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Chapter 6: Problem 27

Solve the given proportion. \(\frac{2}{15}=\frac{24}{x}\)

Short Answer

Expert verified
The value of \( x \) is 180.

Step by step solution

01

Identify the proportion

We have the proportion \( \frac{2}{15} = \frac{24}{x} \). This equation shows that two fractions are equal, and our task is to find the value of \( x \) that makes this true.
02

Cross-Multiply

To solve the proportion, we cross-multiply the fractions. This involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa, resulting in the equation \( 2 \cdot x = 24 \cdot 15 \).
03

Simplify the Equation

Multiply the numbers on the right-hand side: \( 24 \cdot 15 = 360 \). So the equation becomes \( 2x = 360 \).
04

Solve for x

Divide both sides of the equation by 2 to solve for \( x \). Thus, \( x = \frac{360}{2} \).
05

Compute the Result

Calculate \( \frac{360}{2} = 180 \). Therefore, \( x = 180 \).

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Key Concepts

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

Cross-Multiplication
When working with proportions, cross-multiplication is a method that allows us to solve equations involving two equal fractions. In a proportion like \( \frac{2}{15} = \frac{24}{x} \), cross-multiplication involves these steps:

  • Multiply the numerator of one fraction by the denominator of the other fraction.
  • This gives: \( 2 \cdot x \) and \( 24 \cdot 15 \).
  • Equate these products: \( 2x = 360 \).
Cross-multiplying is helpful because it transforms the problem into a linear equation, which is easier to handle. It eliminates the fractions by creating a direct product equation that we can solve for any unknown, such as \( x \). For both students and mathematicians, cross-multiplication is a useful technique whenever proportions are involved, simplifying the process of finding a missing value in a fraction.
Solving Equations
Once you have used cross-multiplication to form an equation, solving it can be straightforward. You often end up with a linear equation, like \( 2x = 360 \), which requires basic algebraic skills. Here are some simple steps to solve equations like this:

  • Identify the operation around the variable \( x \). Here, \( x \) is multiplied by 2.
  • To isolate \( x \), perform the inverse operation, which, in this case, is division.
  • Divide both sides by 2: \( x = \frac{360}{2} \).
Solving equations often focuses on simplifying the expression until you reveal the value of your variable. These basic principles also apply to more complex equations, making this skill an essential part of solving math problems. It helps develop logical thinking and problem-solving strategies essential for tackling various math challenges.
Numerator and Denominator
A fraction consists of two main components: a numerator and a denominator. Understanding these terms is vital when working with proportions. Consider the fraction \( \frac{2}{15} \):

  • The numerator is the top number, which represents part of the whole. Here, 2 is the numerator.
  • The denominator is the bottom number, indicating the total parts into which the whole is divided. Here, 15 is the denominator.
In proportions, the numerators and denominators play a crucial role in setting the relationship between quantities. When solving equations, multiplying or dividing by the numerator or denominator is a key step in manipulating the expressions. Recognizing the role each part plays in forming equivalent fractions is fundamental for solving proportions and further mathematical concepts with confidence.

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