WORLD OF FIGURES: Why it is important to use correct order of arithmetic

A reader sent me this puzzle: “Think of a number; double it; add six; halve that; subtract the number you started with. Your answer is 3”

He wanted me to explain how the answer can be the same (3) whatever number you think of at the beginning.

Well; the reason reveals itself if you write down those instructions mathematically as an equation. It turns out that you are adding a number to 6; then subtracting it (in the final step) and dividing 6 by two.

Since you have added and subtracted in secession, the final result will always be a half of six, that is 3; regardless of the number you think of at the beginning. To see this better, add 10 instead of 6 and the final result will be 5 (a half of 10).

This puzzle demonstrates the conventionally agreed order of arithmetic operations — popularly knowns as Bodmas. The letters stand for Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Given an arithmetic operation, you start by evaluating quantities that are in brackets, followed by orders (that is, powers — squares, cubes, etc) and so on.

The final arithmetic expression in the puzzle above is (2N + 6)/2 - N. Using the Bodmas convention, this simplifies to N + 3 - N = 3. So, it doesn’t matter what number (N) you think of; the answer will always be 3.

Bodmas works quite well, but in certain instances there might be some confusion. For example, what is 7 - 3 + 2? Is it 6 or 2?

According to this convention, we should do the addition first and then the subtraction. Thus: start with 3 + 2 = 5; followed by 7 - 5 = 2. But is this correct?

The way I taught way back in primary school was that the minus sign belongs to the number that comes immediately after it. In that case, subtraction is equivalent to addition of a negative number. Thus 7 - 3 + 2 = 7 + (-3) + 2 = 6.

To find out which of these two calculations is correct, it helps to insert brackets and then open them out. The first calculation is equivalent to 7 - (3 + 2) and the second one is (7 - 3) + 2.

Now we remove these brackets: 7 - (3 + 2) = 7 - 3 - 2; and (7 - 3) + 2 = 7 - 3 + 2. We notice that the first calculation has changed the arithmetic, therefore, it must be wrong!

***

Last month, Kenya Airways published its financial results for the six months ending June 30, 2018.

The report showed that the airline made a loss of Sh4.03 billion, which was an improvement from the previous year when it lost Sh5.67 billion.

After the announcement, someone tried to argue that the airline had actually made a profit of about Sh1.6 billion — this being the improvement recorded.

Well; such an argument is just plain wrong! A loss is a loss … is a loss!

Profit is simply sales minus expenses. If the answer is negative, then it’s a loss.

In this instance, the sales were Sh52 billion and the expenses were Sh56 billion. There is no way that can be called a profit!

www.figures.co.ke Twitter: @MungaiKihanya