Problem 13 - Algebra 1

Question

Find the quotient. $$\frac{16}{\frac{-2}{9}}$$

Step-by-Step Solution

Answer

The quotient is -72.

Step 1: Identify fractions and compute reciprocal

The given value that 16 is divided by is a fraction, \(-\frac{2}{9}\). We find the reciprocal of this fraction by exchanging the numerator and the denominator. The fraction becomes \(-\frac{9}{2}\). Remember that the negative sign stays with the reciprocal too.

Step 2: Perform multiplication

Now, perform multiplication between the original whole number and the reciprocal fraction obtained from the previous step. Therefore, \(16 \times -\frac{9}{2} = -72\)

Step 3: State the result

The quotient of 16 divided by \(-\frac{2}{9}\) is -72.

Key Concepts

Reciprocal of a Fraction

In mathematics, understanding the concept of the reciprocal of a fraction is key when dealing with division involving fractions. The reciprocal of a given fraction is simply another fraction, where the numerator and denominator are swapped. This means if you have a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \) is not zero, the reciprocal is \( \frac{b}{a} \).

For example, if we start with \( \frac{2}{3} \), the reciprocal would be \( \frac{3}{2} \). It's essential to note that the product of a fraction and its reciprocal is always 1. One interesting aspect is dealing with negative fractions. When you calculate the reciprocal of a negative fraction, you must retain the negative sign, so the reciprocal of \( -\frac{2}{9} \) is \( -\frac{9}{2} \). Always remember, the reciprocal is particularly useful when dividing fractions, which leads us into multiplying fractions to find the quotient.

Multiplying Fractions

Multiplying fractions is a fundamental operation in arithmetic, but it's often misperceived as a difficult task. In reality, it's quite straightforward. To multiply two fractions, you simply multiply the numerators together to find the new numerator, and multiply the denominators together to find the new denominator.

Here's a breakdown:

• Multiply the numerators: \( \frac{a}{b} \) times \( \frac{c}{d} \) would result in \( ac \) as the new numerator.
• Multiply the denominators: In the same example, you would multiply \( b \) and \( d \) to get \( bd \) as the new denominator.
• The result will be \( \frac{ac}{bd} \).

This process doesn't change if fractions are mixed numbers or if negative signs are involved. Just make sure to multiply the numerators and denominators separately. In the case of a whole number and a fraction, treat the whole number as a fraction with the number as the numerator and 1 as the denominator. For instance, when multiplying \( 16 \) by \( -\frac{9}{2} \) as in our exercise, you consider 16 to be \( \frac{16}{1} \) and follow the same steps to get \( -72 \).

Arithmetic Operations with Fractions

Arithmetic operations with fractions might sometimes seem confusing, but with a bit of practice, they can become second nature. Dividing fractions, as in our original exercise, is actually similar to multiplying fractions, with a small twist - you use the reciprocal of the divisor. So, while multiplication involves direct operation on numerators and denominators, division requires you to first find the reciprocal of the divisor and then follow the multiplication rule.

When adding or subtracting fractions, it's important to have a common denominator. Once you have that, you can simply add or subtract the numerators. For instance, when adding \( \frac{1}{4} \) and \( \frac{3}{4} \) you would get \( \frac{4}{4} \), which simplifies to 1. Remember to always simplify your answers, which may require finding the greatest common divisor (GCD) to reduce fractions to their simplest form. It's also good to note that operations with mixed numbers usually involve converting them to improper fractions first. The arithmetic of fractions is a vast topic but mastering these foundational skills will allow you to tackle more complex mathematical tasks with confidence.