Question

find the quotient. $$ 12 \div\left(-\frac{1}{5}\right) $$

Short answer

-60

Step-by-step solution

Understand Division with Negative Fraction

Dividing by a negative fraction is the same as multiplying by the negative reciprocal of the fraction. The negative reciprocal of \(-\frac{1}{5}\) is -5.

Multiply

Now proceed and perform the multiplication. Multiply 12 by -5. Use the normal multiplication method that's usually used for whole numbers.

Determine the Answer

After multiplication, you get -60 as the answer. This is your final quotient.

Key concepts

Negative Reciprocal

Understanding the concept of a negative reciprocal is a pivotal step when dividing by a negative fraction. The reciprocal of a number is simply a fraction flipped upside down, meaning we exchange the positions of the numerator and denominator. When dealing with negative fractions, this concept doesn't change but includes the negative sign.

For instance, the negative reciprocal of \(-\frac{1}{5}\) is \(-5\), derived by flipping the fraction to get \(\frac{5}{1}\) and retaining the negative sign. Remember, the negative reciprocal of a negative fraction is a positive whole number, and vice versa. In the context of division, the process of finding the negative reciprocal is integral to solving the problem through multiplication.

Multiplication of Fractions

Multiplying fractions is a straightforward process, although it can sometimes seem tricky when negative signs are involved. To multiply fractions, you simply multiply the numerators together and the denominators together.

However, when one of the fractions is a whole number, like in our exercise where we have 12 multiplied by the negative reciprocal of \(-\frac{1}{5}\) which is \(-5\), it essentially becomes the multiplication of a whole number by another whole number. The principal rule to remember here is that when multiplying a positive number by a negative number, the result is always negative. That's why, in our exercise, 12 multiplied by -5 yields -60.

Quotient of Division

The quotient is the result you get when you divide one number by another. In terms of fractions, when you divide by a fraction, it is equivalent to multiplying by its reciprocal. This concept is a bit counterintuitive but essential for solving division problems involving fractions.

Applying this to a negative fraction simply involves using the negative reciprocal, as we did in our exercise. Instead of dividing 12 by \(-\frac{1}{5}\), we multiply 12 by the negative reciprocal, which is -5. The quotient, in this case, is a whole number, -60. It's important to recognize that the sign of the quotient depends on the signs of the numbers involved in the division—since we divided a positive number by a negative fraction, our final quotient is negative.