By What Percent Will A Fraction Decrease If Its Numerator Is Decreased By $50\%$ And Its Denominator Is Decreased By $25\%$?

Introduction

When dealing with fractions, understanding how changes in the numerator and denominator affect the overall value is crucial. In this article, we will explore the impact of decreasing the numerator by $50\%$ and the denominator by $25\%$ on the fraction's value. We will delve into the mathematical concepts behind this problem and provide a step-by-step solution to determine the percentage decrease.

Understanding the Problem

To begin, let's consider a fraction $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator. We are asked to find the percentage decrease in the fraction's value when the numerator is decreased by $50\%$ and the denominator is decreased by $25\%$.

Decreasing the Numerator by $50\%$

When the numerator is decreased by $50\%$, the new numerator becomes $\frac{1}{2}a$. This is because $50\%$ of the original numerator is subtracted, leaving half of the original value.

Decreasing the Denominator by $25\%$

Similarly, when the denominator is decreased by $25\%$, the new denominator becomes $\frac{3}{4}b$. This is because $25\%$ of the original denominator is subtracted, leaving three-quarters of the original value.

Calculating the New Fraction Value

Now that we have the new numerator and denominator, we can calculate the new fraction value. The new fraction is $\frac{\frac{1}{2}a}{\frac{3}{4}b}$. To simplify this expression, we can multiply the numerator and denominator by the reciprocal of the denominator, which is $\frac{4}{3}$.

Simplifying the New Fraction Value

Multiplying the numerator and denominator by $\frac{4}{3}$, we get:

$$\frac{\frac{1}{2}a}{\frac{3}{4}b} \cdot \frac{4}{3} = \frac{\frac{1}{2}a \cdot 4}{\frac{3}{4}b \cdot 3} = \frac{2a}{3b}$$

Comparing the Original and New Fraction Values

Now that we have the new fraction value, we can compare it to the original fraction value. The original fraction is $\frac{a}{b}$, and the new fraction is $\frac{2a}{3b}$. To find the percentage decrease, we need to find the ratio of the new fraction value to the original fraction value.

Calculating the Percentage Decrease

The ratio of the new fraction value to the original fraction value is:

$$\frac{\frac{2a}{3b}}{\frac{a}{b}} = \frac{2a}{3b} \cdot \frac{b}{a} = \frac{2}{3}$$

This means that the new fraction value is $\frac{2}{3}$ of the original fraction value. To find the percentage decrease, we can subtract 1 from this ratio and multiply by 100.

Finding the Percentage Decrease

The percentage decrease is:

$$\left(1 - \frac{2}{3}\right) \cdot 100\% = \frac{1}{3} \cdot 100\% = 33.33\%$$

Conclusion

In conclusion, when the numerator of a fraction is decreased by $50\%$ and the denominator is decreased by $25\%$, the fraction's value decreases by $33.33\%$. This is a significant decrease, and it highlights the importance of understanding how changes in the numerator and denominator affect the overall value of a fraction.

Final Thoughts

This problem demonstrates the importance of mathematical concepts in real-world applications. By understanding how changes in the numerator and denominator affect the fraction's value, we can make informed decisions in various fields, such as finance, engineering, and science. The solution to this problem also showcases the power of mathematical reasoning and problem-solving skills.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• Khan Academy: Fractions and Decimals
• MIT OpenCourseWare: Mathematics for Computer Science
• Wolfram Alpha: Fraction Calculator
  

Introduction

In our previous article, we explored the impact of decreasing the numerator by $50\%$ and the denominator by $25\%$ on the fraction's value. We found that the fraction's value decreases by $33.33\%$. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What happens if the numerator is decreased by $75\%$ and the denominator is decreased by $25\%$?

A: If the numerator is decreased by $75\%$, the new numerator becomes $\frac{1}{4}a$. If the denominator is decreased by $25\%$, the new denominator becomes $\frac{3}{4}b$. The new fraction value is $\frac{\frac{1}{4}a}{\frac{3}{4}b} = \frac{a}{3b}$. The ratio of the new fraction value to the original fraction value is $\frac{\frac{a}{3b}}{\frac{a}{b}} = \frac{1}{3}$. The percentage decrease is $\left(1 - \frac{1}{3}\right) \cdot 100\% = 66.67\%$.

Q: What happens if the numerator is decreased by $50\%$ and the denominator is increased by $25\%$?

A: If the numerator is decreased by $50\%$, the new numerator becomes $\frac{1}{2}a$. If the denominator is increased by $25\%$, the new denominator becomes $\frac{5}{4}b$. The new fraction value is $\frac{\frac{1}{2}a}{\frac{5}{4}b} = \frac{2a}{5b}$. The ratio of the new fraction value to the original fraction value is $\frac{\frac{2a}{5b}}{\frac{a}{b}} = \frac{2}{5}$. The percentage decrease is $\left(1 - \frac{2}{5}\right) \cdot 100\% = 60\%$.

Q: What happens if the numerator is increased by $25\%$ and the denominator is decreased by $50\%$?

A: If the numerator is increased by $25\%$, the new numerator becomes $\frac{5}{4}a$. If the denominator is decreased by $50\%$, the new denominator becomes $\frac{1}{2}b$. The new fraction value is $\frac{\frac{5}{4}a}{\frac{1}{2}b} = \frac{5a}{2b}$. The ratio of the new fraction value to the original fraction value is $\frac{\frac{5a}{2b}}{\frac{a}{b}} = \frac{5}{2}$. The percentage increase is $\left(\frac{5}{2} - 1\right) \cdot 100\% = 150\%$.

Q: What happens if the numerator and denominator are both decreased by $25\%$?

A: If the numerator is decreased by $25\%$, the new numerator becomes $\frac{3}{4}a$. If the denominator is decreased by $25\%$, the new denominator becomes $\frac{3}{4}b$. The new fraction value is $\frac{\frac{3}{4}a}{\frac{3}{4}b} = \frac{a}{b}$. The ratio of the new fraction value to the original fraction value is $\frac{\frac{a}{b}}{\frac{a}{b}} = 1$. The percentage change is $0\%$.

Conclusion

In this article, we answered some frequently asked questions related to the problem of decreasing the numerator by $50\%$ and the denominator by $25\%$. We found that the fraction's value decreases by $33.33\%$. We also explored other scenarios, such as decreasing the numerator by $75\%$ and the denominator by $25\%$, increasing the numerator by $25\%$ and the denominator by $50\%$, and decreasing both the numerator and denominator by $25\%$. We hope that this article has provided valuable insights and helped to clarify any confusion.

Final Thoughts

This problem demonstrates the importance of mathematical concepts in real-world applications. By understanding how changes in the numerator and denominator affect the fraction's value, we can make informed decisions in various fields, such as finance, engineering, and science. The solution to this problem also showcases the power of mathematical reasoning and problem-solving skills.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• Khan Academy: Fractions and Decimals
• MIT OpenCourseWare: Mathematics for Computer Science
• Wolfram Alpha: Fraction Calculator