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Chapter 5: Problem 176

In the following exercises, simplify. $$[\$ 56.31+0.22(\$ 56.31)] \div 4$$

Short Answer

Expert verified
17.17455

Step by step solution

01

Understand the Expression

The goal is to simplify the expression \[\text{[\text{\textdollar}} 56.31+0.22(\text{\textdollar} 56.31)] \div 4\]. The $$ inside the brackets must be evaluated first.
02

Calculate the Part Inside the Brackets

Calculate the value inside the brackets: \56.31 + 0.22 \times 56.31.\ First, find \0.22 \times 56.31.$$$.
03

Multiply 0.22 and 56.31

Perform the multiplication: \0.22 \times 56.31 = 12.3882.$$.
04

Add the Results Inside the Brackets

Add 56.31 and 12.3882: \56.31 + 12.3882 = 68.6982.$$.
05

Divide by 4

Finally, divide the result by 4: \68.6982 \div 4 = 17.17455.$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Operations
Arithmetic operations are fundamental actions we perform on numbers. They include addition, subtraction, multiplication, and division.

In the original exercise, we encounter all these operations. Let's briefly explain each:
  • Addition (+): Combining two numbers to get a sum.
  • Subtraction (-): Removing one number from another to get the difference.
  • Multiplication (脳): Repeated addition of a number, simplified for ease.
  • Division (梅): Splitting a number into equal parts.

Understanding these operations is crucial because they form the basis of simplifying any algebraic expression.
Order of Operations
The order of operations tells us the order in which we need to perform arithmetic operations to ensure consistency in our calculations. It's often remembered by the acronym PEMDAS:
  • P: Parentheses first
  • E: Exponents (i.e., powers and square roots, etc.)
  • MD: Multiplication and Division (left to right)
  • AS: Addition and Subtraction (left to right)

In our example, we start by evaluating the expression inside the brackets which calls for following the PEMDAS rule step by step:

  • Evaluate the entire expression within the parentheses first.
  • Within those parentheses, perform the multiplication before addition.
  • Finally, perform the division as the last step outside the parentheses.

Understanding PEMDAS ensures our results are mathematically sound and consistent.
Multiplication and Division
Multiplication and division occur together in the order of operations.

  • Multiplication: In the given equation, we first need to multiply 0.22 by 56.31. To perform this, multiply as follows \(0.22 \times 56.31 = 12.3882\).
  • Division: Once you have simplified the expression inside the brackets, namely 56.31 + 12.3882 = 68.6982, you perform the division \(68.6982 \div 4 = 17.17455\).


Multiplication is like adding a number multiple times, and division is like splitting a number into equal parts. Understanding these is crucial as they frequently appear in algebraic simplification exercises.

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Most popular questions from this chapter

In the following exercises, convert each decimal to a fraction or mixed number. $$2.008$$

In the following exercises, convert each decimal to a fraction or mixed number. $$10.25$$

In the following exercises, round each number to the nearest hundredth. $$7.096$$

In the following exercises, order each of the following pairs of numbers, using < or >. $$0.6 \underline\quad0.59$$

In the following exercises, round each number to the nearest hundredth. $$0.697$$
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