What is the reciprocal of 0.8? Write your answer as a decimal.
The reciprocal of a number 𝑥 is one over 𝑥. And another way to think about this is when we take a number and we multiply it by its reciprocal, we get the number of one. It’s really easy to find the reciprocal of a fraction: all you do is you swap the denominator with the numerator. So the reciprocal of two-fifths is five over two and the reciprocal of seven-thirds is three-sevenths.
What this also means is that a reciprocal of five is one-fifth since we can write any whole number as a number over one. So five is five over one. And then, when we swap the numerator and the denominator, we get one-fifth.
0.8 is not in fraction form. We could write it in fraction form and use this rule. But since this is a calculator paper, we might as well use the first rule we wrote. The reciprocal of 𝑥 is one over 𝑥. So the reciprocal of 0.8 is one over 0.8. If we type one divided by 0.8 or one over 0.8 into our calculator, we get a value of 1.25. And that tells us that the reciprocal of 0.8 is 1.25.
Now, alternatively, if we hadn’t have had a calculator, we could have solved this a slightly different way. We can create an equivalent fraction by multiplying by the numerator and the denominator of this fraction by 10. That tells us that one over 0.8 is equivalent to 10 over eight or one divided by 0.8 is the same as working out 10 divided by eight, which we can do using the bus stop method.
We will need to add a zero after the decimal point. We can’t do one divided by eight. So instead, we do 10 divided by eight. We know that’s one with a remainder of two. 20 divided by eight is two with a remainder of four. So we add another zero and we carry this four. 40 divided by eight is five. And we’ve once again shown that the reciprocal of 0.8 is 1.25.
When 𝑥 is rounded to one decimal place, the result is 2.7. b) What is the error interval for 𝑥?
An error interval is a way of representing the upper and lower bound of a rounded number using inequalities. We’ll begin by working out the lower and upper bounds of the number 𝑥. It has been rounded to one decimal place and the result is 2.7. We should consider what the next number down from this would have been and the next number up.
Had this instead been rounded to the number below, that would have been 2.6. And had it actually been rounded to the next number up, it would have been 2.8. To find the lower and upper bound, we’re going to find the halfway point between the rounded number and the two other numbers. Halfway between 2.6 and 2.7 is 2.65.
Now, if you find that a little bit tricky to spot, remember you can find the midpoint of any two numbers by adding those numbers together and dividing by two. Here, 2.6 plus 2.7 all divided by two is 2.65. Halfway between 2.7 and 2.8 is 2.75. So the lower bound of our number 𝑥 is 2.65 and the upper bound is 2.75. But we need to represent this using inequalities to make it an error interval.
Now, if we decided to reverse this process and round 2.65 to one decimal place, we would indeed get to 2.7. 𝑥 could have been 2.65 and some numbers larger than that. However, 2.75 rounded to one decimal places is actually 2.8.
In fact, 𝑥 could not have been 2.75. It could have been 2.749, 2.7499, 2.7499999, and so on. So we say that our value for 𝑥 got really really closer to 2.75, but never quite got there. And that’s why we use this strict inequality. And we’re done. The error interval for 𝑥 is as shown.