A number, 𝐾, has been rounded to three decimal places. The result is 60.752. Write down the error interval for 𝐾 using inequalities.
An error interval is a way of representing the upper and lower bound of a rounded number using inequalities. To begin with then, we should find the upper and lower bounds of the number 𝐾. It has been rounded to three decimal places and the result is 60.752. We should consider what the next number down from this and the next number up from it would have been.
Had 𝐾 actually been rounded to the number below, that would have been 60.751 and had it been rounded to the next number up, that would have been 60.753. To find the lower and upper bounds, we find the halfway point between the rounded number and these two other numbers. Halfway between 60.751 and 60.752 is 60.7515 and halfway between 60.752 and 60.753 is 60.7525.
So the lower bound of 𝐾 is 60.7515 and the upper bound is 60.7525. We need to represent this using inequalities. 60.7515 rounded to three decimal places is actually 60.752. However, 60.7525 rounded to three decimal places is actually 60.753.
In fact, 60.75249 does round to our number of 60.752 as does 60.752499 and 60.7524999 and so on. It gets closer and closer to 60.7525, but it never quite gets there. So we use what is called a strict inequality here. We say that 𝐾 is less than 60.7525. Since it could actually be 60.7515, we say that 𝐾 is greater than or equal to this number.
The error interval for 𝐾 is 𝐾 is greater than or equal to 60.7515 and less than 60.7525.