Problem 69 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$-\frac{9}{4}\left(\frac{1}{2}\right)+\frac{3}{4} \div \frac{5}{6}$$

Step-by-Step Solution

Answer

The simplified value of the given expression is \( -\frac{9}{40} \).

Step 1: Process the parentheses

First, simplify the expression inside the parentheses \( -\frac{9}{4}\left(\frac{1}{2}\right) \). This results in \( -\frac{9}{8} \). Now the expression is \( -\frac{9}{8}+\frac{3}{4} \div \frac{5}{6} \).

Step 2: Perform Multiplication and Division

Next, perform the division operation \( \frac{3}{4} \div \frac{5}{6} \). This is mathematically equivalent to \( \frac{3}{4} \times \frac{6}{5} \) yielding \( \frac{9}{10} \). Therefore, the expression is now \( -\frac{9}{8} + \frac{9}{10} \).

Step 3: Perform Addition and Subtraction

Finally, add the two simplified fractions together, \( -\frac{9}{8} + \frac{9}{10} \). To add these fractions, we must find a common denominator. Multiplying the denominators together (8*10), we find that 80 is the least common denominator. Convert the fractions to have this denominator: \( -\frac{90}{80} + \frac{72}{80} = -\frac{18}{80} \). Reducing this to lowest terms yields \( -\frac{9}{40} \).

Key Concepts

Fraction Simplification

Fraction simplification is all about reducing a fraction to its simplest form. It involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. By doing this, you make the fraction easier to work with and understand. For example, suppose you have the fraction \( \frac{18}{24} \). You would start by finding the GCD of 18 and 24, which is 6. Then, divide both the numerator and the denominator by 6, giving you \( \frac{3}{4} \). This new fraction is simpler and equivalent to the original fraction.

Here are simple steps to simplify any fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Write down the simplified fraction.

Ensuring your fractions are simplified before further operations, such as addition or subtraction, is key to avoiding mistakes and confusion.

Multiplication and Division of Fractions

Multiplying and dividing fractions might seem tricky at first, but once you learn the procedure, it becomes quite straightforward. To multiply fractions, you simply multiply the numerators together and the denominators together. For instance, if you multiply \( \frac{2}{3} \times \frac{4}{5} \), you multiply 2 and 4, and 3 and 5, resulting in \( \frac{8}{15} \).

When dividing fractions, you actually multiply by the reciprocal (or "flip") of the divisor. This is because division is the inverse of multiplication.

• Find the reciprocal of the fraction you are dividing by.
• Multiply the first fraction by this reciprocal.

For example, \( \frac{3}{4} \div \frac{5}{6} \) becomes \( \frac{3}{4} \times \frac{6}{5} \). Applying the rule of multiplication gives \( \frac{9}{10} \), as demonstrated in the solution.

Least Common Denominator

Finding the least common denominator (LCD) is key when adding or subtracting fractions with different denominators. You cannot just add fractions directly if they don't have the same denominator. The least common denominator is the smallest number that is a multiple of each of the denominators involved.

To find the LCD, you can:

• List multiples of each denominator.
• Find the smallest number common in both lists.
• Alternatively, use prime factorization of each denominator, then multiply each factor by the greatest number of times it appears in any one factorization.

In our example with \( -\frac{9}{8} + \frac{9}{10} \), the denominators are 8 and 10. By listing multiples, you find that the smallest common multiple is 80. Once you have the LCD, convert each fraction so that they both have this common denominator and proceed with the addition or subtraction.