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Magazine Chapter 7: Problem 50
\(108 \frac{1}{3} \%\) of what number is 78 ?
Short Answer
Expert verified
The base number is 72.
Step by step solution
01
Understanding the Problem
We need to find a number such that 108 and \(\frac{1}{3}\) percent of it equals 78. This means solving for the base number when the percentage amount is given.
02
Convert Percentage to Fraction
First, convert \(108 \frac{1}{3}\%\) into a fraction. \(108 \frac{1}{3}\%\) is the same as \(\frac{325}{3}\%\), which converts to a fraction by dividing by 100, resulting in \(\frac{325}{300}\).
03
Set Up the Equation
Let's denote the unknown number as \(x\). The equation based on the problem statement is: \(\frac{325}{300} \times x = 78\).
04
Solve for the Unknown
To solve for \(x\), divide both sides of the equation by \(\frac{325}{300}\). This gives us: \(x = \frac{78 \times 300}{325}\). Simplifying this fraction yields the base number.
05
Perform the Calculation
Calculate the right-hand side: \(x = \frac{78 \times 300}{325} = \frac{23400}{325}\). Simplifying further, we get \(x = 72\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fractions
Fractions are an essential concept in mathematics, representing parts of a whole. They consist of a numerator and a denominator, with the numerator above the line indicating how many parts are considered, and the denominator below the line indicating the total number of equal parts. Fractions can be used in various operations such as addition, subtraction, multiplication, and division.
- Proper fractions: where the numerator is less than the denominator, like \( \frac{2}{3} \).
- Improper fractions: where the numerator is greater than or equal to the denominator, like \( \frac{5}{3} \).
- Mixed numbers: a combination of a whole number and a proper fraction, like \( 2 \frac{1}{3} \).
Solving Equations
Solving equations is a fundamental skill in math, allowing us to find unknown values that satisfy a given condition. An equation is a mathematical statement that asserts the equality of two expressions, often involving variables.
When solving equations, the primary goal is to isolate the variable, usually represented by \( x \), on one side of the equation. This often involves:
When solving equations, the primary goal is to isolate the variable, usually represented by \( x \), on one side of the equation. This often involves:
- Adding, subtracting, multiplying, or dividing both sides by the same number.
- Rearranging expressions to simplify the equation.
- Applying inverse operations to undo addition or multiplication.
Prealgebra
Prealgebra lays the foundation for all future algebra courses. It focuses on understanding fundamental mathematical concepts that prepare students for high school algebra. Key topics include:
Emphasizing these concepts ensures a smooth transition to mastering equations and more advanced topics in algebra.
- Understanding operations with integers and rational numbers.
- Grasping the concept of variables as symbols that represent unknown numbers.
- Exploring the properties of numbers and basic arithmetic operations.
Emphasizing these concepts ensures a smooth transition to mastering equations and more advanced topics in algebra.
