Problem 88 - Algebra 1

Question

Evaluate the expression. $$ 5 \cdot \frac{8}{9}-\frac{6}{9}+51 \div 3 $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{187}{9}\) or 20.78 (to two decimal places).

Step 1: Handle Multiplication and Division

Firstly, handle the multiplication and division in the order they appear from left to right. We multiply: 5 times \(\frac{8}{9}\) and then divide: 51 by 3. The expression then becomes: \( \frac{40}{9} - \frac{6}{9} + 17\).

Step 2: Subtraction and Addition

Next, handle the subtraction and addition in the order they appear from left to right. We subtract: \(\frac{40}{9}\) minus \(\frac{6}{9}\), then add: \(\frac{34}{9}\) plus 17. The expression becomes: \(\frac{34}{9} + 17\).

Step 3: Final Addition

Finally, convert 17 to a fraction equivalent by multiplying by \(\frac{9}{9}\), which gives us \(\frac{153}{9}\). Now add the fractions: \(\frac{34}{9} + \frac{153}{9}\) which equals: \(\frac{187}{9}\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating expressions. This order ensures calculations are performed correctly and consistently. In mathematics, this sequence is often remembered using the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the given expression, there are no parentheses or exponents to deal with, so we begin with multiplication and division. Perform these operations as they appear from left to right. In our solved example, this meant multiplying first, followed by division before addressing any addition or subtraction.
It's essential to follow this sequence to arrive at the correct result, as doing these operations in the wrong order can lead to significantly different answers.

Fraction Operations

Working with fractions involves understanding how to properly perform operations like addition, subtraction, multiplication, and division. When multiplying fractions, multiply the numerators together and the denominators together. For example, with 5 multiplied by \(\frac{8}{9}\), think of 5 as \(\frac{5}{1}\). This becomes \( \frac{5 \times 8}{1 \times 9} = \frac{40}{9}\).
When adding or subtracting fractions, make sure they have a common denominator before proceeding. In our exercise, both \(\frac{40}{9}\) and \(\frac{6}{9}\) already have the same denominator, allowing direct subtraction: \(\frac{40}{9} - \frac{6}{9} = \frac{34}{9}\). This simplifies the calculation process and keeps everything neat and organized.

Multiplication and Division

Multiplication and division are operations that go hand in hand and are processed from left to right according to the order of operations. In the given expression, you start with multiplication: 5 times \(\frac{8}{9}\). As mentioned earlier, this treats 5 as a fraction \(\frac{5}{1}\), resulting in \(\frac{40}{9}\).
Next is division, where 51 is divided by 3 to yield 17. Think of division as multiplying by the reciprocal instead, which often helps conceptualize division with fractions.For example:

• 51 divided by 3 can be viewed as 51 times \(\frac{1}{3}\)
• This makes it clearer why the result is 17 (since \(\frac{51}{1} \times \frac{1}{3} = \frac{51}{3} = 17\))

Mastering these concepts ensures accurate results and builds a strong foundation for more complex mathematical operations.