Problem 68 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$\frac{\frac{3}{5}-\frac{7}{10}}{\frac{1}{2}}$$

Step-by-Step Solution

Answer

-1/5

Step 1: Subtraction within the numerator

First, subtract the two fractions in the numerator: \(\frac{3}{5}-\frac{7}{10}\). To do this, find a common denominator, which is 10 in this case, then subtract the fractions accordingly: \(\frac{6}{10}-\frac{7}{10} = -\frac{1}{10}\)

Step 2: Simplification

Now that we have the new numerator, the expression then becomes: \(\frac{-1/10}{1/2}\). To simplify the fraction division, we multiply the numerator by the reciprocal of the denominator: \(-\frac{1}{10} * \frac{2}{1} = -\frac{2}{10}\)

Step 3: Final Simplification

Finally, we simplify the fraction \(-\frac{2}{10}\) to its simplest form: \(-\frac{1}{5}\)

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them is a fundamental skill. Simplifying, also known as reducing, involves rewriting a fraction in its simplest form, where the numerator and denominator share no common factors other than 1. To achieve this, you divide both the numerator and the denominator by their greatest common divisor (GCD).

For example, take the fraction \(-\frac{2}{10}\). We notice that both 2 and 10 can be divided by 2. Dividing the numerator and denominator by 2, the fraction reduces to \(-\frac{1}{5}\). It's important to always simplify fractions as a final step, because it makes them easier to work with in future calculations and allows you to clearly see the ratio that the fraction represents.

Common Denominators

Another crucial concept in fraction operations is the use of common denominators. This is necessary when adding or subtracting fractions, just as seen in the exercise with \(\frac{3}{5}-\frac{7}{10}\). To combine these fractions, they must have the same denominator. Consider the denominators 5 and 10; the smallest number that both can divide into without a remainder is 10, making it the least common denominator (LCD).

To adjust the fractions to have this common denominator, we multiply the numerator and denominator of \(\frac{3}{5}\) by 2, ending up with \(\frac{6}{10}\). Now that both fractions have the denominator of 10, it's straightforward to perform the subtraction: \(\frac{6}{10} - \frac{7}{10} = -\frac{1}{10}\). Always remember to convert fractions to a common denominator before attempting to add or subtract them.

Fraction Division

The division of fractions is tackled differently than addition or subtraction. To divide one fraction by another, as in the example \(\frac{-1/10}{1/2}\), we multiply the first fraction (the dividend) by the reciprocal of the second fraction (the divisor). The reciprocal of a fraction is simply flipping its numerator and denominator.

So, in our case, we convert the division into a multiplication problem: \(-\frac{1}{10} \times \frac{2}{1}\). Carry out the multiplication normally, across the numerator and denominator, which gives us \(-\frac{2}{10}\). Don't forget to simplify the result, as mentioned previously, and in this example, we get \(-\frac{1}{5}\) as the answer. This process is essential because it transforms division into multiplication, which is generally a more straightforward operation when dealing with fractions.