Problem 59 - 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. $$15\left(\frac{2}{3}+\frac{3}{5}\right)$$

Step-by-Step Solution

Answer

The solution is 19.

Step 1: Find a common denominator

Before combining the fractions inside the parentheses, find the least common denominator (LCD) of the two fractions, \(\frac{2}{3}\) and \(\frac{3}{5}\). The denominators are 3 and 5, so the LCD is 15.

Step 2: Convert fractions to have the same denominator

Convert \(\frac{2}{3}\) and \(\frac{3}{5}\) to have the denominator of 15. Multiply the numerator and the denominator of \(\frac{2}{3}\) by 5 to get \(\frac{10}{15}\), and multiply the numerator and the denominator of \(\frac{3}{5}\) by 3 to get \(\frac{9}{15}\).

Step 3: Add the fractions

Add the fractions \(\frac{10}{15}\) and \(\frac{9}{15}\) inside the parentheses: \(\frac{10}{15} + \frac{9}{15} = \frac{19}{15}\).

Step 4: Multiply by the number outside the parentheses

Now, multiply \(15\) by \(\frac{19}{15}\). The "15" cancels out the denominator: \(15 \times \frac{19}{15} = 19\).

Step 5: Verification Step: Alternative solution using distribution

Apply the distributive property: \(15 \cdot \frac{2}{3} + 15 \cdot \frac{3}{5}\). Simplify each term separately: \(15 \cdot \frac{2}{3} = 10\) and \(15 \cdot \frac{3}{5} = 9\). Add these results: \(10 + 9 = 19\). This verifies our previous solution.

Key Concepts

Understanding the Least Common Denominator

When you're working with fractions, especially those with different denominators, finding the least common denominator (LCD) is essential.
This concept ensures that fractions are easy to add or subtract, as they are expressed with a common denominator.

To find the LCD:

• Identify the denominators of the fractions you are working with. For example, in \(\frac{2}{3}\) and \(\frac{3}{5}\), the denominators are 3 and 5.
• List out the multiples of each denominator until you find a common multiple. Here, multiples of 3 are 3, 6, 9, 12, 15, and so on, while multiples of 5 are 5, 10, 15, etc.
• The smallest number that appears in both lists is the least common denominator, which is 15 in this case.

The LCD facilitates operations on the fractions, enabling you to manage them uniformly.

The Distributive Property Explored

The distributive property is a useful algebraic property that allows you to simplify expressions in a more manageable way.
Instead of handling everything at once, you distribute one term across others in a sequence.

In mathematical terms, if you have \(a(b + c)\), it can be expanded to \(ab + ac\). This property is particularly handy because it helps divide complex operations into simpler parts.

For instance, given the problem \(15(\frac{2}{3} + \frac{3}{5})\), you can use the distributive property to:

• Calculate \(15 \times \frac{2}{3}\) and then \(15 \times \frac{3}{5}\).
• This breaks down potentially complicated arithmetic into two independent operations: \(10 + 9 = 19\).

Utilizing this method often makes calculations less prone to error and, importantly, more intuitive for learning.

The Basics of Fraction Addition

Adding fractions seems challenging at first, but it gets easier once the fractions share a common denominator.
This enables you to combine the numerators directly to find your answer.

With fractions like \(\frac{2}{3}\) and \(\frac{3}{5}\), once you rewrite them as \(\frac{10}{15}\) and \(\frac{9}{15}\), adding becomes simple:

• The denominators remain the same: 15.
• Add the numerators together: 10 + 9 = 19.
• Resulting in a sum of \(\frac{19}{15}\).

Make sure to consistently simplify fractions if needed, though \(\frac{19}{15}\) here is already in simplest form.
This technique ensures easy calculation and accurate results when working with different denominators.

Fundamentals of Fraction Multiplication

Multiplying fractions requires a straightforward approach, a contrast to addition or subtraction that often requires finding a common denominator.
Simply multiply the numerators and then multiply the denominators.

To solve the entire expression \(15 \times \frac{19}{15}\), consider the numerator:

• Multiply the whole number (15) by the fraction's numerator (19), leading to \(285\).
• This over the product of the denominators gives you \(\frac{285}{15}\).
• The denominator of 15 cancels with the 15 in the multiplication, leaving the simple result of 19.

Multiplication is often a direct, efficient path to solving problems involving fractions, and by paying attention to simplification opportunities, your results are both tidy and precise.