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Order of operations

 Algebraic notation (the language of maths) can look complicated, but in fact it makes things clearer and easier to understand —at least for a mathematician!

For example, read this problem and notice the words in red: This is the same problem in maths language:

DATA: OPERATIONS: As you can see, the problem says that each coach is divided into three, but to find out the number of seats in a coach we actually have to multiply by three! Then it says that two thirds of the seats are empty in the second coach, so we have to multiply by one third! (or divide by three). That's because we can express the amount of occupied seats in different ways —as a number, a percentage, a fraction, or something more complex like "all the window seats".

The words that mean that we have to add, subtract, multiply or divide —words like "each", "and", "wins", "loses", "finds", "pays", "earns"— can be very different in different problems. Besides, the language we use every day is full of synonyms, long sentences, strange words,... Maths language helps us operate with all those data without having to read the problem again and again.

However, to find the solution it is essential that we do the operations in the correct order:

Problem: 50%
1/3
     3 · 4 · 12 · 50 / 100 + 3 · 4 · 12 · 1 / 3
Wrong:  [( 3 · 4 · 12 · 50 / 100 ) + 3] · (4 · 12 · 1 / 3)
Right:  [(3 · 4 · 12) · (50 / 100)] + [(3 · 4 · 12) · (1 / 3)]

In the first solution, the wrong one, we are just doing all the operations from left to right, so that we are doing the addition before we do the multiplications that appear after it:

[ ( 3 · 4 · 12 · 50 / 100 ) + 3 ] · ( 4 · 12 · 1 / 3) =
= ( 72 + 3 ) · 4 · 12 · 1 / 3 =
= 75 · 4 · 12 · 1 / 3 =
= 1,200 passengers!! 		(72)
(75)
(1,200)

The result is that in the second line we are adding together the passengers in the first coach (72) with the number of compartments in the second coach (3)! We are mixing passengers and compartments, but they are data of a different kind, and you can't add different types of data in this problem! That doesn't make sense!

So, in the second solution, the correct one, we calculate the number of seats in each coach (3 compartments · 4 rows · 12 seats = 144 seats), and then we calculate how many passengers there are in each coach (50%, or half, in the first coach, 1/3 in the second one), and finally we add up the number of passengers in both coaches, by doing the multiplications and divisions first, and the additions and subtractions last:

[ ( 3 · 4 · 12 ) · ( 50 / 100 ) ]
= ( 144 / 2 )
= 72
+
+
+
[ (3 · 4 · 12) · ( 1 / 3 ) ]
( 144 / 3 )
48
=
=
=
120 passengers

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