IT’S DAY 3 of Maths Week, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!
WE HAVE ALL lived the nightmare of splitting the bill in a restaurant.
Who had what? Who didn’t have a starter? Do you have that friend that always seems to do well out of it every time? You know, the one that was hogging the wine who helpfully suggests “we should just split the drinks bill evenly, that would be the easiest”.
In maths, it is often useful to consider the simplest situation and then increase the complexity to see if we can find a method.
The simplest situation is with two people (unless you are in the habit of splitting bills with yourself).
A. Imelda and Sinéad go to Belfast for a weekend. They agree to split the expenses equally – Sinéad booked the train and paid €86 and the concert tickets for €120. Imelda booked the accommodation for €330 and paid €46 for drinks. Overall, Imelda ended up spending more than Sinéad. How should they settle up afterwards? In other words, how much should Sinéad give Imelda so that they each contributed the same amount?
B. Can you make a general rule that would allow two people to settle fairly?
C. How can we extend to three people? Tom, Dick and Harry spend a weekend in Sligo. They too agree to split the expenses equally, and different people pay for petrol, meals, accommodation and so on. Assume that Tom spends the most and Harry spent the least. How much should Dick and Harry give to Tom so that everyone contributes the same amount? Does the solution for Imelda and Sinéad help us here? If not, can it be adapted to work?
D. How can this be extended to four? For instance a camping trip in Waterford taken by Saoirse, Ruth, Colin, and Brendan?
Come back tomorrow for the answers to today’s puzzle.
Sunday’s puzzle: the answer
1. Mary is 8 and John is 2. She is 4 times John’s age. In 4 years, she will be 12 and John will be 6. This puzzle can be solved by trial and error or, trial and improvement.
Guess an age try it out. For instance, you might guess Mary is 16 then John would be 4, but in 4 year’s time she would be 20 and he would be 8. More than twice.
If we then try 20 for Mary, we see that the gap is getting wider. This suggests that if we
go smaller and we will find the answer.
It can also be solved using algebra:
If we represent Mary’s age with the letter M and John’s age with the letter J, we can
write the first piece of information, Mary is four times John’s age, as:
M=4J
And our second piece of information, in four years she will be twice John’s age, as:
M+4 = 2(J+4)
We can replace M in the second equation with 4J as they are equal and we get
4J + 4 = 2(J+4)
Which can be resolved as follows
4J + 4 = 2J + 8
Subtract 2J from each side
2J + 4 = 8
Subtract 4 from each side
2J = 4
Therefore, J = 2, which is John’s age
And so, M = 4J = 8 which is Mary’s age.
2. My friend’s birthday is on 31 December. The conversation took place on the 1 January. So on 30 December – day before yesterday – she was 22. Today she is 23. Later this year (31 December), she will be 24, and next year on 31 December, she will reach 25.
Maths Week Ireland is coordinated by SETU with partners across the island of Ireland. This year over 400,000, north and south will take part and these puzzles give you a chance to participate.
The Maths Week puzzles this year are presented by Colm Mulcahy, professor emeritus of Mathematics at Spelman College, USA, and adjunct professor with Calmast at South East Technological University. Colm is chairperson of the Martin Gardner Foundation USA, and the curator of the website mathsireland.ie
To embed this post, copy the code below on your site
have your say