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Chapter 2: Problem 41

Solve each equation. $$ \frac{x}{24}=\frac{15}{16} $$

Short Answer

Expert verified
x = 22.5

Step by step solution

01

Understand the Problem

The given equation is \ \ \( \frac{x}{24} = \frac{15}{16} \). We need to solve for \( x \).
02

Cross-Multiply

To eliminate the fractions, cross-multiply the terms. This means multiplying the numerator of one fraction by the denominator of the other and vice versa: \ \ \( 16x = 15 \times 24 \).
03

Simplify the Equation

Next, perform the multiplication on the right-hand side: \ \ \( 16x = 360 \).
04

Solve for \( x \)

Finally, divide both sides of the equation by 16 to isolate \( x \): \ \ \( x = \frac{360}{16} \). \ Simplify the fraction: \( x = 22.5 \).

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Key Concepts

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

Cross-Multiplication
Cross-multiplication is a useful technique to solve equations involving fractions. It helps in eliminating the fraction by multiplying the numerator of one fraction with the denominator of the other. For instance, in the equation \( \frac{x}{24} = \frac{15}{16} \), we cross-multiply to get:
  • Multiply the numerator of the left fraction (x) by the denominator of the right fraction (16).
  • Multiply the numerator of the right fraction (15) by the denominator of the left fraction (24).
This gives us \(16x = 15 \times 24\). Now, the fractions are gone, making it easier to solve the equation.
Fractions
Understanding fractions is crucial for solving many algebraic equations. A fraction represents a part of a whole and is expressed as \(\frac{numerator}{denominator}\). For example, in our problem, \(\frac{x}{24}\) indicates that x is divided into 24 parts. When equations involve fractions, like \( \frac{x}{24} = \frac{15}{16} \), you might need to eliminate the fractions to simplify the equation. This often involves techniques like cross-multiplication. When you encounter fractions:
  • Make sure both sides of the equation have a common denominator or
  • use cross-multiplication to eliminate the fractions.
Dealing with fractions requires care, as small mistakes can lead to incorrect answers.
Isolate Variable
To solve an equation means to find the value of the variable that makes the equation true. A key step in many solutions is isolating the variable, which means getting the variable alone on one side of the equation. In our solved problem, we started with \( \frac{x}{24} = \frac{15}{16} \). After cross-multiplying, we got \(16x = 360\). To isolate x, we:
  • Divide both sides of the equation by 16.
  • This gives us \( x = \frac{360}{16} \).
Simplifying \( \frac{360}{16} \), we get \( x = 22.5\). By isolating the variable, we can determine its value clearly and accurately.

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