Problem 92 - Prealgebra

Question

Simplify. $$15\left(\frac{4}{5}-\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Distribute the 15

Begin by considering the expression: \( 15\left(\frac{4}{5} - \frac{1}{3}\right) \). Distribute the 15 to both terms inside the parentheses. This gives us \( 15 \times \frac{4}{5} - 15 \times \frac{1}{3} \).

Step 2: Simplify Each Term Separately

Calculate each term separately. First, simplify \( 15 \times \frac{4}{5} \):\( 15 \times \frac{4}{5} = \frac{15 imes 4}{5} = \frac{60}{5} = 12 \).Now, simplify \( 15 \times \frac{1}{3} \):\( 15 \times \frac{1}{3} = \frac{15 imes 1}{3} = \frac{15}{3} = 5 \).

Step 3: Subtract the Simplified Terms

Now subtract the second term from the first: \( 12 - 5 = 7 \).

Key Concepts

Understanding Fractions

Fractions are a fundamental concept in prealgebra and are used to represent parts of a whole. A fraction consists of two numbers: a numerator and a denominator. The numerator is the number above the line, showing how many parts you have. The denominator is the number below the line, indicating how many equal parts the whole is divided into. For example, in the fraction \(\frac{4}{5}\), "4" is the numerator and "5" is the denominator.

When dealing with fractions, it's crucial to understand how to perform basic arithmetic operations such as addition, subtraction, multiplication, and division. To multiply fractions, you multiply the numerators together and the denominators together. For example, to multiply \(\frac{2}{3}\) and \(\frac{4}{5}\), you do \(\frac{2 \times 4}{3 \times 5} = \frac{8}{15}\).

• Multiplying by whole numbers can be thought of as multiplying the whole number by the numerator while keeping the denominator the same.
• Fractions can also be simplified by finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by this number.
• Simplifying makes fractions easier to work with and compare.

The Distributive Property

The distributive property is a valuable tool in algebra, especially when simplifying expressions involving parentheses. It states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results. Mathematically, this can be written as \(a(b + c) = ab + ac\). The same rule applies when there is subtraction: \(a(b - c) = ab - ac\).

In the exercise, we have 15 distributed across the terms inside the parentheses: \((\frac{4}{5} - \frac{1}{3})\). This means mupltiplying 15 by each fraction separately. For example:

• Applying the distributive property gives us \(15 \times \frac{4}{5} - 15 \times \frac{1}{3}\).
• This allows each multiplication to be simplified individually before any subtraction occurs.

The property helps break down complex expressions into simpler steps, making them easier to solve. It's especially helpful when working with fractions, as it allows us to manage each part independently before combining the results.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining the same value. This is a crucial skill in algebra as it helps in solving equations and understanding complex expressions more clearly. The goal is to simplify the expression to make it easier to work with.

• In the given exercise, you have already used the distributive property to break down \(15(\frac{4}{5} - \frac{1}{3})\) into simpler parts.
• Each fractional multiplication was simplified by multiplying across the fractions and reducing where possible.

For example, \(15 \times \frac{4}{5}\) simplifies as follows: multiply the whole number by the numerator, then divide by the denominator, \(\frac{60}{5} = 12\). Similarly, handle \(15 \times \frac{1}{3}\) to get \(\frac{15}{3} = 5\).

Once you have both simplified terms, subtraction is straightforward: from 12, subtract 5 to get 7. Simplifying expressions in this way helps ensure calculations are correct and efficient, preparing you for solving more advanced math problems later.