Jayne organised a charity dinner to raise money. Her costs for the event were 1995 pounds, and she wanted to raise an additional 3795 pounds to donate to charity. Jayne set a fixed price for entry into the event, and 312 people attended the dinner. Given that she achieved her fundraising target, estimate a value for the minimum amount that each person had to pay towards the dinner.
In maths, we estimate a solution by rounding each number in the question to one significant figure. This can also be a really useful technique to allow you to check your answers in a non-calculator paper.
There are three numbers we’re interested in. We know that the costs for the event were 1995 pounds, and Jayne wanted to raise an additional 3795 pounds to donate. We also know that 312 people attended the dinner. We’ll begin by rounding each of these numbers to one significant figure.
The first significant figure in any number is the first digit that isn’t zero. In 1995, that’s one; in 3795, it’s three; and in the number 312, it’s three again. Once we’ve decided which is the first significant figure, we look to the number immediately to its right. This is sometimes called the deciding digit. In the number 1995, the deciding digit is nine.
There are two rules for this deciding digit. If this digit is five or larger, we round up. If it is less than five, we round down. By considering the value of the deciding digit in the number 1995 relative to the number five, this allows us to decide whether 1995 is closer to 2000 or 1000. Nine is larger than five. This means that 1995 is closer to 2000 than it is to 1000. This means that 1995 rounds to 2000, correct to one significant figure.
Let’s now look at 3795. The first significant figure is three, and the deciding digit is seven. Seven is greater than five. This means 3795 is closer to 4000 than it is to 3000. And we say that 3795 correct to one significant figure is 4000.
We’ll repeat this for the number 312. The deciding digit here is one. This time, one is less than five. This means 312 is closer to 300 than it is to 400, and we round to 300, correct to one significant figure.
We can now use these numbers to estimate a value for the amount that each person had to pay towards the dinner. We can add the values of 2000 pounds and 4000 pounds to calculate the total cost required to cover the dinner. 2000 plus 4000 is 6000.
We’re sharing these costs between 300 people, so we’re going to divide 6000 by 300. We can treat 6000 divided by 300 as a fraction and then simplify to make the numbers easier. If we divide both the numerator and the denominator by 10, it becomes 600 over 30. And if we divide by 10 again, it then becomes 60 over three or 60 divided by three. Six divided by three is two, so 60 divided by three is 20.
Now because we simplified this fraction, we don’t need to do anything with our answer. 6000 divided by 300 gives us the same answer as 60 divided by three. And the minimum amount that each person has to pay is 20 pounds.
Part b) Is your estimate an overestimate or an underestimate? Give a reason for your answer.
We rounded 1995 pounds up to 2000, and we also rounded 3795 pounds up to 4000, but we rounded 312 down to 300. That means 6000 was an overestimate, but 300 was an underestimate. What we did is divide an overestimated number by an underestimated number. We divided a number that was too big by a number that was too small. Essentially, we’re sharing a larger number than required into a smaller number of pieces. This means that the size of each piece will be larger than we expect, and our answer will be an overestimate.
In fact, if we think about this a different way, these 312 people will be paying 20 pounds we estimated. If we worked out 20 pounds multiplied by 312, we’ll actually get a larger number than 6000. The estimate is an overestimate.