Video Transcript
Which of the following division expressions has a quotient of one and one-third? a) Six and five-ninths divided by two and eight-ninths. b) Nine and two-sevenths divided by three and four-fifths. c) Eight and one-fourth divided by nine and six-sevenths. d) Nine and three-fifths divided by two and one-sixth. Or e) Four and two-thirds divided by three and one-half.
One way to solve this problem would be to convert all of these mixed numbers into improper fractions and then divide. But it could be helpful to eliminate some of the five before we start doing division. We can eliminate some answer choices by estimating the quotient of these five problems. We’ll estimate the quotient by rounding all of these values to the nearest whole number. Six and five-ninths rounds to seven, because five out of nine is more than one-half. Two and eight-ninths rounds to three, because eight-ninths is greater than one-half. Seven divided by three is seven-thirds. And we can break that up into six over three plus one over three. Six over three simplifies to two. We can say that the quotient of option a is about two and one-third.
We’ll do the same thing for option b. Nine and two-sevenths rounds down to nine, because two-sevenths is less than one-half. Three and four-fifths rounds up to four because four-fifths is greater than one-half. Nine divided by four can be written as nine-fourths. We can break that up into two parts, eight-fourths plus one-fourth. And we know that eight-fourths is equal to two. And so the quotient of option b is about two and one-fourth. In option c, eight and one-fourth rounds down to eight. One-fourth is less than one-half. Nine and six-sevenths rounds up to 10. Six-sevenths is greater than one-half. And here we have eight divided by 10. We can write that as eight-tenths. But notice, eight-tenths is less than one. This is because we have a smaller number eight and one-fourth. And it’s being divided by a larger number, nine and six-sevenths.
Whenever we have a smaller number being divided by a larger number, the result will always be less than one. And so we can say for certain that eight and one-fourth divided by nine and six-sevenths cannot be equal to one and one-third. We know nine and three-fifths rounds up to 10. Two and one-sixth rounds down to two. And 10 divided by two equals five. So we can estimate that nine and three-fifths divided by two and one-sixth will be about five. Four and two-thirds rounds up to five. Two-thirds is greater than one-half. Three and one-half rounds up to four. Because if we have a value that’s equal to half, we round up. And so to estimate, we need to divide five by four, which will give us five-fourths. We can break that into four-fourths plus one-fourth. Four-fourths is equal to one. So our estimate for four and two-thirds divided by three and one-half is one and one-fourth.
At this point, we’ve eliminated option c. But we still have four options. We have estimates two and one-third, two and one-fourth, five, and one and one-fourth. We can be fairly confident in eliminating option d, as our estimate is five. And five is very far off from one and one-third. From there, we still have three options. It’s clear to us that option e, one and one-fourth, is closest to one and one-third as a and b have estimate quotients of two and one-third and two and one-fourth. The best option for us here is to go ahead and calculate four and two-thirds divided by three and one-half exactly.
To calculate four and two-thirds divided by three and one-half, we need to convert both of these mixed numbers to improper fractions. First, four and two-thirds. We multiply the whole number value four by the denominator three. And then we add the two that was in the numerator. Four times three plus two is the numerator for our improper fraction. And the denominator stays the same. Four times three is 12. And 12 plus two is 14. Four and two-thirds written as an improper fraction is fourteen-thirds. We need to find the improper fraction for three and one-half. We can use the method that I’ve just shown of multiplying the whole number by the denominator and adding the numerator. But we can also write three with the denominator of two. We do that by saying three, written as a fraction out of two, is six over two. Six divided by two is three. And if we add six-halves plus one-half, we get seven-halves. Three and one-half written as an improper fraction is seven over two.
If we want to divide these two improper fractions, we keep the 14 over three. Change our division to multiplication and take the reciprocal of our divisor. 14 over three divided by seven over two is equal to 14 over three times two over seven. We can simplify 14 divided by seven equals two. And then we need to multiply two times two equals four. Three times one equals three. Fourteen-thirds divided by seven-halves equals four-thirds. If we wanna convert this four-thirds back into a mixed number, we could say four-thirds equals three-thirds plus one-third. And we know three-thirds is the whole number one. Four-thirds is equal to one and one-third. And so we can say that four and two-thirds divided by three and one-half equals one and one-third.
And so we can say that option e has a quotient of one and one-third.
