Kayla has memorized the value of 𝜋 to eight decimal places: 3.14159265. A common approximation for 𝜋 is twenty-two sevenths. Show that the percentage error between Kayla’s memorized version of 𝜋 and the approximation twenty-two sevenths is 0.04 percent to two decimal places.
To find the percentage error, we’re going to use this formula. We begin by finding the actual error. This is the difference between the approximation and the actual value of 𝜋. We divide that by the original. Here that’s the correct value of 𝜋. And then, we multiply this by 100.
If we type twenty-two sevenths into our calculator, we get 3.142857. Those dots above the one and the seven show that everything between and including these two numbers recurs. The digit in the ten thousandths column of twenty-two sevenths is two and the corresponding digit in the value of 𝜋 is one. This means twenty-two sevenths is larger than the value of 𝜋.
To find the error then, we’re going to subtract the smaller value of 𝜋 from the approximation of twenty-two sevenths. Now it wouldn’t have mattered if we done this the other way round. But we would have ended up with a negative value for the percentage at the end and we would have needed to have change that to a positive.
So to find the percentage error, we’re going to take this error twenty-two sevenths minus 3.14159265. We’re going to divide that by the value of 𝜋 and then multiply that by 100. if we do that, we get 0.04025. Let’s round this to two decimal places as required.
The second digit after the decimal point is the four. The digit immediately to its right is called the deciding digit. Remember if that deciding digit is less than five, we round the number down. If it’s five or above, we round the number up.
Zero is less than five. So we’re going to round down to four. This becomes 0.04 percent. And we have shown that the percentage error between Kayla’s memorized version of 𝜋 and the approximation is 0.04 percent to two decimal places.
Now, it’s worth being aware that there is a slightly different method we could have used. We could have found what percentage of the original the approximation is. We would have divided the approximation by the actual value of 𝜋 and multiplied by 100. That would have given us 100.0402 percent, which correct to two decimal places is 100.04 percent.
This shows us that twenty-two sevenths is 100.04 percent of the correct value of 𝜋. To find the percentage error or the percentage difference from the original, we then subtract 100 percent. And that once again gives us 0.04 percent.