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Magazine Chapter 6: Problem 52
If \(p\) is \(120 \%\) of \(q,\) then \(q\) is what percent of \(p ?\)
Short Answer
Expert verified
83.33%
Step by step solution
01
Understand the Given Information
Given that \( p \) is 120% of \( q \). Express this relationship mathematically as \( p = 1.2q \).
02
Rearrange the Equation
We need to express \( q \) in terms of \( p \). From the equation \( p = 1.2q \), divide both sides by 1.2 to get \( q = \frac{p}{1.2} \).
03
Calculate the Percentage
To find what percent \( q \) is of \( p \), we use the formula: \[ \text{Percentage} = \left( \frac{\text{part}}{\text{whole}} \right) \times 100 \]Here, \( q \) is the part and \( p \) is the whole. Substitute \( q = \frac{p}{1.2} \) into the formula: \[ \text{Percentage} = \left( \frac{\frac{p}{1.2}}{p} \right) \times 100 \]
04
Simplify the Expression
Simplify the fraction: \[ \text{Percentage} = \left( \frac{1}{1.2} \right) \times 100 \]Calculate: \[ \frac{1}{1.2} = \frac{1}{1.2} = 0.8333 \]So, \[ \text{Percentage} = 0.8333 \times 100 = 83.33 \]%
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Math Equations
A math equation is essentially a statement that asserts the equality of two expressions. In this exercise, we deal with the equation: \( p = 1.2q \). This equation conveys that the value of \( p \) (some number) is 120% of \( q \) (another number). Using equations like this helps us break down and solve problems in a structured and logical manner.
Math equations are powerful tools in problem-solving.
They help us find unknown values and relationships between variables. If you understand and manipulate them, solving complex problems becomes much easier.
Math equations are powerful tools in problem-solving.
They help us find unknown values and relationships between variables. If you understand and manipulate them, solving complex problems becomes much easier.
Percentage Calculation
Percentage calculation is a method used to express a number as a fraction of 100. It simplifies comparisons between different quantities.
In our exercise, we start with the information that \( p \) is 120% of \( q \). We need to find what percent \( q \) is of \( p \).
Here's the breakdown:
In our exercise, we start with the information that \( p \) is 120% of \( q \). We need to find what percent \( q \) is of \( p \).
Here's the breakdown:
- Step 1: We express the percentage relationship mathematically: \( p = 1.2q \) (120% is the same as 1.2 in decimal.)
- Step 2: Then, rearrange it to express \( q \) in terms of \( p \): \( q = \frac{p}{1.2} \)
- Step 3: Use the percentage formula: \[ \text{Percentage} = \left( \frac{\text{part}}{\text{whole}} \right) \times 100 \] Here, \( q \) is the part, and \( p \) is the whole.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to find the value of unknown variables. In this exercise, we start with the given equation \( p = 1.2q \).
To find \( q \) in terms of \( p \), follow these steps:
Algebraic manipulation is a fundamental skill in mathematics.
It simplifies complex equations and reveals insights hidden within mathematical relationships.
To find \( q \) in terms of \( p \), follow these steps:
- Step 1: We divide both sides of the equation by 1.2: \( p = 1.2q \rightarrow q = \frac{p}{1.2} \)
Algebraic manipulation is a fundamental skill in mathematics.
It simplifies complex equations and reveals insights hidden within mathematical relationships.
Understanding Percentages
Understanding percentages is vital in mathematics. Percentages represent parts of a whole and are often used in various real-world contexts.
In our exercise, we're given that \( p \) is 120% of \( q \).
The final goal is to determine what percent \( q \) is of \( p \). Here鈥檚 how we understand it:
Understanding percentages helps in many areas like finance, statistics, and everyday calculations.
In our exercise, we're given that \( p \) is 120% of \( q \).
The final goal is to determine what percent \( q \) is of \( p \). Here鈥檚 how we understand it:
- Step 1: We use the simplified equation: \( q = \frac{p}{1.2} \). This tells us that \( q \) is a fraction of \( p \).
- Step 2: We express this fraction in percentage form: \[ \text{Percentage} = \left( \frac{1}{1.2} \right) \times 100 = 83.33\% \]
Understanding percentages helps in many areas like finance, statistics, and everyday calculations.
