Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{14} \div \frac{1}{7}$$

Step-by-Step Solution

Answer

The solution to \(\frac{1}{14} \div \frac{1}{7}\) is \(\frac{1}{2}\).

Step 1: Write the Problem

The problem is given as \(\frac{1}{14} \div \frac{1}{7}\). It must be rewritten as a multiplication problem. In division of fractions, the reciprocal of the divisor (the second fraction) is found, and the problem is changed to multiplication.

Step 2: Find the Reciprocal

The reciprocal of a fraction is found by swapping its numerator and denominator. So, the reciprocal of \(\frac{1}{7}\) is \(\frac{7}{1}\).

Step 3: Rewrite the Problem as Multiplication

Now, re-write the division problem as a multiplication problem using the reciprocal of the divisor. So we get \(\frac{1}{14} \times \frac{7}{1}\).

Step 4: Multiply the Fractions

When multiplying fractions, multiply across the numerators and the denominators. \(\frac{1 \times 7}{14 \times 1} = \frac{7}{14}\).

Step 5: Simplify the Result

The fraction \(\frac{7}{14}\) can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 7. Hence, \(\frac{7}{14} = \frac{1}{2}\).

Key Concepts

Reciprocal in Mathematics

When you divide one fraction by another, you're actually using a concept called the "reciprocal." The reciprocal of a fraction is simply what you get when you flip the fraction upside down. You take the numerator (the top part) and swap it with the denominator (the bottom part). For example, the reciprocal of \( \frac{1}{7} \) is \( \frac{7}{1} \). This is because after flipping the numerator and denominator, the places are swapped.

Using reciprocals makes dividing fractions much easier, because you transform the division problem into a multiplication problem. This technique is universally applied in mathematics to simplify complex calculations. So, whenever you see a division involving fractions, remember to find the reciprocal of the divisor.

Simplifying Fractions

After performing mathematical operations involving fractions, such as multiplication or division, it's always important to simplify the result. Simplifying a fraction means reducing it to its most basic form. In this reduction process, you are looking for the greatest common divisor (GCD) of the numerator and the denominator.

For example, with \( \frac{7}{14} \), you would look for the largest number that evenly divides both 7 and 14. The GCD here is 7. Divide both the numerator and the denominator by 7 to get the simplified fraction \( \frac{1}{2} \).

This step ensures that your final answer is clear and as manageable as possible. It's a fundamental practice in math that helps in making calculations more straightforward.

Multiplication of Fractions

Multiplying fractions is much simpler compared to adding or subtracting them. There are no common denominators to worry about here. You simply multiply the numerators together to get the new numerator, and the denominators together to get the new denominator.

Consider the fraction multiplication in \( \frac{1}{14} \times \frac{7}{1} \). Here, you multiply the numerators \(1 \times 7\), getting 7, and the denominators \(14 \times 1\), resulting in 14. So, the multiplication of these fractions results in \( \frac{7}{14} \).

• Always remember to multiply straight across.
• Check your answer to see if it can be simplified.

Multiplying fractions is straightforward and can often lead to great insights when solving problems in mathematics, ensuring clear and concise solutions.