Problem 11 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$2 \frac{1}{6} \div\left(-1 \frac{1}{5}\right)$$

Step-by-Step Solution

Answer

The quotient of \(2 \frac{1}{6} \div \left(-1 \frac{1}{5}\right)\) is \(-\frac{65}{36}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. For the mixed number 2 \(\frac{1}{6}\), multiply the whole number 2 by the denominator 6 (giving 12), and add the numerator 1 to get 13. The improper fraction is \(\frac{13}{6}\). For \(-1 \frac{1}{5}\), multiply the whole number 1 by the denominator 5 (giving 5), and add the numerator 1 to get 6. The improper fraction is \(-\frac{6}{5}\).

Step 2: Rewrite the Division as Multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \(2 \frac{1}{6} \div \left(-1 \frac{1}{5}\right)\) becomes \(\frac{13}{6} \times \left(-\frac{5}{6}\right)\).

Step 3: Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together. So, \(\frac{13}{6} \times \left(-\frac{5}{6}\right) = \frac{13 \times (-5)}{6 \times 6} = \frac{-65}{36}\).

Step 4: Simplify the Result

Check if the resulting fraction \(\frac{-65}{36}\) can be simplified. Since 65 and 36 do not have common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is a way to represent numbers that are between whole numbers, giving a clearer picture of quantities in terms of both whole parts and fractions.
For example, in the exercise given, we have the mixed numbers 2 \(\frac{1}{6}\) and \( -1 \frac{1}{5}\). Here, the first number consists of 2, which is the whole part, and \(\frac{1}{6}\), which is the fractional part.

To work comfortably with mixed numbers during calculations, they must be converted to improper fractions. It's a simple two-step process:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator to the result.

In the case of the mixed number 2 \(\frac{1}{6}\), multiply 2 by 6 to get 12, then add 1 to get 13. The improper fraction you'll use in the calculation is \(\frac{13}{6}\).
Understanding mixed numbers helps in visualizing the problem, especially in real-life situations like measuring ingredients or dividing objects.

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator. They often come from converting mixed numbers to a form that is easier to manipulate in mathematical calculations.
In our example, after converting the mixed numbers, we end up with improper fractions \(\frac{13}{6}\) and \(-\frac{6}{5}\). The negative sign here indicates direction, just like with whole numbers.

Working with improper fractions can seem less intuitive initially, but they are extremely useful as they allow for easier multiplication and division. Instead of juggling whole numbers and fractions simultaneously, everything is in a uniform fraction form.

• To use them in operations like multiplication, follow the straightforward approach: multiply numerators together and denominators together.
• Don’t forget the sign! Always pay attention to the negative or positive sign as it affects the final outcome.

Having converted mixed numbers into improper fractions, division calculations become more streamlined and less prone to errors.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1. This step helps in presenting the fraction in its most concise and understandable format.

For our division result \(\frac{-65}{36}\), check for any common factors between 65 and 36.
A quick inspection or usage of factorization reveals they have none in common. Thus, it already represents the simplest form possible.

• Verification of simplification is essential in ensuring final answers are presented clearly and accurately.
• When simplifying, always look out for the greatest common divisor (GCD) to divide both numerator and denominator. This gives the fraction in its lowest terms.

Simplifying is a vital part of fraction work, allowing for easier interpretation of results in a wide array of mathematical problems.