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Chapter 11: Problem 31

Divide and then reduce your answers to lowest terms. $$ \frac{1}{60} \div \frac{1}{2}= $$

Short Answer

Expert verified
\( \frac{1}{60} \div \frac{1}{2} = \frac{1}{30} \)

Step by step solution

01

Convert Division into Multiplication

To divide fractions, change the division sign to multiplication and invert the divisor (the second fraction). Thus, \( \frac{1}{60} \div \frac{1}{2} \) becomes \( \frac{1}{60} \times \frac{2}{1} \).
02

Multiply the Fractions

Multiply the numerators and the denominators. Numerator: \(1 \times 2 = 2\) Denominator: \(60 \times 1 = 60\) So, \( \frac{1}{60} \times \frac{2}{1} = \frac{2}{60} \).
03

Simplify the Resulting Fraction

To reduce \( \frac{2}{60} \) to its lowest terms, find the greatest common divisor (GCD) of 2 and 60, which is 2. Divide both the numerator and the denominator by their GCD:\( \frac{2 \div 2}{60 \div 2} = \frac{1}{30} \).

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Key Concepts

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

Fraction Division
Dividing fractions might seem tricky at first, but it's really a transformation to multiplication. Here's how you do it: take the fraction you're dividing by (called the divisor) and flip it upside down. This means you swap its numerator and denominator. This flipped fraction is known as the reciprocal. Now, instead of dividing by the original fraction, you multiply by its reciprocal. For example, in the problem \( \frac{1}{60} \div \frac{1}{2} \), you change it to a multiplication problem: \( \frac{1}{60} \times \frac{2}{1} \). That simple twist makes dividing fractions straightforward!
Fraction Multiplication
Once converted into a multiplication problem, dealing with fractions becomes much more straightforward. All you need to do is multiply straight across: multiply the numerators together and the denominators together.
  • Multiply Numerators: If you have \( \frac{1}{60} \times \frac{2}{1} \), multiply the numerators: 1 and 2, giving you 2.
  • Multiply Denominators: Do the same with the denominators: 60 and 1, resulting in 60.
When you put it together: \( \frac{1 \times 2}{60 \times 1} = \frac{2}{60} \). Multiplication is simple because you don鈥檛 need to worry about finding a common denominator like in addition or subtraction of fractions. Just multiply and go!
Simplifying Fractions
After multiplying, the final step is to simplify your fraction to its lowest terms. Simplifying fractions means making them as simple as possible. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD). For \( \frac{2}{60} \), the GCD of 2 and 60 is 2. Divide the top and bottom by 2:
  • Numerator: \( 2 \div 2 = 1 \)
  • Denominator: \( 60 \div 2 = 30 \)
So, \( \frac{2}{60} \) simplifies to \( \frac{1}{30} \). Simplifying fractions is important because it gives us the cleanest, easiest form of the fraction to work with in future calculations.

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