0.30 recurring is equal to 10 over 33. Use this fact to show that 0.030 recurring is equal to one over 33.
Let’s consider what has happened to get from 0.30 recurring to 0.030 recurring. If we look carefully at the position of each of the digits in each number on the place value grid, we can see that each digit has moved to the right exactly one space. When we move digits to the right one space, that’s the same as saying we’re dividing by 10. So that means 0.30 recurring has been divided by 10 to give us 0.030 recurring. And this means, to write 0.030 recurring in fraction form, we need to take the fraction form for 0.30 recurring and divide that by 10. That’s 10 over 33 divided by 10.
Now when combining arithmetic with fractions and integers, it’s always sensible to give your integers a denominator. And we can say that the denominator is one. 10 is the same as 10 one wholes. When we divide two fractions, we keep the first fraction the same. We then change the divide to a multiply and we find the reciprocal of the second fraction. The reciprocal of 10 over one is one over 10. We could at this stage multiply the numerator by the numerator and the denominator by the denominator.
However, if we look carefully, we can see that we can cross-cancel by a factor of 10. 10 divided by 10 is one. Now we can multiply the two numerators. One multiplied by one is one. And then the denominators, 33 multiplied by one is 33. And we have indeed shown 0.30 recurring is equal to one over 33.
Hence, or otherwise, convert 0.130 recurring to a fraction. Give your answer in its simplest form.
This word “hence” is a surefire sign that we need to use what we did in part a. So let’s see if we can work out how we can get to 0.130 recurring with what we did in part a. In fact, we can see that the number that we dealt with in part a, 0.030 recurring, is almost identical to the number we’ve just been given. However, there’s a one in the tens column, so that tells us that 0.130 recurring must be equal to 0.1, one-tenth, plus 0.030 recurring.
We know that 0.1 is the same as one-tenth. And we showed that 0.030 recurring is the same as one over 33. To add fractions, we need to find a common denominator. It’s not instantly obvious what the lowest common denominator for 10 and 33 is. But if we multiply 10 by 33 and 33 by 10, we know that we’ve definitely got a common denominator.
We trying to create equivalent fractions though, fractions that have the same value. So when we multiply 10 by 33, we need to do the same to the numerator. One multiplied by 33 is 33. Similarly, with the second fraction, when we multiply the denominator by 10, we need to do the same to the numerator. One multiplied by 10 is 10.
And of course, we’re adding these fractions. And when we have fractions with the same denominator, we simply add the numerators. And 0.130 recurring is equal to 43 over 330.
Now we were asked to check that our answer is in its simplest form. 43 is a prime number, so it can only be divided by one and 43. To check whether the fraction can be simplified then, we need to double-check whether 330 can be divided by 43, whether it’s a multiple of 43. In fact, if we list out the first few numbers in the 43 times tables, we can see that some of the multiples are 301 and 344. But 330 isn’t included. So 43 over 330 is indeed in its simplest form.