How can you use the order of operations to solve real-world problems involving rational numbers?

Answer 1

The order of operations is a fundamental concept in mathematics that provides a standard procedure for evaluating expressions. It is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Using this framework can help you solve real-world problems involving rational numbers by ensuring that calculations are performed in the correct sequence. Here’s how you can apply it:

Steps to Solve Real-World Problems Involving Rational Numbers

1. Identify the Problem: Begin by clearly understanding the problem. Determine what you are trying to find and what information is given.

2. Translate the Problem into an Expression: Convert the words of the problem into a mathematical expression or equation. Identify operations such as addition, subtraction, multiplication, or division involved in the situation.

3. Use Parentheses for Clarity: Use parentheses to group operations that should be done first. This is particularly useful when dealing with fractions or when different parts of an expression need to be calculated independently before combining results.

4. Follow the Order of Operations: Apply the order of operations to your expression:

   • Parentheses: Solve expressions within parentheses first.
   • Exponents: Next, evaluate any exponents (if applicable).
   • Multiplication and Division: Perform multiplication and division from left to right.
   • Addition and Subtraction: Finally, carry out addition and subtraction from left to right.

5. Calculate Carefully: Ensure that each step is computed accurately, particularly when dealing with rational numbers (fractions and decimals).

6. Interpret the Result: After arriving at the final answer, interpret it in the context of the problem to ensure it makes sense. Check that the units and scale are appropriate for the situation.

Example Problem

Example: A recipe requires \( \frac{1}{4} \) cup of sugar for one batch of cookies. How much sugar is needed for 5 batches, and what would be the cost if sugar costs $2.50 per cup?

1. Identify the Problem: We need to find the total amount of sugar for 5 batches and the total cost.

2. Translate into Expressions:

   • Amount of sugar needed: \( \text{Sugar} = \frac{1}{4} \times 5 \)
   • Cost of sugar: \( \text{Cost} = \text{Sugar} \times 2.50 \)

3. Use Parentheses: The expression for sugar can be calculated as: \[ \text{Sugar} = \frac{1}{4} \times 5 = \frac{5}{4} \text{ cups (or 1.25 cups)} \]

4. Follow the Order of Operations:

   • Calculate sugar: \( \frac{5}{4} \) is already simplified.
   • Then calculate cost: \[ \text{Cost} = 1.25 \times 2.50 \]

5. Calculate:

   • \( 1.25 \times 2.50 = 3.125 \) dollars.

6. Interpret the Result: The total cost for sugar for 5 batches of cookies is $3.13.

Conclusion

By applying the order of operations correctly, we can solve real-world problems involving rational numbers with confidence. This systematic approach helps to avoid mistakes in calculations, leading to accurate answers that are crucial for effective decision-making in everyday situations.