Find six-sevenths divided by three-sevenths.
To answer this question, we’ll firstly look at the standard method for dividing two fractions. We will also look at a shortcut that can be used when the denominators are the same. Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of a fraction is the fraction upside down. The numerator becomes the denominator and vice versa. You might have seen this method referred to as KCF. We keep the first fraction the same. We change the division sign to a multiplication sign. And we flip the second fraction.
Three over seven becomes seven over three, as this is the reciprocal. Six over seven divided by three over seven is the same as six over seven multiplied by seven over three. In order to multiply fractions, we can just multiply the numerators and then, separately, multiply the denominators. In this question, seven multiplied by six is 42. Seven multiplied by three is 21. This means that our fraction becomes 42 over 21. As 21 is a half of 42, this is equal to two.
You might have noticed, however, that we could cross simplify or cross cancel before multiplying the fractions. We can divide both of the sevens by seven, so they become one. Three and six have a common factor of three. Six divided by three is two, and three divided by three is one. We’re then left with two multiplied by one on the top and one multiplied by one on the bottom. This is equal to two over one, which simplifies to two. Six-sevenths divided by three-sevenths is equal to two.
As mentioned at the start of this question, there is a shortcut we can use when the denominators are the same. Our first number is six-sevenths. And we need to divide this into three-sevenths. There are two lots of three-sevenths in six-sevenths. Therefore, six-sevenths divided by three-sevenths is equal to two. This leads us to a rule that if the denominators are the same, we can just divide the numerators. In this question, the numerators were six and three. Six divided by three is equal to two, which gives us the same answer as the traditional method.