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Chapter 2: Problem 41

Divide the following fractions and mixed numbers. Reduce to lowest terms. $$\frac{15}{30} \div 10=$$

Short Answer

Expert verified
\( \frac{1}{20} \) is the simplified answer.

Step by step solution

01

Simplify the Fraction

First, simplify the fraction \( \frac{15}{30} \). This can be simplified by finding the greatest common divisor (GCD) of 15 and 30. The GCD is 15. So, \( \frac{15}{30} \) simplifies to \( \frac{1}{2} \) after dividing both numerator and denominator by 15.
02

Convert the Whole Number to a Fraction

Convert the whole number 10 into a fraction. You do that by expressing 10 as \( \frac{10}{1} \).
03

Change the Division to Multiplication

When dividing fractions or a fraction by a whole number, change the division sign to a multiplication sign and use the reciprocal of the divisor. Thus, \( \frac{1}{2} \div \frac{10}{1} \) becomes \( \frac{1}{2} \times \frac{1}{10} \).
04

Multiply the Fractions

Multiply the fractions: \( \frac{1}{2} \times \frac{1}{10} = \frac{1 \times 1}{2 \times 10} = \frac{1}{20} \).
05

Simplify the Result

Finally, check if the resulting fraction \( \frac{1}{20} \) can be simplified further. Since 1 and 20 do not have a common divisor other than 1, the fraction is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Simplification
Simplifying a fraction means reducing it to its lowest terms so that it still represents the same value with smaller numbers. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).
The denominator is the bottom part of a fraction, while the numerator is the top part.
  • For example, in the fraction \( \frac{15}{30} \), 15 is the numerator and 30 is the denominator.
  • We find the GCD of 15 and 30, which is 15.
You divide the numerator and denominator by the GCD to get the simplified fraction:
\[ \frac{15 \div 15}{30 \div 15} = \frac{1}{2} \]
This gives us the fraction \( \frac{1}{2} \) which is simplified or reduced to its lowest terms.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides two or more integers without leaving a remainder.
Finding the GCD of two numbers helps simplify fractions easily.
For the fraction \( \frac{15}{30} \):
  • List the factors of 15: 1, 3, 5, 15
  • List the factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
  • The common factors are 1, 3, 5, and 15, with 15 being the greatest.
Thus, the GCD of 15 and 30 is 15, which helps in simplifying the fraction to \( \frac{1}{2} \). Understanding GCD is an essential skill for fraction simplification.
Reciprocal of a Fraction
A reciprocal of a fraction is simply switching the positions of the numerator and the denominator.
If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
Reciprocals are incredibly useful in dividing fractions.
  • For instance, to divide \( \frac{1}{2} \) by \( \frac{10}{1} \), you take the reciprocal of \( \frac{10}{1} \) to get \( \frac{1}{10} \).
  • Then, you change the division to multiplication: \( \frac{1}{2} \div \frac{10}{1} \) becomes \( \frac{1}{2} \times \frac{1}{10} \).
This step transforms the division problem into a multiplication one, often making it easier to solve.
Multiplying Fractions
Multiplying fractions is a straightforward process and often simpler than dividing. To multiply fractions, you multiply the numerators together and the denominators together.
Here鈥檚 how you do it step-by-step:
  • Take the simplified fractions \( \frac{1}{2} \) and \( \frac{1}{10} \).
  • Multiply the numerators: \( 1 \times 1 = 1 \).
  • Multiply the denominators: \( 2 \times 10 = 20 \).
The result is \( \frac{1}{20} \).
After multiplying, always check if you can simplify the fraction further. However, \( \frac{1}{20} \) is already in its simplest form as there is no common factor between 1 and 20 other than 1.
This process of multiplying fractions ensures you reach the final result effectively.

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