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Chapter 8: Problem 18

Translate to an equation and solve. What number is \(105 \%\) of \(84 ?\)

Short Answer

Expert verified
Answer: 88.2

Step by step solution

01

Understand the problem

To find \(105\%\) of the number \(84\), we can write this in the form of an equation. Let's say the number we're trying to find is x. So, the equation we will have is: $$x=(105\%)(84)$$
02

Convert percentage to decimal

We'll need to convert the percentage to a decimal in order to solve the equation. To do that, divide the percentage by 100: $$105\% = \frac{105}{100} = 1.05$$ Now, our equation becomes: $$x=(1.05)(84)$$
03

Solve the equation

Now, we'll simply multiply 1.05 by 84 to find the value of x: $$x = 1.05 \times 84 = 88.2$$ So, \(105\%\) of \(84\) is equal to \(88.2\).

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Key Concepts

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

Converting Percentages to Decimals
When working with percentages, it's often necessary to convert them into decimals to simplify calculations, especially when equations or multiplication are involved. The process involves a simple division. To convert a percentage to a decimal:
  • Take the percentage value.
  • Divide it by 100.
This is because the word 'percent' essentially means 'per hundred'. For instance, to convert 105% into a decimal:
  • Start with 105%.
  • Divide by 100 to get 1.05.
Now, you can use this decimal in further calculations. This conversion is crucial in solving many mathematical problems where direct operations with percentages are cumbersome.
Solving Equations
Solving equations involves finding the value of an unknown variable that makes the equation true. In our problem, the equation is aimed at determining what number is 105% of 84. Here, we need to solve:
\[ x = 1.05 imes 84 \] Steps to solve this are:
  • Identify the equation and the unknown variable (here, it's \(x\)).
  • Perform the operations defined, which typically include multiplication, division, addition, or subtraction.
  • Solve for \(x\) by simplifying the expression, which involves straightforward calculations.
Solving such problems often requires a clear understanding of how the operations affect the value of \(x\), resulting in an accurate and reliable solution.
Mathematical Multiplication
Multiplication is a basic arithmetic operation that combines equal groups. In our equation, after converting the percentage into a decimal, multiplication helps in calculating what 105% of 84 is.
When multiplying a decimal by a whole number, as we have here with \(1.05 \times 84\), you can follow these steps:
  • Ignore the decimal and multiply the numbers as if they were whole numbers: 105 by 84.
  • Calculate the product, which in this case is 8,820.
  • Count the number of decimal places in the decimal number (here, it's two in 1.05) and place the decimal point in the product accordingly: 88.20.
This shows how a decimal multiplier affects the size of the outcome, and why positioning the decimal point accurately in the result is important for achieving the correct answer of 88.2.

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Most popular questions from this chapter

According to a report by the Nielsen Company, the average American spends 31.5 hours per week watching television. What percent of the week does the average American spend watching television?

Write each percent as a decimal number. $$1.65 \%$$

Hunter answered 48 questions correctly out of a total of 54 questions. What percent of the total questions did Hunter answer correctly?

Translate to an equation and solve. Sabrina scored \(90 \%\) on a test with 30 total questions. a. How many questions did she answer correctly? b. What percent did she answer incorrectly? c. How many questions did she answer incorrectly?

Write each decimal number as a percent. $$0.8$$
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