Calculate nine tenths divided by seven and one-fifth. Give your answer in simplest form.
Immediately we remember that dividing fractions means multiplying by the reciprocal. Before we do anything else, we’ll set our problem to be a multiplication problem, with the reciprocal of seven and one-fifth. I need to find the reciprocal of seven and one-fifth. But seven and one-fifth is a mixed number. Before I can find it’s reciprocal, I need to write it as an improper fraction.
To write seven and one-fifth as an improper fraction, I multiply seven by the denominator of five and then I’ll add one, which is the current numerator of one-fifth. Seven and one-fifth is the same thing as thirty-six fifths. We need to find the reciprocal of thirty-six fifths. The reciprocal of thirty-six fifths is five thirty-sixths. So we’ll change our division symbol to multiplication, and we’ll flip thirty-six fifths and turn into five thirty-sixths.
Now I need to multiply nine-tenths times five thirty-sixths. I could multiply nine times five equals forty-five and ten times thirty-six equals three hundred and sixty. But then I would need to simplify forty-five over three hundred and sixty, and that’s not such an easy task.
In this case, it will be easier for us to simplify before we multiply. I recognise that five and ten are both divisible by five. Five divided by five equals one, and ten divided by five equals two. I noticed that nine and thirty-six are both divisible by nine. Nine divided by nine equals one and thirty-six divided by nine equals four. If we simplify before we multiply, our multiplication becomes a lot easier. One-half times one-fourth equals one-eighth. Forty-five over three hundred and sixty actually simplifies to one-eighth. They’re the same value. But our question is looking specifically for the simplest form of our fraction, which is one-eighth. Nine-tenths divided by seven and one-fifth equals one-eighth.