Calculate three and two-thirds multiplied by one and five thirteenths. Write your answer in its simplest form, as a mixed number.
Now just because our answer is going to be a mixed number doesn’t mean we need to work with mixed numbers throughout. In fact, whenever we’re performing fraction arithmetic, we need to work with improper fractions; those are top heavy. Let’s begin by converting three and two-thirds into an improper fraction. The first thing we do is we multiply the integer, that’s the whole number, by the denominator. In this case, that’s three multiplied by three, which is nine. We then add the numerator to this number; that’s two add nine, which is 11. This number is the numerator to our improper fraction. The denominator stays the same. So three and two-thirds is the same as eleven-thirds.
Let’s repeat this process for one and five thirteenths. Multiplying the integer by the denominator gives us one multiplied by 13, which is 13. We then add the numerator of five and get 18. And since the denominator stays the same, one and five thirteenths as an improper fraction is 18 over 13. So three and two-thirds multiplied by one and five thirteenths is the same as eleven-thirds multiplied by eighteen thirteenths.
Our next step would usually be to multiply the numerator of the first fraction by the numerator of the second and then repeat that with the denominators. However, we can perform something called cross cancelling. We look diagonally across at the numbers and see if any of them share a common factor. 18 and three share a factor of three, so we can divide both of these numbers by three. Three divided by three is one, and 18 divided by three is six. 11 and 13 don’t share any common factors apart from one. That’s called coprime.
So we multiply the numerator of the first fraction by the new numerator of the second. That’s 11 multiplied by six which is 66. And for the denominators, we multiply one by 13, which is 13. Our answer is in its simplest form, but it’s not yet a mixed number. We need to reverse the process from earlier. This time, we divide the numerator by the denominator. That’s 66 divided by 13. The first six multiples of 13 are 13, 26, 39, 52, 65, and 78. So we can see that
16  divided by 13 is five with a remainder of one. We then use this remainder as the numerator for our fraction part in our mixed number. The denominator still stays the same. And that means that sixty-six thirteenths is equivalent to five and one thirteenth. Three and two-thirds multiplied by one and five thirteenths is five and one thirteenth.