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Magazine Chapter 3: Problem 25
Find the following quotients. $$\left(\frac{3}{4} \div 2 \frac{1}{2}\right) \div 3$$
Short Answer
Expert verified
The quotient is \(\frac{1}{10}\).
Step by step solution
01
Convert Mixed Number to Improper Fraction
Change the mixed number, \(2 \frac{1}{2}\), into an improper fraction. Multiply the whole number by the denominator (2*2) and add the numerator. This equals 4 + 1 = 5, so the improper fraction is \(\frac{5}{2}\).
02
Perform the First Division
Divide \(\frac{3}{4}\) by \(\frac{5}{2}\) by multiplying \(\frac{3}{4}\) by the reciprocal of \(\frac{5}{2}\). Therefore, multiply \(\frac{3}{4}\) by \(\frac{2}{5}\), which equals \(\frac{3 \times 2}{4 \times 5} = \frac{6}{20}\).
03
Simplify the Resulting Fraction
Simplify \(\frac{6}{20}\) by dividing the numerator and the denominator by their greatest common divisor (2). This gives \(\frac{3}{10}\).
04
Perform the Second Division
Divide \(\frac{3}{10}\) by 3. To do this, multiply \(\frac{3}{10}\) by the reciprocal of 3, which is \(\frac{1}{3}\). This results in \[\frac{3}{10} \times \frac{1}{3} = \frac{3 \times 1}{10 \times 3} = \frac{3}{30}\].
05
Simplify the Final Result
Simplify \(\frac{3}{30}\) by dividing the numerator and the denominator by their greatest common divisor (3). This gives the final simplified fraction, \(\frac{1}{10}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mixed Numbers
A mixed number is a combination of a whole number and a fraction. It is a way of expressing numbers that are between two whole numbers. For example, the mixed number \(2 \frac{1}{2}\) represents the quantity that is the sum of 2 and a half, or an additional \(\frac{1}{2}\).
- To convert a mixed number to an improper fraction, follow these steps:
- Multiply the whole number by the denominator of the fraction.
- Add the result to the numerator.
- Place this sum over the original denominator.
Reciprocal
Reciprocals are key in fraction division. Understanding them can make division less confusing. A reciprocal of a fraction is simply flipping its numerator and denominator.
- For instance, the reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\).
- Likewise, the reciprocal of a whole number, like 3, is \(\frac{1}{3}\).
Simplifying Fractions
Simplifying, or reducing fractions, is the process of making them as simple as possible. This means shortening the fraction to its smallest equivalent expression, while keeping its value.
- To simplify, identify the greatest common divisor (GCD) of the numerator and denominator.
- Divide both the numerator and the denominator by this number.
Improper Fractions
Improper fractions might seem unusual because the numerator is larger than the denominator. This means the fraction represents a value greater than one.
- Converting mixed numbers to improper fractions is useful, especially in mathematical operations.
- For instance, \(2 \frac{1}{2}\) gets converted into \(\frac{5}{2}\), clarifying the calculation process.
