Among all the pairs of numbers whose sum is six, find the pair with the largest
There are two keywords in this problem that we need to know what they mean before we
can work out the answer, and they are sum and product. Sometimes in maths, you might hear the word sum used to describe a calculation. I’ve just completed a page of sums. But really that’s not the correct way to use the word sum. The word sum means something very definite. The sum of two or more numbers is what we get when we add them together. It’s the total of some numbers.
So, the first part of our problem could read, among all the pairs of numbers whose
total is six. The product of a pair of numbers is what we get when we multiply them together. It’s the answer to a multiplication. Let’s read our problem using our new phrases. Among all the pairs of numbers whose total is six, find the pair with the largest
answer when multiplied.
To start with, we need to find all the pairs of numbers whose total is six. So that we don’t get mixed up, let’s call one number 𝑥 and one number 𝑦. When we’re trying to find a list of numbers like this. It’s sometimes a good idea to start either at the very bottom or the very top and to
work our way methodically through them. So, let’s start with the very lowest possible value of 𝑥 and go from there.
The lowest number that 𝑥 could be is zero. And if it makes a pair of numbers whose total is six when we add it to 𝑦, 𝑦 must be
six. Zero plus six equals six. The next highest number is one. One plus five equals six. You’ll see a pattern here. Two plus four equals six. Three plus three equals six. Four plus two equals six. But we’ll stop there because now we’re starting to repeat ourselves.
The numbers two and four, if we multiply them together have the same product as four
and two. And so, there’s no point writing them down twice. The only pairs of numbers that will make different products are these four. So, our problem asks to look among these pairs whose total is six and find the pair
that has the largest product, the largest answer when we multiply them together
Let’s multiply each pair to see what the answers are. Zero multiplied by six equals zero. One multiplied by five equals five. Two fours are eight. And the product of three and three equals nine. So, the pair with the largest product is when both numbers are three.
First, we found all the pairs of numbers whose sum was six. We found four different pairs. Then, we multiplied the two numbers in each pair together to see what the products
were. And among all the pairs of numbers whose sum was six, the pair with the largest
product was when 𝑥 equals three and 𝑦 equals three.