Problem 60 - Prealgebra

Question

There are two ways to work the problems below. You can combine the fractions inside the parentheses first and then multiply, or you can apply the distributive property first, then add. $$15\left(\frac{4}{5}-\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The result of the expression is 7.

Step 1: Combine the Fractions

First, we need to find a common denominator for the fractions \(\frac{4}{5}\) and \(\frac{1}{3}\). The least common multiple of 5 and 3 is 15. Rewrite each fraction with a denominator of 15: \(\frac{4}{5} = \frac{12}{15}\) and \(\frac{1}{3} = \frac{5}{15}\). Now, subtract the second fraction from the first: \(\frac{12}{15} - \frac{5}{15} = \frac{7}{15}\).

Step 2: Multiply by the Whole Number

Now that we have combined the fractions inside the parentheses, let's multiply the result by 15. We have: \(15 \times \frac{7}{15}\). When we multiply, the 15 in the numerator and the 15 in the denominator cancel each other out, leaving us with 7.

Key Concepts

Understanding Fractions

Fractions are a way of expressing numbers that are not whole. They consist of a numerator, which is the top part, and a denominator, which is the bottom part. For example, in \(\frac{4}{5}\), 4 is the numerator and 5 is the denominator. The denominator tells us into how many equal parts a whole is divided, while the numerator tells us how many of those parts we are considering.

• Simple fractions have both a numerator and a denominator that are whole numbers.
• Improper fractions have a numerator larger than their denominator, like \(\frac{7}{4}\).
• Mixed numbers include both a whole number and a fraction, such as \(2\frac{1}{2}\).

Converting fractions to have the same denominator helps in addition or subtraction. For example, converting \(\frac{4}{5}\) and \(\frac{1}{3}\) to have a common denominator of 15 allows straightforward arithmetic: \(\frac{12}{15} - \frac{5}{15} = \frac{7}{15}\). This process is fundamental in simplifying expressions and solving problems.

The Distributive Property Explained

The distributive property is a useful algebraic principle that helps to simplify expressions and solve equations. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. Mathematically, this is expressed as \(a(b + c) = ab + ac\).
When applied to fractions, the distributive property allows you to simplify expressions by distributing a whole number to each term inside a parenthesis. For example, with the expression \(15\left(\frac{4}{5} - \frac{1}{3}\right)\), you can distribute 15 to both \(\frac{4}{5}\) and \(-\frac{1}{3}\).

• First, apply the multiplication separately: \(15 \times \frac{4}{5} = 12\) and \(15 \times \frac{1}{3} = 5\).
• Then subtract the two results: \(12 - 5 = 7\).

Using the distributive property provides an alternate solution path that can sometimes be more intuitive, especially when dealing with complex expressions.

Finding the Least Common Multiple

The Least Common Multiple (LCM) of two numbers is the smallest multiple that is shared between them. It is particularly useful for handling fractions with different denominators, as it allows you to find a common denominator.
To calculate the LCM of numbers like 5 and 3, consider the multiples of each:

• Multiples of 5 are 5, 10, 15, 20...
• Multiples of 3 are 3, 6, 9, 12, 15...

The smallest multiple they both share is 15, making 15 the LCM. Using the LCM as a common denominator is the trick to adding or subtracting fractions. It standardizes the fractions so they can be easily compared and combined.