Write an expression that matches this description: the quotient of the sum of three and four and the difference between 12 and nine.
So in this question, what we’re asked to do is to change the words in the description into a mathematical calculation or a series of operations. And since we’re asked for an expression, that means we won’t actually need to carry out the calculation. So let’s start by looking at some of the different words in the description.
So starting with quotient, a quotient is the answer that we get when two numbers are divided. We could also say that the quotient is what we get when we have a dividend — that’s the number we’re dividing — divided by the divisor — that’s the number that does the dividing. So, for example, if we have the quotient of 12 and six, that would be two because 12 is our dividend divided by six our divisor to give us our quotient, two. The important thing to note here is that if we have a statement with a quotient, we’re looking for the quotient of something and something. So we have two values of which to find the quotient.
The other two mathematical words in our description are sum and difference. So remember that sum means adding and difference means subtracting. So in the calculation that we’re going to create from the description, we know that there will be three different operators: division, addition, and subtraction. So let’s have a look at our description again: the quotient of the sum of three and four and the difference between 12 and nine.
So let’s have a look at the first part of the statement. We know that if we’ve got a statement regarding quotient, we need the quotient of something and something. In our first part of the statement, we have the quotient of the sum of three and four. But that would mean we would need to divide the sum of three by four. But since the sum of three doesn’t make any sense, then that can’t be right. So what would happen if we extended the first missing value in our quotient statement to be this sum of three and four? This would mean that the second value in our quotient statement would be the difference between 12 and 9 This would mean that we would have the sum of three and four divided by the difference between 12 and nine.
Since we know that sum means addition, this means that we can take the sum of three and four and write it as three plus four. Next, we have to change the word difference into an operation. So remembering that difference means subtract, we can write this as 12 take away nine. This would mean that a calculation for the description would be three plus four divided by 12 minus nine.
But let’s check if somebody else would perform this calculation in the same way that we were given in our description. And to do this, we’re going to have to remember the rules for the order of operations, which can be described as PEMDAS standing for Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction. In the case of our calculation, out of addition, division, and subtraction, the operator with the highest order of precedence is division. But this would mean that the first calculation to be evaluated would be four divided by 12.
In fact, if we wanted to describe the calculation that we’ve written in words, it would be the sum of three and the difference of the quotient of four and 12 and nine, which isn’t the same as what we were asked. So what we really need is some way so that if we were to give the calculation to someone else, they would work like the sum of three and four and the difference of 12 and nine first before dividing, which we can do with brackets.
So if we put a set of parentheses around three and four and another set around 12 minus nine, this would be calculated before division, since in PEMDAS, our order of operations, parentheses is first, which gives us the answer for the description, the quotient of the sum of three and four difference between 12 and nine.