Problem 104 - Elementary Algebra

Question

\(\frac{1}{2}(11-4)-\frac{3}{4}(15-12)\)

Step-by-Step Solution

Answer

\(\frac{5}{4}\)

Step 1 - Simplify Inside Parentheses

First, simplify the expressions inside the parentheses. For the first part, simplify \(11 - 4\) and for the second part, simplify \(15 - 12\). \(11 - 4 = 7\) and \(15 - 12 = 3\).

Step 2 - Substitute Simplified Values

Next, substitute the simplified values back into the original expression to get \(\frac{1}{2} \cdot 7 - \frac{3}{4} \cdot 3\).

Step 3 - Perform Multiplication

Now, perform the multiplications: \(\frac{1}{2} \cdot 7 = \frac{7}{2}\) and \(\frac{3}{4} \cdot 3 = \frac{9}{4}\). So the expression becomes \(\frac{7}{2} - \frac{9}{4}\).

Step 4 - Create Common Denominator

To subtract the fractions, find a common denominator. The lowest common denominator for 2 and 4 is 4. Convert \(\frac{7}{2}\) to \(\frac{14}{4}\). Now the expression is \(\frac{14}{4} - \frac{9}{4}\).

Step 5 - Subtract Fractions

Finally, subtract the fractions. \(\frac{14}{4} - \frac{9}{4} = \frac{5}{4}\).

Key Concepts

Order of operations

In mathematics, it’s crucial to follow the order of operations correctly. This ensures we get the right answer every time. The order of operations can be remembered with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Let’s see how this applies in our example exercise \(\frac{1}{2}(11-4)-\frac{3}{4}(15-12)\):

• First, perform the operations inside the parentheses: 11-4 and 15-12. This gives us 7 and 3 respectively.
• Then, handle any multiplication or division from left to right: \(\frac{1}{2} \cdot 7\) and \(\frac{3}{4} \cdot 3\).
• Finally, perform any addition or subtraction from left to right. In this case, we subtract the fractions we obtained in the previous step.

By following these steps, we ensure our solution remains accurate.

Fractions

Fractions represent parts of a whole. They're written in the form \(\frac{a}{b}\), where 'a' is the numerator (the number of parts we have), and 'b' is the denominator (the total number of equal parts).

In the given exercise, we encounter fractions multiple times: \(\frac{1}{2}\) and \(\frac{3}{4}\). Here are some key points to understand fractions better:

• Multiplying fractions: Multiply the numerators together and the denominators together. For example, multiplying \(\frac{1}{2} \) by 7 gives us \(\frac{1 \cdot 7}{2} \).
• Dividing fractions: Involves multiplying by the reciprocal of the divisor. The reciprocal of \(\frac{3}{4} \) is \(\frac{4}{3} \).
• Subtracting fractions: Requires a common denominator, as we'll see in the next section.

Understanding these basic operations with fractions is crucial for simplifying expressions and solving problems.

Common denominators

To add or subtract fractions, they must share a common denominator. This ensures the pieces being added or subtracted are of the same size.

Let's look at our example: \(\frac{7}{2} - \frac{9}{4}\). To subtract these fractions, we need a common denominator. Here’s how to do it:

• Identify the least common denominator (LCD) for the fractions. For 2 and 4, the LCD is 4.
• Convert each fraction to an equivalent fraction with this common denominator. For \(\frac{7}{2} \), multiply both the numerator and the denominator by 2 to get \(\frac{14}{4}\).
• Now, we can easily subtract the fractions: \(\frac{14}{4} - \frac{9}{4} = \frac{5}{4} \).

By understanding and finding common denominators, we can perform addition or subtraction of fractions accurately.