Problem 10 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$-\frac{8}{9} \div 3 \frac{1}{5}$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{5}{18}\).

Step 1: Convert Mixed Number to Improper Fraction

First, you need to convert the mixed number \(3 \frac{1}{5}\) into an improper fraction. To do this, multiply the whole number (3) by the denominator (5), which equals 15. Then add the numerator (1), which gives you 16. So, \(3 \frac{1}{5}\) becomes \(\frac{16}{5}\).

Step 2: Rewrite the Division as Multiplication

Change the division problem to a multiplication problem by multiplying by the reciprocal of the divisor. So, the division \(-\frac{8}{9} \div \frac{16}{5}\) becomes \(-\frac{8}{9} \times \frac{5}{16}\).

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together: \(-\frac{8}{9} \times \frac{5}{16} = \left(-8 \times 5 \right) / \left(9 \times 16\right) = -40/144\).

Step 4: Simplify the Fraction

Simplify \(-\frac{40}{144}\) by finding the greatest common divisor (GCD) of 40 and 144, which is 8. Divide both the numerator and the denominator by 8: \(-\frac{40}{144} = -\frac{5}{18}\).

Key Concepts

Division of Fractions

Dividing fractions might initially seem challenging, but it actually becomes straightforward once you understand the process.
To divide fractions, you follow three main steps:

• Convert mixed numbers to improper fractions if necessary.
• Change the division to multiplication by using the reciprocal of the divisor.
• Simplify your answer at the end.

Let's put this into practice with the example \(-\frac{8}{9} \div 3 \frac{1}{5}\). After converting \(3 \frac{1}{5}\) into an improper fraction, you rewrite the division as multiplication: \(-\frac{8}{9} \times \frac{5}{16}\). The division of fractions boils down to multiplying fractions and then simplifying the result.

Improper Fractions

Understanding improper fractions is key when working with divisions involving mixed numbers.
An improper fraction has a numerator larger than or equal to its denominator.This happens when you convert a mixed number into a fraction.
For example, converting \(3 \frac{1}{5}\) to an improper fraction involves multiplying the whole number (3) by the denominator (5) and adding the numerator (1).
This gives us \(\frac{{16}}{{5}}\), an improper fraction because the numerator (16) is larger than the denominator (5). Remembering this process helps ensure you work accurately with mixed numbers in division.

Reciprocal

The reciprocal is a crucial concept in division of fractions.
In simple terms, the reciprocal of a fraction is what you multiply by to get one.This means you swap the numerator and the denominator.
When dividing fractions, like in the expression \(-\frac{8}{9} \div \frac{16}{5}\), finding the reciprocal of the divisor lets you turn the division into multiplication.
The reciprocal of \(\frac{16}{5}\) is \(\frac{5}{16}\). By using the reciprocal, you're simplifying division problems to multiplication problems, which are easier to handle.

Simplifying Fractions

Simplifying fractions is the final and crucial step in fraction multiplication and division.
Simplification makes your answer neat and easier to understand.To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator.
For \(-\frac{40}{144}\), the GCD is 8.
Dividing both the numerator and the denominator by 8, you simplify it to \(-\frac{5}{18}\).
The process ensures your fraction is in its simplest form, which is important for clear communication of your results.