Problem 89 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5}{2}\).

Step 1: Distribute the Numbers

First, distribute the numbers 3, 4, and 2 to the fractions \( \frac{1}{2} \), \( \frac{2}{3} \), and \( \frac{5}{6} \) respectively.\[3 \cdot \frac{1}{2} = \frac{3}{2}\]\[4 \cdot \frac{2}{3} = \frac{8}{3}\]\[2 \cdot \frac{5}{6} = \frac{10}{6}\]

Step 2: Simplify Fractions if Possible

Make sure each fraction is in its simplest form:\[\frac{10}{6} = \frac{5}{3} \text{ (Divide numerator and denominator by 2)}\]

Step 3: Find a Common Denominator

The fractions \(\frac{3}{2}\), \(\frac{8}{3}\), and \(\frac{5}{3}\) need a common denominator for addition and subtraction. The least common denominator is 6:\[\frac{3}{2} = \frac{9}{6}\]\[\frac{8}{3} = \frac{16}{6}\]\[\frac{5}{3} = \frac{10}{6}\]

Step 4: Perform the Operations

Now add and subtract the fractions:\[\frac{9}{6} + \frac{16}{6} - \frac{10}{6}\]Combine the fractions:\[\frac{9 + 16 - 10}{6} = \frac{15}{6}\]

Step 5: Simplify the Result

Simplify the fraction \(\frac{15}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:\[\frac{15}{6} = \frac{5}{2}\]

Key Concepts

Numerical Expressions

A numerical expression is a mathematical phrase that includes numbers, operations, and sometimes parentheses, but it does not include an equality or inequality symbol. In this exercise, the numerical expression is \(3\left(\frac{1}{2}\right)+4\left(\frac{2}{3}\right)-2\left(\frac{5}{6}\right)\). These expressions are used to represent a specific value based on the operation performed.

When simplifying numerical expressions, it’s essential to follow the order of operations to correctly evaluate the expression. Start with any calculations inside parentheses, followed by exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). In our case, multiplication of whole numbers by fractions came first. This step is vital in simplification before any further computations can occur.

Common Denominator

A common denominator is a shared multiple of the denominators of several fractions. Finding a common denominator is crucial when adding or subtracting fractions so that the fractions can be combined seamlessly.

In the exercise, fractions such as \(\frac{3}{2}\), \(\frac{8}{3}\), and \(\frac{5}{3}\) need a common denominator for the addition and subtraction process to proceed.

The least common denominator (LCD) for these is determined by identifying the smallest multiple that each original denominator divides into evenly.

Here, the LCD found is 6, allowing us to rewrite each fraction:

• \(\frac{3}{2} = \frac{9}{6}\)
• \(\frac{8}{3} = \frac{16}{6}\)
• \(\frac{5}{3} = \frac{10}{6}\)

Mastering how to find a common denominator simplifies the process of fraction operations.

Distributive Property

The distributive property is a helpful mathematical rule used in algebra to simplify expressions. It states that a number multiplied by the sum or difference of two numbers can be distributed into separate products. This property is expressed as:

• \(a(b + c) = ab + ac\)
• \(a(b - c) = ab - ac\)

In this exercise, the distributive property was used to distribute numbers 3, 4, and 2 across fractions:

• \(3 \times \frac{1}{2} = \frac{3}{2}\)
• \(4 \times \frac{2}{3} = \frac{8}{3}\)
• \(2 \times \frac{5}{6} = \frac{10}{6}\)

Distributing the numbers first simplifies the expression, setting the stage for further simplification by combining like terms and reaching the final simplified fraction.

Fraction Operations

Fraction operations come into play when you add, subtract, multiply, or divide fractions. In this exercise, we primarily dealt with adding and subtracting fractions.

After ensuring all fractions have a common denominator, they can be added or subtracted by combining the numerators while keeping the denominator the same.

For instance, the operation was:

• \(\frac{9}{6} + \frac{16}{6} - \frac{10}{6} = \frac{15}{6}\)

This result is then simplified by finding the greatest common divisor for the numerator and the denominator. In this case, both 15 and 6 have a greatest common divisor of 3, leading us to a simplified result of \(\frac{5}{2}\).

By mastering these operations, you can simplify fraction expressions efficiently, making it easier to interpret and use them in various mathematical problems.