Calculate one and three-fifths divided by four-fifths all divided by five and seven- eighths. Give your answer in simplest form.
So where do we start? We need to follow the order of operations. First we take care of what’s inside the parentheses, then exponents, then multiply or divide whichever comes first left to right, and then add or subtract whatever comes first left to right.
So let’s begin working inside these parentheses. So we need to divide these fractions. Now one of these fractions is a mixed number. So let’s change it to an improper fraction and then we can divide the fractions. So let’s change it to an improper fraction. Well first, what is an improper fraction?
It’s where the numerator is larger than the denominator but it looks like a normal fraction, just a number divided by a number. So one, we can rewrite with having a denominator of five by making it five over five. So it’s one and three-fifths. So we need to make it five over five and three-fifths.
So when add fractions, we add the numerators and keep the common denominator. So we get eight-fifths. So let’s go ahead and replace the one and three-fifths with eight-fifths. So now we still need to simplify inside the parentheses. We need to divide those two fractions.
When we divide, we actually change it to a multiplication problem and we need to flip the second fraction, make it a reciprocal. So we’re going to multiply by the reciprocal of the second fraction. So now let’s multiply. Now when we multiply, we can simplify first. We could multiply the numerators together and the denominators and simplify. But five goes into five once, and four goes into eight twice, and four goes into four once.
So the numerator would be two times one and the denominator would be one times one. So we have two divided by five and seven-eighths. Now let’s say we weren’t comfortable with simplifying how we did. So we could’ve multiplied straight across and got 40 for the numerator and 20 for the denominator. But 40 divided by 20 is two. So we could’ve done it that way.
Now here we have another mixed number. So let’s change it to an improper fraction. How can we rewrite five so has a denominator of eight? Well, we can take eight times five to be 40, and 40 divided by eight is five. So we can make 40 eighths, the five. And now we need to add the seven-eighths. So we add the numerators to be 47, and we keep our denominator of eight.
So we have 47 eighths. So now we have two divided by 47 eighths. We can rewrite two to be two over one and then divide by 47 eighths. So just like we did before, change the division sign to multiplication and then flip the second fraction. We multiply by the reciprocal, so eight 47ths.
So there is nothing on the numerator that simplifies a something on the denominator. So we need to multiply two times eight, which is 16. And on the denominator, one times 47 is 47. So we get a final answer of 16 47ths.