Dividing fractions with whole numbers and mixed numbers

Dividing mixed numbers is very similar to multiplying mixed numbers. You just add one step—after changing the divisor into an improper fraction, you then find its reciprocal and multiply. Let's work through a "word problem" example.

The SuperQuik Market has just installed new scanners for its check-out lanes. They claim the average time to check out a customer is 2 ½ minutes. How many customers, on average, can they check out in half an hour?

To solve this problem, we have to know that half an hour is the same as 30 minutes. Then we can divide 30 by 2 ½.

First step: Write the whole number and the mixed number as improper fractions.

Second step: Write the reciprocal of the divisor, 2/5, and multiply.

Third step: Simplify, if possible. Notice that we can simplify our problem at this step, to make our calculations easier. Five goes evenly into 30, so we can divide both 5 and 30 by 5, to give 1 and 6.

Fourth step: Perform the simple multiplication of the numerators and the denominators. We find that the market can check out 12 customers in 30 minutes with its new scanners.

Fifth step: Put the answer in lowest terms, and check the answer. Our answer is already in lowest terms, so there is nothing left to do but check the answer, to be sure it makes sense. We can use estimation and rounding to do our check. If we round 2 ½ minutes to 3 minutes and divide 3 into 30, we get 10 customers in 30 minutes. So it is reasonable that 2 more customers per half hour, or 12 customers, can be checked, since 2 ½ minutes per customer is less than 3 minutes per customer.

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Video transcript

- [Tutor] Let's see if we can figure out what four and four fifths divided by one and one half is and I encourage you to pause the video and see if you can figure it out on your own and I'll give you a hint, see if you can rewrite these mixed numbers as what is sometimes called improper fractions. Alright, now let's do this together, so how can we rewrite four and four fifths? Well, four and four fifths is the same thing as, if we take the four, that's the same thing as four plus four fifths, four plus four fifths and four is the same thing as 20 fifths, so 4 is the same thing as 20 fifths and then plus four fifths, well, what are you going to get? Your going to get, you add the numerators, you get 24 fifths, 24 fifths, another way to think about it is take this denominator, take the fifths, multiply by four, you get 20 fifths, plus the four fifths that you already have is 24 fifths and so this is the same things as 24 fifths divided by, use the same idea, one and a half is the same thing as one plus one half, one is the same thing as two halves plus one half and so that's going to be, add the numerators, that's going to be three halves, so just like this, we're able to rewrite our expression as 24 fifths divided by three halves and now the key realization is is that that is the same thing as 24 fifths times the reciprocal of three halves, so times, pause the video, what's the reciprocal of three halves? Well, the reciprocal of three halves, you just swap the numerator and the denominator, it's going to be two over three, now what is this going to be? Well, there's a couple of ways to do it, you could just straight up multiply the numerators and you would get 48 and then multiply the denominators and you would get 15, so you get 48 over 15, but you might be able to rewrite that in sometimes what's called a more simplified way, but another way of thinking about this is you could just say, well this is the same thing as 24 times two, times two over five times three and simplify things, before you even multiply them out, five times five times three and you realize that look, 24 and three are both divisible by three, so let me divide them both by three, so 24 divided by three is eight and three divided by three is equal to one and then you could multiply the numerators and the denominators and so, and so you get in the numerator, eight times two is 16, in the denominator, you get a five, so you get 16 fifths and then if you wanna express that as a mixed number, 16 over five, well, five goes into 16 three times with one left over, so this is three and one fifth and one thing to appreciate, right over here, I simplified that the 24 and the three of this step, sometimes you'll see people simplifying at this step, so they'll say, hey look, eventually I'm gonna have a 24 in the numerator and a three in the denominator, so let me divide both of those by three, so they'll say 24 divided by three is eight and then three divided by three is one and this is sometimes called a cross reduction, but this is all that's going on right over here.

How do you divide fractions involving mixed numbers?