Problem 4 - Technical Mathematics with Calculus

Question

Combined Operations with Exact Numbers. Perform each computation by calculator. $$(89-74+95)(87-49)$$

Step-by-Step Solution

Answer

The result of the computation \[(89-74+95)(87-49)\] is 4180.

Step 1: Evaluate the Parentheses

First, we need to simplify the expressions within the parentheses separately. This means performing the addition and subtraction for each set of parentheses before multiplying the results. Start with the first set: calculate 89 minus 74 and add 95 to the result.

Step 2: Simplify the First Parenthetical Expression

Simplify the expression inside the first set of parentheses: \[89 - 74 + 95 = 15 + 95 = 110\].

Step 3: Simplify the Second Parenthetical Expression

Simplify the expression inside the second set of parentheses: \[87 - 49 = 38\].

Step 4: Multiply the Results

Now multiply the results from Step 2 and Step 3: \[110 \times 38\].

Step 5: Perform the Multiplication

Calculate the product of 110 and 38 to get the final result: \[110 \times 38 = 4180\].

Key Concepts

Order of Operations

Understanding the order of operations is crucial when tackling problems involving combined arithmetic calculations. It is a mathematical convention that helps to ensure consistency and clarity in solving equations. To solve the exercise \( (89-74+95)(87-49) \) accurately, one must follow the standard order: parentheses, exponents, multiply and divide (from left to right), and then add and subtract (from left to right). This rule is often remembered by the acronym PEMDAS.

In our exercise, we start by addressing the operations within the parentheses, as these have the highest priority. We perform any addition or subtraction inside the parentheses before dealing with any multiplication or division outside of them. Once the parenthetical expressions are simplified, we can then proceed to multiply the results, following the correct sequence prescribed by the order of operations.

Parenthetical Expressions

Parenthetical expressions are parts of a math problem that are set off by parentheses. They play a significant role in the order in which operations should be carried out. It's like completing tasks inside bubbles before popping them in sequence and moving on to the rest of the equation.

When we encounter an expression like \( (89-74+95)(87-49) \), our priority is to simplify the numbers within the parentheses, treating each set as its own mini problem. As demonstrated in our solution, we simplify \(89 - 74 + 95\) first. It's like untangling a knot: we must carefully work through each operation to avoid mixing up the numbers. The same attention is given to the second set of parentheses, \(87 - 49\). After simplifying these 'bubbles', the values obtained from each are then ready to be used in the subsequent operations.

Arithmetic Operations

Arithmetic operations consist of the basic processes we use in everyday math: addition, subtraction, multiplication, and division. Our problem \( (89-74+95)(87-49) \) combines several of these operations. Once we've simplified the parenthetical expressions, it's time to use multiplication.

As seen in Step 4, after simplifying both parenthetical expressions, we perform the multiplication of 110 by 38. This is a straightforward calculation, but it requires careful attention to detail to ensure we correctly align our numbers and carry any values as needed. Arithmetic operations may seem simple on the surface, but each step needs to be executed with precision to reach the correct final result, which in this case is 4180.