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Chapter 2: Problem 35

Determine whether each proportion is true or false. $$ \frac{\frac{1}{2}}{5}=\frac{1}{10} $$

Short Answer

Expert verified
The proportion is true as both sides are equal.

Step by step solution

01

Evaluate the left-hand side

First, simplify the left-hand side of the equation. The left-hand side is \ \ \ \frac{\frac{1}{2}}{5} \ \ Multiply the numerator and denominator by the reciprocal of the denominator: \ \( \frac{1}{2} \times \frac{1}{5} = \frac{1}{10} \)
02

Compare both sides of the equation

Now compare the simplified left-hand side with the right-hand side. We have: \ \ \( \frac{1}{10} = \frac{1}{10} \)
03

State the conclusion

Since both sides of the equation are equal, the proportion is true.

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Key Concepts

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

Algebraic Fractions
One common type of problem in algebra involves fractions. These are called algebraic fractions.
Algebraic fractions have a polynomial in the numerator, denominator, or both. For instance, in the given problem, the fraction is \(\frac{\frac{1}{2}}{5}\).
Simplifying algebraic fractions involves working with both the top and bottom parts of the fraction separately.
Always try to simplify fractions by multiplying or dividing both the numerator and the denominator by the same value.
This makes it easier to compare with other fractions and perform operations on them.
Comparative Analysis
In mathematical problems involving proportions, comparing two sides of an equation is often necessary.
This technique helps determine if both sides are equal or not.
Here鈥檚 a step-by-step way to do a comparative analysis for proportions:
  • Simplify each side separately.
  • Ensure both sides are reduced to their simplest form.
  • Compare the final simplified forms.
  • Check for equality. If both simplified forms are the same, your proportion holds true; if not, it is false.

    • In this exercise, we simplified the left-hand side to \( \frac{1}{10} \) and compared it with the right-hand side, which is also \( \frac{1}{10} \). Since both sides match, we concluded that the proportion is true.
Proportion Verification
Proportions represent two ratios that are equal.
Verifying a proportion involves checking if the two ratios or fractions being compared have the same value. We achieve this through simplification and comparison.
For example, in the exercise, we started with \( \frac{\frac{1}{2}}{5} \), simplified it to \( \frac{1}{10} \), and then compared it with the right-hand side to verify the proportion.
Follow these steps for proportion verification:
  • Simplify both ratios or fractions completely.
  • Compare the simplified forms side by side.
  • If both sides are equal, the proportion is verified as true.

    • If they are not equal, the proportion is false.
    Verifying proportions helps you understand and solve many algebraic problems, reinforcing your skills in algebra and fractions.

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Most popular questions from this chapter

Solve each formula for the specified variable. \(P=2(a+b)\) for \(a\)

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Solve each formula for the specified variable. \(M=C(1+r)\) for \(r\)

Solve each inequality. Graph the solution set, and write it using interval notation. \(5<1-6 m<12\)
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