Problem 7 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. [Examples 1–3] $$2\left(1 \frac{1}{2}\right)+5\left(6 \frac{2}{5}\right)$$

Step-by-Step Solution

Answer

The simplified expression is 35.

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. For \(1 \frac{1}{2}\), multiply the whole number 1 by the denominator 2 and add the numerator 1, which gives us \(\frac{3}{2}\). Similarly, for \(6 \frac{2}{5}\), multiply 6 by 5 and add 2, giving us \(\frac{32}{5}\).

Step 2: Apply Order of Operations: Multiplication

Next, we apply the order of operations, starting with multiplication. We need to multiply \(2\) by \(\frac{3}{2}\) and \(5\) by \(\frac{32}{5}\). Calculating these gives us \(2 \times \frac{3}{2} = 3\) and \(5 \times \frac{32}{5} = 32\).

Step 3: Apply Order of Operations: Addition

Add the results from the multiplication step. We have the two results: 3 and 32. So we perform the addition \(3 + 32 = 35\).

Key Concepts

Mixed Numbers

Mixed numbers are a way to express quantities that involve both a whole number and a fractional part. They look something like this: \(1 \frac{1}{2}\). In mixed numbers, the number to the left of the fraction is called the whole number, and the fraction itself represents the part less than one.
To work with mixed numbers in operations like multiplication or division, it's often necessary to convert them into improper fractions first. An improper fraction is a single fraction where the numerator (top number) is greater than the denominator (bottom number). This is done by

• Multiplying the whole number by the denominator of the fraction part.
• Adding the resulting product to the numerator of the fraction part.
• Writing the sum over the original denominator.

For example, to convert the mixed number \(1 \frac{1}{2}\) into an improper fraction, you multiply 1 by 2 and add 1, resulting in \(\frac{3}{2}\). This conversion ensures that our calculations using mixed numbers are accurate and more straightforward.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This might seem unusual if you are used to seeing fractions that represent less than one whole. However, improper fractions are incredibly useful for calculations, especially when dealing with mixed numbers.
To convert a mixed number to an improper fraction, the formula involves simple multiplication and addition:

• Multiply the integer part of the mixed number by the denominator of the fractional part.
• Add this product to the numerator of the fractional part.
• Place this sum as the new numerator over the original denominator.

As an example, the mixed number \(6 \frac{2}{5}\) converts to \(\frac{32}{5}\) by multiplying 6 by 5, and then adding 2. This results in a single fraction, which simplifies further calculations, making operations like multiplication and addition easier.

Multiplication

In mathematics, multiplication is a fundamental operation indicated by the symbol \(\times\). When you multiply fractions, including improper ones, follow these steps:

• Multiply the numerators together to find the new numerator.
• Multiply the denominators together to find the new denominator.
• Simplify the resulting fraction if possible.

For example, multiplying fractions like \(2 \times \frac{3}{2}\) involves multiplying 2 by 3 for the numerator, giving you 6, and the denominator remains 2, so you get 3, simplified from \(\frac{6}{2}\).
When performing operations with mixed numbers, always convert them to improper fractions first to ensure precise results. Multiplication, when done correctly, leads to the intermediate values needed before combining terms in further operations like addition.

Addition

Addition is the process of finding the total or sum by combining two or more numbers. Once you've converted mixed numbers to improper fractions and completed any necessary multiplications, addition usually comes next.
For example, if you have results like 3 and 32 from previous multiplication steps, adding them is straightforward:

• Align the numbers to be added vertically if necessary.
• Simply add them together, such as \(3 + 32\ = 35\).

This result is your final answer. The addition step wraps up the order of operations, ensuring you get the correct and complete sum from complex expressions involving multiple operations. It's essential to follow the correct order of operations: first managing multiplication and division operations before heading to addition and subtraction to maintain accuracy.