Question Video: Adding Two Fractions and Dividing the Result by a Whole Number, a Mixed Number, or a Fraction

Calculate ((14/15) + (10/5)) ÷ (1(2/5)), giving your answer in its simplest form.


Video Transcript

Calculate fourteen fifteenths plus ten-fifths divided by one and two-fifths, giving your answer in its simplest form.

Remember, when we’re performing calculations involving mixed numbers, we always begin by turning those mixed numbers into improper fractions. Here we have one and two-fifths. And we know that to convert a mixed number into an improper fraction, we begin by multiplying the integer part by the denominator. Here that’s one times five, which is five. We then take that number and we add it to the numerator of the proper-fraction part. That gives us five plus two, which is equal to seven. This number forms the numerator part of our improper fraction. And the denominator is the same as the denominator in the proper fraction. So one and two-fifths is equal to seven-fifths.

Now, we’re also going to apply the order of operations. And we’re going to begin by performing the calculation inside the pair of parentheses. That’s fourteen fifteenths plus ten-fifths. Now we might also even notice that ten-fifths or 10 divided by five is equal to two. And then we might look to create a mixed number by adding two and fourteen fifteenths. But of course then we will need to convert that back into an improper fraction.

So let’s recall how we actually add fractions. We create a common denominator. So what we’re going to do is multiply both the numerator and denominator of our second fraction by three to give us a denominator of 15. When we do, we find that ten-fifths is equivalent to thirty fifteenths. And so we get fourteen fifteenths plus thirty fifteenths. And of course now we have that common denominator; we just add the numerators. And we get forty-four fifteenths. And so our calculation now becomes forty-four fifteenths divided by seven-fifths.

And we know that there are a couple of ways that we can divide fractions. Let’s look at the first method. That involves creating a common denominator. Once again, that denominator is actually going to be 15. And so we’re going to multiply the numerator and denominator of our second fraction by three. And so we get forty-four fifteenths divided by twenty-one fifteenths.

Now that the denominators are equal, we simply divide the numerators. We can write 44 divided by 21 as 44 over 21. And since we’re asked to give our answer in its simplest form, we’re going to finally turn this back into a mixed number. 44 divided by 21 is two with a remainder of two. So two forms the integer part, and then another two forms the numerator of the proper-fraction part. The denominator remains unchanged, so 44 over 21 is two and two twenty-oneths.

Now, of course, we do have one second method, so we’ll briefly consider that. In the second method, we simply multiply by the reciprocal of the second fraction, by the divisor. So forty-four fifteenths divided by seven-fifths is equal to forty-four fifteenths times five-sevenths. Then we could multiply the numerators and separately multiply the denominators. But we might notice that we can divide both five and 15 by five. And so now we do 44 times one to get 44 and three times seven to get 21. And once again we find that forty-four fifteenths divided by seven-fifths is 44 over 21, which we’ve seen is equal to two and two over 21.

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