Why it is important to use correct order of arithmetic
By MUNGAI KIHANYA
The Sunday Nation
Nairobi,
30 September 2018
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 the financial results for the six months ending on 30th
June 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
Sh5.67B.
After the
announcement, some one tried to argue that the airline had actually made
a profit of about Sh1.6B – 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 Sh52B and the expenses were Sh56B. There
is no way that can be called a profit!
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