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Chapter 11: Problem 24

Solve using the multiplication principle. Don't forget to check! $$ \frac{4}{5} x=16 $$

Short Answer

Expert verified
x = 20

Step by step solution

01

Understand the given equation

The given equation is \ \( \frac{4}{5} x = 16 \). This equation states that four-fifths of a number equals 16.
02

Multiply both sides by the reciprocal

To isolate \( x \), multiply both sides of the equation by the reciprocal of \( \frac{4}{5} \), which is \( \frac{5}{4} \). Thus, the equation becomes: \ \( x = 16 \times \frac{5}{4} \).
03

Simplify the right side

Calculate \( 16 \times \frac{5}{4} \). First, multiply the numerators: \ 16 \times 5 = 80. Then divide by the denominator: \ 80 \div 4 = 20. \ Therefore, \ x = 20.
04

Check the solution

Substitute \( x = 20 \) back into the original equation to verify. \ Substitute: \ \frac{4}{5} \times 20 = 16. \ Calculate: \ \frac{4 \times 20}{5} = \frac{80}{5} = 16. \ Since both sides equal, the solution \( x = 20 \) is correct.

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Key Concepts

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

Solving Equations
Solving equations is a crucial skill in algebra. When faced with an equation like \(\frac{4}{5} x = 16\), your goal is to find the value of \(x\) that makes the equation true. This involves performing operations that maintain the balance of the equation.
Multiplication Principle
The multiplication principle is a key technique in algebra. It states that you can multiply both sides of an equation by the same non-zero number without changing the equation's equality. For example, in the equation \(\frac{4}{5} x = 16\), you can multiply both sides by the reciprocal of \( \frac{4}{5}\) to isolate \(x\).
Reciprocals
A reciprocal is essentially the 'flipped' version of a number. The reciprocal of \( \frac{4}{5}\) is \( \frac{5}{4}\). Multiplying a number by its reciprocal results in 1. In our example, multiplying both sides of \( \frac{4}{5} x = 16\) by \( \frac{5}{4}\) helps us isolate \(x\).
Checking Solutions
After solving for \(x\), it鈥檚 important to check your work. Substitute the value of \(x\) back into the original equation. For \( x = 20\), substitute it back into \( \frac{4}{5} x = 16\) to ensure both sides are equal. This verifies that your solution is correct.
Basic Algebra
Basic algebra involves understanding and using fundamental concepts like variables, constants, and arithmetic operations. Techniques like the multiplication principle and reciprocals are essential tools. These concepts form the foundation for more advanced topics in mathematics and are used to solve equations like \( \frac{4}{5} x = 16\).

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