Problem 38 - Algebra and Trigonometry

Question

Perform the indicated operations. \(\frac{\frac{1}{12}}{\frac{1}{8}-\frac{1}{9}}\)

Step-by-Step Solution

Answer

6

Step 1: Simplify the Denominator

The denominator of the expression is \(\frac{1}{8} - \frac{1}{9}\).First, find a common denominator for \(8\) and \(9\), which is \(72\).Rewrite the fractions: \(\frac{1}{8} = \frac{9}{72}\) and \(\frac{1}{9} = \frac{8}{72}\).Then, subtract the smaller fraction from the larger: \(\frac{9}{72} - \frac{8}{72} = \frac{1}{72}\).

Step 2: Simplify the Entire Expression

Now, the expression is \(\frac{\frac{1}{12}}{\frac{1}{72}}\).When dividing one fraction by another, multiply the first by the reciprocal of the second:\[ \frac{1}{12} \times \frac{72}{1} = \frac{1 \times 72}{12 \times 1} \].Simplify the fraction: \(\frac{72}{12} = 6\).

Key Concepts

Common Denominator

When dealing with fraction operations, finding a common denominator is a crucial step, especially when subtracting or adding fractions. A common denominator is a shared multiple of the denominators you are working with, which allows you to combine the fractions easily.

Here's what you can do to find the common denominator:

• List the multiples of each denominator. In this case, for 8 and 9, start like this: 8, 16, 24... and 9, 18, 27...
• Identify the smallest multiple that both denominators share. For 8 and 9, this is 72, as it's the first number both lists have in common.
• Once found, rewrite each fraction by multiplying both the numerator and denominator to get an equivalent fraction with the common denominator. For example, \(\frac{1}{8} = \frac{9}{72}\) and \(\frac{1}{9} = \frac{8}{72}\).

Now, with the same denominators, you can easily subtract: \(\frac{9}{72} - \frac{8}{72} = \frac{1}{72}\).

Finding common denominators is essential and simplifies the process of performing further operations on fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD).

When we deal with the fraction \(\frac{72}{12}\) from our exercise, we look for a number that both 72 and 12 can be evenly divided by. The GCD here is 12.

• To simplify, divide both the numerator and the denominator by this GCD: \(\frac{72 \div 12}{12 \div 12} = \frac{6}{1}\).

This results in the simplest form of the fraction, which is just 6. Simplifying fractions is a valuable skill because it makes the numbers smaller, easier to work with, and helps reach the final answer more effectively. Make a habit of always checking if you can break down a fraction further after any operation.

Reciprocal

The reciprocal of a fraction is basically flipping the numerator and the denominator. It's incredibly handy, particularly when dividing fractions.

To find a reciprocal, simply swap the roles of the numerator and the denominator:

• For \(\frac{1}{12}\), the reciprocal is \(\frac{12}{1}\).
• For \(\frac{1}{72}\), you get \(\frac{72}{1}\).

In operations like dividing fractions, you multiply by the reciprocal of the divisor. Consider the expression we were working on, \(\frac{\frac{1}{12}}{\frac{1}{72}}\). Instead of dividing, we'll multiply by the reciprocal of \(\frac{1}{72}\), which is \(\frac{72}{1}\). That’s why our expression becomes \(\frac{1}{12} \times \frac{72}{1} = \frac{72}{12}\).

The concept of reciprocals streamlines complex calculations and is a vital part of solving fraction operations.