IN HONOUR OF Maths Week, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!

Algebra, or ‘The reunion of broken parts’

Arithmetic deals with specific numbers, performing different operations on them to get a solution. In some cases, values are not immediately known and algebra gives us powerful tools for finding unknown values.

Algebra uses symbols to represent unknown values (often x and y) and uses rules and reasoning to find the values represented by the symbols. While the Babylonians used algebra 4,000 years ago, it was in Medieval Persia that modern algebra developed and from there transmitted to Europe.

The seminal work was the 9th Century book by mathematician Al-Khwarizmi called “Al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābala”. Translations of this book also helped spread Hindu-Arabic numerals to Europe. The word algebra derived from the Arabic word “al-jabr”, which means the reunion of broken parts.

In algebra, we typically express problems in mathematical equations using symbols to represent unknown values. Equations are mathematical statements that show the known relationships on both sides of an equal sign. Equations can be manipulated in many ways, but we must keep both sides equal for them to remain valid. Therefore whatever we do to one side of an equation we must do to another.

The step up from arithmetic where concrete numbers are used to algebra with its symbols can be daunting for many learners. The following puzzles can be solved by formal algebra or by trial and error guided by reason.

1. I buy 10 pieces of fruit for €10. The apples cost 80c each and oranges cost €1.30 each. How many apples and how many oranges did I buy?
2. Two coffees and a brownie costs €8.50, while two brownies and one coffee costs €8. How much does (a) a coffee cost and (b) a brownie cost?
3. Michael’s age is 4 times his son’s age. In 5 years’ time it will be 3 times his son’s age. How old is Michael Now?
4. My age is four times what it was four years ago minus twice what it will be in four years’ time. What is my age?
5. A group of friends meet in a pub and each buys a round. Two drink half pints and the rest drink pints. The total volume purchased is 12 pints. How many in the group?

Tuesday’s puzzles: The answers

There may be more than one way of doing these.

1. 24, 12, 8, 6, 4, 3, 2, 1.
2. b) 9+7 = 16 +6+2 c) (5 x 5) – 2 + 1
3. a) 8 *(6 ÷ (4 – 2)) b) (8 – (6 ÷ 2)) x 4 = 6/2 =3, 8-3 = 5*4 =20
   c) (8 – (6 ÷ 4)) x 2 6/4 = 1.5, 8-1.5 = 6.5* 2= 13
4. d) (6 – (2 ÷ 8)) x 4
5. £16 16s 4½d

Come back tomorrow for the answers to today’s puzzle. 

The puzzles this week have been compiled for The Journal by Eoin Gill of Maths Week Ireland and South-East Technological University (SETU).