Evaluate two and two-fifths times negative seven and one-fourth divided by negative four and five-sixths, giving the answer in its simplest form.
First, we copy down our expression. And then following the order of operations, we’ll need to calculate what’s inside the brackets first, which would be multiplying two and two-fifths by negative seven and one-fourth. However, when we’re working with mixed numbers, we always want to convert them to their improper fraction form. To calculate two and two-fifths as an improper fraction, we multiply two by five and get 10. And then we add the numerator of two so that we have 12. And the denominator doesn’t change; it stays five.
The improper fraction for negative seven and one-fourth will be seven times four plus one in the numerator. Seven times four is 28 plus one is 29. The denominator remains four, and the whole fraction is negative, which gives us 12 over five times negative 29 over four. And now we’re able to multiply these values together. We see that four and 12 are both divisible by four, so we can simplify a bit. Then, we have three times negative 29 over five. Three times negative 29 is negative 87, which we write as a fraction over five. And then we’re ready to bring down the rest of the problem.
But what we want to do here is we’ll also need to rewrite negative four and five-sixths as an improper fraction. To do that, we’ll say four times six is 24 plus five is 29, which gives us a numerator of 29, a denominator of six, and the whole value is negative. So far, we’ve taken our expression and simplified it to say negative 87 over five divided by negative 29 over six. And when we divide by a fraction, we multiply by its reciprocal. So we can say negative 87 over five divided by negative 29 over six is equal to negative 87 over five times six over negative 29.
Now, remember that 87 was equal to three times negative 29, and that means negative 29 and negative 87 are both divisible by negative 29. Negative 29 divided by negative 29 is one. Negative 87 divided by negative 29 is three, which means we have three times six over five. Three times six is 18, and the denominator is five. 18 and five do not share any common factors apart from one, which means eighteen-fifths is in its simplest form. The expression we started with can be simplified to eighteen-fifths.