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Magazine Chapter 4: Problem 49
Divide the mixed fractions and express your answer as a mixed fraction. $$8 \div 2 \frac{2}{9}$$
Short Answer
Expert verified
The answer is \(3 \frac{3}{5}\).
Step by step solution
01
Convert Mixed Fraction to Improper Fraction
Convert the mixed fraction \(2 \frac{2}{9}\) to an improper fraction. To do this, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For \(2 \frac{2}{9}\), this becomes: \((2 \times 9) + 2 = 18 + 2 = 20\), so the improper fraction is \(\frac{20}{9}\).
02
Rewrite Division as Multiplication
Division by a fraction is the same as multiplying by its reciprocal. Hence, rewrite \(8 \div \frac{20}{9}\) as \(8 \times \frac{9}{20}\).
03
Multiply the Fractions
Rewrite 8 as a fraction: \(\frac{8}{1}\). Now, multiply the fractions: \(\frac{8}{1} \times \frac{9}{20} = \frac{72}{20}\).
04
Simplify the Fraction
Simplify \(\frac{72}{20}\) by finding the greatest common divisor (GCD) of 72 and 20, which is 4. Divide both the numerator and the denominator by 4: \(\frac{72 \div 4}{20 \div 4} = \frac{18}{5}\).
05
Convert Improper Fraction to Mixed Number
Convert \(\frac{18}{5}\) to a mixed number. Divide 18 by 5: 18 divided by 5 is 3 with a remainder of 3. So, \(\frac{18}{5}\) as a mixed number is \(3 \frac{3}{5}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Improper Fractions
When you're working with mixed numbers like \(2 \frac{2}{9}\), converting them to improper fractions is an essential step in operations such as division. An improper fraction is simply a fraction where the numerator is greater than the denominator. To convert a mixed number to an improper fraction, follow these steps:
- Multiply the whole number by the denominator.
- Add the result to the numerator of the fractional part.
- Place the result over the original denominator.
- Multiplying 2 (the whole number) by 9 (the denominator), which gives 18.
- Adding 2 (the original numerator) to 18, resulting in 20.
- Writing 20 over the original denominator of 9, resulting in the improper fraction \(\frac{20}{9}\).
Reciprocal Multiplication
When dividing fractions, an important concept to understand is "reciprocal multiplication". Instead of dividing by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is simply obtained by swapping its numerator and denominator.
For instance, the reciprocal of \(\frac{20}{9}\) is \(\frac{9}{20}\).
So, to compute \(8 \div \frac{20}{9}\), you would convert it into:
For instance, the reciprocal of \(\frac{20}{9}\) is \(\frac{9}{20}\).
So, to compute \(8 \div \frac{20}{9}\), you would convert it into:
- \(8 \times \frac{9}{20}\)
Simplifying Fractions
After performing operations like multiplication, the resulting fraction often needs to be simplified to be more manageable and easily understood. Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
Let's simplify \(\frac{72}{20}\):
Let's simplify \(\frac{72}{20}\):
- First, find the GCD of 72 and 20, which is 4.
- Divide both 72 and 20 by 4.
- This gives you \(\frac{18}{5}\), which is the simplest form of \(\frac{72}{20}\).
Mixed Numbers
Mixed numbers are a combination of whole numbers and fractions, and they offer a neat way to express improper fractions as a more understandable number. After performing arithmetic that results in an improper fraction, converting it into a mixed number can help clarify the result.
Here's how you convert an improper fraction like \(\frac{18}{5}\) into a mixed number:
Here's how you convert an improper fraction like \(\frac{18}{5}\) into a mixed number:
- Divide the numerator by the denominator to get the whole number part. For \(\frac{18}{5}\), dividing 18 by 5 gives 3.
- The remainder becomes the new numerator. In this case, 3 is the remainder, so it becomes \(3\frac{3}{5}\).
