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Magazine Chapter 2: Problem 44
Divide the following fractions and mixed numbers. Reduce to lowest terms. $$\frac{7}{8} \div 14=$$
Short Answer
Expert verified
\( \frac{1}{16} \)
Step by step solution
01
Convert Whole Number to Fraction
First, we need to convert the whole number 14 into a fraction. Any whole number can be written as a fraction by putting it over 1. So we can write 14 as \( \frac{14}{1} \).
02
Rewrite the Division as Multiplication
To divide by a fraction, we multiply by its reciprocal. Therefore, we rewrite the division problem of \( \frac{7}{8} \div \frac{14}{1} \) as \( \frac{7}{8} \times \frac{1}{14} \).
03
Multiply the Fractions
Multiply the numerators together and the denominators together. Multiply the numerators: 7 \( \times \) 1 = 7. Multiply the denominators: 8 \( \times \) 14 = 112. So, \( \frac{7}{8} \times \frac{1}{14} = \frac{7}{112} \).
04
Reduce the Fraction to Lowest Terms
To reduce \( \frac{7}{112} \), find the greatest common divisor (GCD) of 7 and 112. The GCD is 7. Divide the numerator and the denominator by 7: \( \frac{7}{112} \div \frac{7}{7} = \frac{1}{16} \). Thus, the fraction in its lowest terms is \( \frac{1}{16} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mixed Numbers
Mixed numbers are a combination of whole numbers and fractions. For example, the mixed number 2\(\frac{3}{4}\) consists of the whole number 2 and the fraction \(\frac{3}{4}\). Understanding mixed numbers is crucial in arithmetic operations like addition, subtraction, multiplication, and division involving fractions.
- Converting Mixed Numbers: To simplify calculations, mixed numbers are often converted into improper fractions. For instance, to convert 2\(\frac{3}{4}\) to an improper fraction, multiply the whole number by the denominator and add the numerator (2 \times 4 + 3 = 11), then place over the original denominator: \(\frac{11}{4}\).
- Reverse Conversion: It's equally essential to know how to convert an improper fraction back to a mixed number. Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder over the denominator forms the fractional part. For \(\frac{11}{4}\), divide 11 by 4 to get 2 R3, so it becomes 2\(\frac{3}{4}\).
Reciprocal
A reciprocal is a concept where you take a fraction and flip the numerator and the denominator. This operation might sound simple, but it's incredibly useful, especially in fraction division. When dividing by a fraction, you multiply by its reciprocal. For instance, to find the reciprocal of \(\frac{14}{1}\), flip it to get \(\frac{1}{14}\).
- Basic Usage: The reciprocal of a number always gives a product of 1 when multiplied with the original number. For example, \(\frac{14}{1} \times \frac{1}{14} = 1\).
- Uses in Division: When facing division problems with fractions, such as \(\frac{7}{8} \div \frac{14}{1}\), it is converted into a multiplication problem \(\frac{7}{8} \times \frac{1}{14}\). This often leads to simpler calculations.
Reducing Fractions
Reducing fractions involves simplifying a fraction to its smallest possible form. This step is significant to ensure the fraction is as straightforward and minimal as possible, representing the same value. In our exercise, \(\frac{7}{112}\) was reduced to \(\frac{1}{16}\).
- Process: To reduce fractions, divide both the numerator and the denominator by their greatest common divisor (GCD).
- Why Reduce: It ensures fractions are in their simplest form, making them easier to understand and work with. For example, comparing \(\frac{1}{16}\) is more straightforward than \(\frac{7}{112}\).
Greatest Common Divisor
The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. This concept is essential in reducing fractions and other operations involving numbers.
- How to Find the GCD: There are various methods to find the GCD, such as Prime Factorization, the Euclidean algorithm, and listing all divisors. In the exercise, the GCD of 7 and 112 is 7.
- Using GCD to Reduce Fractions: Divide both the numerator and the denominator by their GCD to simplify the fraction. Like in our example: \(\frac{7}{112} \div \frac{7}{7} = \frac{1}{16}\).
