Problem 90 - Algebra for College Students

Question

Simplify each numerical expression. $$2\left(\frac{3}{8}\right)-5\left(\frac{1}{2}\right)+6\left(\frac{3}{4}\right)$$

Step-by-Step Solution

Answer

\( \frac{11}{4} \)

Step 1: Distribute Multiplication

Distribute each multiplication inside the expression: \[ 2 \times \left(\frac{3}{8}\right) - 5 \times \left(\frac{1}{2}\right) + 6 \times \left(\frac{3}{4}\right) \] Evaluate each multiplication term separately in the next steps.

Step 2: Calculate Each Product

Calculate each of the multiplication operations from Step 1:For the first term, \[ 2 \times \frac{3}{8} = \frac{6}{8} = \frac{3}{4} \] For the second term, \[ -5 \times \frac{1}{2} = -\frac{5}{2} \] For the third term, \[ 6 \times \frac{3}{4} = \frac{18}{4} = \frac{9}{2} \]

Step 3: Combine the Simplified Terms

Combine the results from Step 2: \[ \frac{3}{4} - \frac{5}{2} + \frac{9}{2} \] Convert all terms to have a common denominator of 4: \[ \frac{3}{4} - \frac{10}{4} + \frac{18}{4} \]

Step 4: Perform the Addition and Subtraction

Simplify the expression by performing the addition and subtraction:Combine \[ \frac{3}{4} + \frac{18}{4} = \frac{21}{4} \] Then subtract: \[ \frac{21}{4} - \frac{10}{4} = \frac{11}{4} \]

Step 5: Simplify the Result if Needed

The result of \( \frac{11}{4} \) is already in its simplest form, as 11 is a prime number and does not share any factors with 4 other than 1.Thus, the simplified expression is \( \frac{11}{4} \).

Key Concepts

Distributive Property

The distributive property is a valuable tool in algebra that allows us to simplify expressions by multiplying a single term by terms inside a set of parentheses. In our example, we used the distributive property to break down the original expression:

• Multiply: \( 2\left( \frac{3}{8} \right) \)
• Multiply: \(-5\left( \frac{1}{2} \right) \)
• Multiply: \(6\left( \frac{3}{4} \right) \)

Using this technique, we distribute each factor across the corresponding fraction, simplifying the calculations that follow. The beauty of the distributive property is that it helps us manage complicated expressions by focusing on easily workable parts.

Common Denominator

Having a common denominator is essential when adding or subtracting fractions. This means that each fraction has the same number as the denominator, allowing for straightforward arithmetic operations. For the expression \( \frac{3}{4} - \frac{5}{2} + \frac{9}{2} \), the denominators initially differ. To address this:

• We notice \( \frac{5}{2} \) and \( \frac{9}{2} \) need adjustment to match \( 4 \).
• Convert by finding equivalent fractions: Convert \( \frac{5}{2} \) to \( \frac{10}{4} \) and \( \frac{9}{2} \) to \( \frac{18}{4} \).

After conversion, all terms can seamlessly combine. This approach makes sure our calculations are both smooth and accurate.

Prime Numbers

Prime numbers play a special role in mathematics, especially when simplifying fractions. A prime number is one with only two positive divisors: 1 and itself. In the final expression \( \frac{11}{4} \), we determine that it is in its simplest form because 11 is a prime number and shares no common factors with 4 except for 1.Understanding prime numbers:

• They cannot be divided evenly except by 1 and themselves.
• This quality is crucial in identifying irreducible fractions.

By recognizing a numerator as prime, we quickly confirm that the fraction is in its simplest form, as no further simplification is possible.

Numerical Expressions

Numerical expressions involve combinations of numbers and operation signs (+, −, ×, ÷). They require evaluation based on the order of operations. Let's see how it applies to our original numerical expression:

• The expression is \( 2\left( \frac{3}{8} \right) - 5\left( \frac{1}{2} \right) + 6\left( \frac{3}{4} \right) \).

Initially, we tackle each operation separately using the distributive property. Then, we manage the fractions with common denominators. Lastly, we add or subtract the simplified fractions as guided by arithmetic rules.To solve a numerical expression effectively:

• Address each operation step-by-step.
• Use mathematical properties (like distributive) strategically.

This method ensures accurate computations and simplifies complex expressions efficiently.