turtle-doves; taking these from 30 there remain 9 for the number of sparrows. So there are 9 sparrows worth 3 pounds, and 10 turtle-doves worth 5 pounds and 11 doves worth 22 pounds. So from these three kinds of birds we shall have 30 for 30 pounds as was required.”

Source: Mac Tutor, Maths History at University of St Andrews

That’s a bit of a head-wrecker. Luckily, we can do it using secondary school algebra and a
little detective work.

Let us call the number of sparrows S, the number of turtle-doves T and the number of
doves D.

Then we can write:

S + T + D = 30 (1)

1/3S + 1/2T + 2D = 30 (2)

If we multiply both sides of equation (2) by 6 we will get rid of the fractions:

6/3S + 6/2T + 6x2D = 30 x 6

2S + 3T + 12D = 180 (3)

Multiply equation (1) by 2

2S + 2T + 2D = 60

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And subtract from equation (3)

2S + 3T + 12D = 180

-(2S + 2T + 2D = 60)

T + 10D = 120 (4)

It looks like we are out of information and we will have to resort to trial and error to continue.

However, we have some more information that will narrow it down. We know that we have to have whole birds (ie positive integers) and there can’t be zero birds of any type.

If we divide (4) by 10 we get

T/10 + D = 12

T/10 must be a positive integer so T must be a multiple of 10.

It can only be 10 or 20 as we have only 30 birds.

Checking T = 10:

T/10 + D = 12

1 + D = 12

D = 11

As S+T+D=30, then

S = 9.

You can check if T = 20 gives another valid solution. 

Come back tomorrow at 7.30pm for the answers to today’s questions and a brand new challenge.  

These puzzles were prepared exclusively for The Journal by Eoin Gill, co-founder and co-ordinator of Maths Week Ireland.