The product of two numbers is 2,159. Given that one of the numbers is 121, estimate the other number by rounding to the nearest 10.
Let’s begin by recalling what we mean by the word product. We say that the product of two numbers is the number we get after we multiply those two numbers together. In this question, one of the initial numbers that we’re going to multiply together is 121. And we know the outcome of the product. It’s 2,159. So, let’s let the other number be equal to 𝑥. We can then say that since 2,159 is the product of our two numbers, then 121 times 𝑥 is equal to 2,159.
So, how are we going to work out the value of 𝑥? Well, we’re going to perform an inverse operation. The opposite to multiplying is dividing, so to work out what 𝑥 is we’re going to divide by 121. If we think of this formally as an equation, we need to divide both sides of the equation by 121. But we can simply think about it as reversing the process of multiplying. And so, 𝑥 is 2,159 divided by 121, which we can write as a fraction as shown.
Now, the question is telling us to estimate the value of the other number. We use estimation to predict the answer to a calculation. And we often do it when we don’t necessarily need to know the exact value. We do this by rounding the numbers we’ve been given. Quite often, we’ll around each number to one significant figure. In this example, though, we’ve been told to round to the nearest 10.
So, let’s begin by rounding 2,159 to the nearest 10. The number in our tens column is five. And so, we look to the digit immediately to its right. We call this the deciding digit. If the deciding digit is five or above, that tells us we round our number up. The five becomes a six. And to keep the size of our number, we add a zero as a placeholder. And we can say that 2,159 correct to the nearest 10 is 2,160.
Let’s repeat this process for the number 121. This time the number in the tens column is a two. And the deciding digit is a one. Because the deciding digit is less than five, this tells us to round our number down. Essentially, we’re saying that 121 is closer to 120 than it is to 130. And so, 121 correct to the nearest 10 is 120.
And this means then that our number 𝑥 must be roughly equal to 2,160 divided by 120. Now, this is where it can be nice to write our division as a fraction. We see that both the numerator and denominator of our fraction can be divided by 10. And so, doing so, we get 216 divided by 12. We can then look for further common factors in the numerator and denominator of our fraction. A really straightforward one is two. And so, if we divide the numerator and denominator of our fraction by two, it becomes 108 over six.
Now, this is an equivalent fraction to 2,160 over 120, so we’ll still get the same answer. But what we’ve done is create a much simpler division problem. Let’s use the bus stop method to calculate 108 divided by six. We can’t easily calculate one divided by six. So, we add a zero here, and instead we calculate 10 divided by six. 10 divided by six is one with the remainder of four. So, we put a one at the top and carry the four. Then, we work out 48 divided by six. Well, we know that eight sixes make 48, so we put an eight here.
We can, therefore, say that 108 divided by six is simply 18. And so, an estimate to the other number in our calculation is 18. Now, note that at this stage, we could have done two different steps. We could have performed some long division, or we could have divided by a different factor, such as a factor of four. The main purpose is just to make the sum that you’re doing a little bit easier. And so, it really doesn’t matter which method you choose.