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Chapter 7: Problem 30

Solve. 7.2 is \(6 \frac{1}{4} \%\) of what number?

Short Answer

Expert verified
115.2

Step by step solution

01

Understand the Problem

We are given that 7.2 is \(6 \frac{1}{4} \%\) of some unknown number. Our task is to find this number.
02

Convert the Percentage to a Decimal

First, convert \(6 \frac{1}{4} \%\) to a decimal. The fractional part \(\frac{1}{4}\) is equivalent to 0.25. Thus, \(6 \frac{1}{4} = 6 + 0.25 = 6.25\). As a decimal, this is \(0.0625\), because we move the decimal point two places to the left when converting from a percentage to a decimal.
03

Set Up the Equation

According to the problem, 7.2 is \( \frac{6.25}{100} \) times some number \(x\). We can write this relationship as an equation: \(7.2 = 0.0625 \times x\).
04

Solve for x

To find \(x\), divide both sides of the equation by 0.0625:\[x = \frac{7.2}{0.0625}\]Calculate the division to find the value of \(x\).
05

Calculate the Result

Carry out the division \( \frac{7.2}{0.0625} \). This equals 115.2. Therefore, \(x = 115.2\).

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Key Concepts

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

Decimal Conversion
Decimal conversion is a crucial step when dealing with percentage problems in mathematics. Percentages like \(6 \frac{1}{4}\%\) must be converted to decimal form to solve equations effectively. Here's how you can do it:
  • First, recognize the mixed number format: \(6 \frac{1}{4}\%\).
  • The fraction \(\frac{1}{4}\) translates to 0.25 because 1 divided by 4 gives 0.25.
  • Add the whole number part and the decimal fraction: \(6 + 0.25 = 6.25\).
  • Since it's a percentage, divide by 100 to convert it to a decimal: \(6.25\% = 0.0625\).
By learning this conversion method, you'll be more comfortable handling any percentage encountered in math problems. Remember, converting percentages to decimals is essential for setting up equations that solve for unknowns.
Mathematical Equations
Mathematical equations form the backbone of problem-solving in mathematics. Once we convert percentages to decimals, we can set up equations to find unknown values. For this problem:
  • We need to express the relationship described in the problem as an equation: \(7.2 = 0.0625 \times x\).
  • Here, "7.2 is \(6 \frac{1}{4}\%\) of a number" translates to multiplying by the decimal equivalent of the percentage to find the unknown number \(x\).
  • This equation involves multiplication, making it a straightforward linear equation.
Creating equations allows us to structure our mathematical thinking and systematically approach finding unknowns by isolating variables. Mastery of forming equations is a critical skill in mathematical problem solving.
Problem Solving Steps
Solving mathematical problems often involves a series of logical steps, allowing us to break down and simplify complex concepts. Here鈥檚 a breakdown of the steps used in this exercise:
  • Understand the problem: Clearly define what鈥檚 given and what needs to be found.
  • Convert percentages to decimals: Essential for creating manageable equations.
  • Set up the equation: Represent the problem with a mathematical equation, such as \(7.2 = 0.0625 \times x\).
  • Solve for the unknown: In our case, divide by 0.0625 to find \(x = \frac{7.2}{0.0625}\).
  • Calculate the result: Performing the division gives \(x = 115.2\).
By following clear and detailed steps, even seemingly difficult problems can be solved easily. Practicing this process helps build confidence and improve problem-solving skills in mathematics. Each step logically leads to the next, ensuring no aspect of the problem is overlooked.

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Most popular questions from this chapter

Solve. What number is \(89 \%\) of \(62 ?\)

For each exercise, determine whether the number, \(y\), is (a) equal to \(45,\) (b) greater than 45 , or \((\) ) less than 45 . $$ 100 \% \text { of } y \text { is } 45 $$

Write each fraction as a decimal and then write each decimal as a percent. Round the decimal to three decimal places (nearest thousandth) and the percent to the nearest tenth percent. $$ \frac{21}{79} $$

Perform each indicated operation. See Sections 4.3 through \(4.5,4.7,\) and 5.2 through \(5.4 .\) $$2 \frac{4}{5}-3 \frac{9}{10}$$

Compare the following accounts: Account 1: \(\$ 1000\) is invested for 10 years at a simple interest rate of \(6 \% .\) Account 2: \(\$ 1000\) is compounded semiannually at a rate of \(6 \%\) for 10 years. Discuss how the interest is computed for each account. Determine which account earns more interest. Why?
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