If one ninety-ninth equals 0.010101 and two ninety-ninths equals 0.020202. Find three ninety-ninths without using a calculator and giving the answer to six decimal places.
In this problem, we need to find the value of three ninety-ninths. And we’re asked to give our answer to six decimal places. The fact that the answer is going to have six decimal places suggests to us that it might be complicated. If we were trying to write the value of three one hundredths, this would be a lot easier. But three ninety-ninths! This is more tricky. And we’re not allowed to use a calculator. The question tells us we’re not. So what can we do to find the answer?
Well, the question gives us some facts about similar fractions. And we can use these decimals to help us to find the answer for three ninety-ninths. Firstly, we’re told that one ninety-ninth has a value of 0.010101. We might think we can see a pattern to help us straight away. But in maths, it’s important that we take several examples. So we can find a pattern across more than one number. So this is why it’s important. We’re also given the value of two ninety-ninths. We’re told this is equal to 0.020202. So knowing what we know about one ninety-ninth and also two ninety-ninths, what is three ninety-ninths as a decimal?
There are two ways that we can use the facts that we’ve been given to help. Firstly, what do we notice about the decimals? Well, each number starts with zero point something, which we’d expect. The decimals are less than one. We can see that each number is a recurring decimal. It has a part that repeats itself again and again. In the fraction one ninety-ninth, the numerator is one. And the part that repeats itself is zero followed by one or zero followed by the numerator. And so we have 0.010101.
And the same is true of our second fraction. The denominator is still 99. This time, the recurring part is a zero, followed by a different numerator. This time it’s two. And so we have 0.020202. The fraction we need to find also has 99 as the denominator. But this time, the numerator is three. And so we’d expect it to follow the same pattern. We start with a zero point, because we know that it’s less than one. And then we’d expect the recurring part of the decimal to feature a zero and also the numerator, which we know is a three. And so we get 0.030303. And we’ll stop there because we’re told to give the answer to six decimal places. And we have.
We used the recurring pattern that we found in the fractions we’ve been given to find the value of three ninety-ninths. We did say there was another way to find the answer. This method we just need to know the first fact. If we know that one ninety-ninth is worth 0.010101, then to find three ninety-ninths, we just need to add that decimal three times or even multiply it by three.
If one ninety-ninth equals 0.010101 and two ninety-ninths equal 0.020202, then the value of three ninety-ninths equals 0.030303.