Problem 21 - Fundamentals of Mathematics

Question

Use a calculator to find each value. $$55(84-26)[phone]) $$

Step-by-Step Solution

Answer

The final value is 6070.

Step 1: Simplify the Parentheses

Begin by simplifying the expressions within the parentheses. Calculate each one separately. Start with the first parenthetical expression: \( 84 - 26 = 58 \). Now calculate the second parenthetical expression: \( 512 - 488 = 24 \).

Step 2: Replace Simplified Values

Substitute the results from Step 1 back into the original expression. This changes the expression to: \( 55 \times 58 + 120 \times 24 \).

Step 3: Perform Multiplications

Calculate the products in the expression. First, find \( 55 \times 58 = 3190 \). Then calculate \( 120 \times 24 = 2880 \).

Step 4: Add the Results

Finally, sum the two products from Step 3. Add \( 3190 + 2880 = 6070 \).

Key Concepts

Order of Operations

In mathematics, the order of operations is a standard that defines the rules for which operations to perform first in a mathematical expression. This ensures that everyone solves the problem the same way and reaches the same result. The sequence is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

It's important to evaluate expressions in this order. In our example, we started with parentheses, simplifying each expression inside them first. Only after did we perform multiplication, and mathematically concluded with addition. Following the order ensures clarity and consistency.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining equivalence. In our step-by-step solution, simplifying targets the contents within parentheses first—these are sub-expressions that need calculation. This means efficiently performing subordinate arithmetic tasks to transform the expression.
For instance, in our exercise, simplifying within the parentheses meant calculating:

• \( 84 - 26 = 58 \)
• \( 512 - 488 = 24 \)

By tackling these simpler tasks, the expression transforms and becomes more manageable for subsequent operations.

Multiplication

Multiplication is the operation of scaling one number by another. In our scenario, multiplication comes after simplifying the brackets as part of our order of operations.
When we had:

• \( 55 \times 58 \)
• \( 120 \times 24 \)

each multiplication task took its resolved segment from the simplified expression. This stage of the process is crucial as it directly affects the final sum. Solving these multiplication tasks yielded:

• \( 3190 \) for the first product
• \( 2880 \) for the second product

These results become components in our final addition step.

Addition

After completing multiplication, our last operation follows: adding the products to finalize the expression. Addition combines the values, resulting in the ultimate sum.
We performed:

• \( 3190 + 2880 \)

This addition yields the total: 6070. Whether working manually or using tools like calculators, addition must be handled correctly to avoid errors in the final outcome. It's the culmination of all previous work, building up from simplifying to multiplication.