Order of Operations with Integers

Look at two plus three times four. If you work straight across the page you get five times four, which is twenty. If you do the multiplying first you get two plus twelve, which is fourteen. Only one of these can be the accepted answer, and it is fourteen. The picture shows why that choice makes sense: two loose counters plus three bags holding four counters each really is fourteen counters. Maths could not function if the same string of symbols meant different things to different people, so everybody agrees on one order of operations. Your calculator uses it, your textbook uses it, and every exam marker uses it. Learning it properly now saves you a lot of lost marks later.

Here is the agreed order, and it is a ladder you climb down. Rung one is brackets: whatever sits inside brackets gets done first. Rung two is indices, which means squares, cubes and roots. Rung three is multiplying and dividing. Rung four is adding and subtracting. Notice that rungs three and four each hold two operations sharing the same level. That is the part people get wrong. Multiplying does not outrank dividing, and adding does not outrank subtracting. When two operations share a rung you simply work left to right, the way you read. So four plus five in brackets is nine, times two is eighteen, and three plus four squared is three plus sixteen, which is nineteen.

This is the rung that catches people out. Twenty divided by five times two. Multiplying and dividing sit on the same rung, so you go left to right: twenty divided by five is four, then four times two is eight. If you had multiplied first you would get twenty divided by ten, which is two, and that is wrong. The same thing happens with adding and subtracting. Ten minus four plus three means ten minus four is six, then six plus three is nine. Doing the addition first gives ten minus seven, which is three, and that is wrong too. So whenever two operations share a rung, forget which one looks more important and just take the one further left first.

Brackets sit at the top of the ladder, so clear them before anything else, and remember that the answer inside a bracket is often negative. Negative three times the bracket four minus seven: inside the bracket four minus seven is negative three, so now you have negative three times negative three, and two matching signs give positive nine. Next, negative eight plus two in a bracket is negative six, and negative six divided by negative three is positive two. The trickiest layout is a subtraction in front of a product. Five minus three times negative four. Do the multiplying first to get negative twelve, so the sum reads five minus negative twelve, which is five plus twelve, seventeen. Write that middle line down every time.

Now the full ladder on one question: twenty-four divided by negative four, plus three times negative two. There are no brackets and no indices, so you start at rung three. Both the division and the multiplication live there, so take the one further left first. Twenty-four divided by negative four is negative six. Then three times negative two is also negative six. Rung three is finished, so drop to rung four: negative six plus negative six is negative twelve. Indices work the same way. In two times the bracket five minus eight, all squared, the bracket gives negative three, squaring that gives positive nine, and two times nine is eighteen. Finish every question by asking whether your sign looks sensible.