Question

find the quotient. $$ -20 \div\left(-2 \frac{1}{2}\right) $$

Short answer

The quotient of -20 divided by -2 1/2 is 8.

Step-by-step solution

Convert mixed number to improper fraction

Convert the mixed fraction -2 1/2 into an improper fraction. The conversion can be done using the formula: (whole part * denominator) + numerator/denominator.\nSo, -2 1/2 = -2*2 - 1/2 = -5/2.

Division of number by a fraction

To divide a number by a fraction, multiply that number by the reciprocal of the fraction. Here, reciprocal of -5/2 is -2/5.\n\nSo, -20 ÷ (-5/2) becomes -20 * (-2/5).

Perform multiplication

The operation now is multiplication. Multiply -20 by -2/5.\n\n -20 * (-2/5) = 8.

Key concepts

Improper Fraction

An improper fraction is a type of fraction where the numerator (the top number) is larger than or equal to the denominator (the bottom number). Unlike proper fractions, which represent values less than one, improper fractions represent a value equal to or greater than one. For example, in the fraction \( \frac{5}{2} \), the numerator 5 is greater than the denominator 2, making it improper. Improper fractions can be easily mistaken for mixed numbers because they both represent values greater than one, but improper fractions remain solely in fraction form. Understanding improper fractions is a fundamental step in various mathematical operations, especially when dealing with division and multiplication of fractions.

In the given problem, the mixed number \( -2 \frac{1}{2} \) must be converted into an improper fraction for easier computation during division, and it becomes \( -\frac{5}{2} \).

Mixed Number Conversion

A mixed number is a representation of a whole number combined with a proper fraction. Converting a mixed number to an improper fraction is necessary to perform operations like division or multiplication. This process involves a simple formula: \[ (\text{whole part} \times \text{denominator}) + \text{numerator} = \text{improper fraction}. \]

For the conversion, multiply the whole number by the denominator and then add the numerator to the result. Following this formula ensures accurate results when using fractions in mathematical equations. In our example, the mixed number \( -2 \frac{1}{2} \) is converted by multiplying -2 (the whole number) by 2 (the denominator) and then subtracting 1 (the numerator) to get \( -\frac{5}{2} \) as the improper fraction.

Multiplicative Inverse

The multiplicative inverse, also known as the reciprocal, is a number which, when multiplied by the original number, gives a product of one. To find the reciprocal of a fraction, you simply swap the numerator and the denominator. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \), provided neither a nor b is zero.

When dividing fractions, instead of performing the division directly, we multiply by the multiplicative inverse of the divisor. In our exercise, the reciprocal (or multiplicative inverse) of \( -\frac{5}{2} \) is \( -\frac{2}{5} \). This reversed fraction is then used to turn the division problem into a multiplication problem, facilitating simpler computation.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are the building blocks of mathematics. With fractions, these operations have specific rules. For example, to add or subtract fractions, one needs a common denominator. To multiply fractions, you multiply the numerators together and the denominators together. And as seen in our exercise, to divide by a fraction, you multiply by its reciprocal.

This fundamental knowledge allows students to approach problems systematically and to perform complex calculations with ease. The exercise we reviewed involves converting a mixed number to an improper fraction, finding the multiplicative inverse, and finally, multiplying it by another number to find the quotient, which in this case, equals 8.