Problem 88 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the value of each expression. $$\frac{9}{10}-\left(\frac{1}{4}-\frac{7}{10}\right)$$

Step-by-Step Solution

Answer

The value of the expression is \(1\frac{7}{20}\).

Step 1: Solve within the brackets

According to the order of operations, the first step is to solve the operation within the brackets, \(\frac{1}{4}-\frac{7}{10}\). To do this, you need to find a common denominator for the two fractions. Here, the common denominator for \(1/4\) and \(7/10\) is 20. So, rewrite the fractions as \(\frac{5}{20}-\frac{14}{20}\) and now subtract the fractions within the brackets to get: \(-\frac{9}{20}\). So the new expression to evaluate becomes \(\frac{9}{10}-(-\frac{9}{20})\).

Step 2: Simplify the expression

Next, simplify the expression by resolving the minus signs. The term \(-(-\frac{9}{20})\) becomes \(+\frac{9}{20}\). Therefore, the expression simplifies to \(\frac{9}{10}+\frac{9}{20}\).

Step 3: Addition of fractions

Now, add these two fractions. Since the denominators are not the same, first find a common denominator. The common denominator for \(9/10\) and \(9/20\) is 20, so rewrite the fractions as \(\frac{18}{20}+\frac{9}{20}\). Add the fractions to get \(\frac{27}{20}\).

Step 4: Simplify the fraction

The final step is to simplify the fraction. Given that this is an improper fraction, convert it to a mixed number. Divide 27 by 20 to get 1 and a remainder of 7. So, the simplified and final answer is \(1\frac{7}{20}\).

Key Concepts

Order of Operations

When tackling mathematical expressions, it's important to follow the order of operations. This ensures you get the correct result every time. You may remember the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

The order of operations tells us to solve any math inside parentheses first. In our problem, we started by dealing with the operation inside the brackets first. This method ensures that we handle complex expressions in a structured manner. Once the innermost operations are done, we move step by step according to the hierarchy set by the operations guidelines. Remember, getting the sequence right is crucial for accurate results.

Common Denominator

Fractions with different denominators can't be directly added or subtracted unless they are rewritten with a common denominator. A common denominator is a shared multiple of the denominators of each fraction.

For example, in the expression \( \frac{1}{4} - \frac{7}{10} \), the denominator 4 and 10 do not match. Their smallest common multiple is 20. To convert the fractions, \( \frac{1}{4} \) was rewritten as \( \frac{5}{20} \) and \( \frac{7}{10} \) was rewritten as \( \frac{14}{20} \). This conversion lets us subtract these fractions easily.

Finding the common denominator allows us to handle fraction operations effectively and should be a standard step when dealing with sums or subtractions in fraction problems.

Improper Fractions

An improper fraction is where the numerator is larger than the denominator. These fractions often arise during operations like addition or subtraction of fractions.

In the expression\( \frac{9}{10} + \frac{9}{20} \), after adding, we obtained \( \frac{27}{20} \). Since 27 is larger than 20, \( \frac{27}{20} \) is an improper fraction. Although improper fractions are mathematically correct, it's often useful to convert them into a mixed number for better understanding or simplified reporting.

Mixed Numbers

A mixed number combines a whole number and a fraction. It is a more intuitive way to express quantities larger than one whole.

To convert an improper fraction to a mixed number involves simple division. For \( \frac{27}{20} \), divide 27 by 20. This gives a quotient of 1 with a remainder of 7. Hence, \( \frac{27}{20} \) becomes the mixed number \( 1\frac{7}{20} \).

Mixed numbers are very useful in everyday measurements and reportings, making the results easier to grasp, especially for larger quantities.