The Numerator Of A Fraction Is 3 More Than The Denominator. If The Numerator And The Denominator Are Increased By 5, The Fraction Becomes $\frac{10}{7}$. Find The Fraction.
Problem Description
We are given a fraction where the numerator is 3 more than the denominator. When the numerator and the denominator are increased by 5, the fraction becomes $\frac{10}{7}$. Our goal is to find the original fraction.
Step 1: Define the Variables
Let's denote the denominator as x. Since the numerator is 3 more than the denominator, the numerator can be represented as x + 3.
Step 2: Formulate the Equation
When the numerator and the denominator are increased by 5, the new fraction becomes $\frac{x + 3 + 5}{x + 5} = \frac{x + 8}{x + 5}$. We are given that this new fraction is equal to $\frac{10}{7}$. Therefore, we can set up the equation $\frac{x + 8}{x + 5} = \frac{10}{7}$.
Step 3: Solve the Equation
To solve the equation, we can cross-multiply to get rid of the fractions. This gives us the equation $(x + 8) \cdot 7 = (x + 5) \cdot 10$. Expanding both sides of the equation, we get $7x + 56 = 10x + 50$.
Step 4: Isolate the Variable
To isolate the variable x, we can subtract 7x from both sides of the equation, which gives us $56 = 3x + 50$. Then, we can subtract 50 from both sides to get $6 = 3x$.
Step 5: Solve for x
To solve for x, we can divide both sides of the equation by 3, which gives us $x = 2$.
Step 6: Find the Original Fraction
Now that we have found the value of x, we can find the original fraction. The denominator is x, which is 2, and the numerator is x + 3, which is 5. Therefore, the original fraction is $\frac{5}{2}$.
Conclusion
In this problem, we were given a fraction where the numerator is 3 more than the denominator. When the numerator and the denominator are increased by 5, the fraction becomes $\frac{10}{7}$. We were able to find the original fraction by setting up an equation and solving for the variable x. The original fraction is $\frac{5}{2}$.
Example Use Case
This problem can be used to demonstrate the concept of algebraic equations and how to solve them. It can also be used to show how to work with fractions and how to manipulate them to solve problems.
Real-World Application
This problem can be applied to real-world situations where fractions are used to represent ratios or proportions. For example, in cooking, a recipe may call for a certain ratio of ingredients, which can be represented as a fraction. By understanding how to work with fractions and how to solve problems involving them, individuals can better understand and apply mathematical concepts to real-world situations.
Tips and Tricks
When working with fractions, it's essential to remember that the numerator and denominator can be manipulated to solve problems. By adding or subtracting the same value from both the numerator and denominator, we can create equivalent fractions. Additionally, by multiplying or dividing both the numerator and denominator by the same value, we can also create equivalent fractions.
Common Mistakes
One common mistake when working with fractions is to forget to simplify the fraction after solving the problem. It's essential to simplify the fraction to ensure that it's in its simplest form. Another common mistake is to forget to check if the fraction is in its simplest form before solving the problem.
Final Answer
The final answer is:
Frequently Asked Questions
Q: What is the original fraction if the numerator is 3 more than the denominator and the fraction becomes $\frac{10}{7}$ when the numerator and the denominator are increased by 5?
A: The original fraction is $\frac{5}{2}$.
Q: How do I solve the equation $\frac{x + 8}{x + 5} = \frac{10}{7}$?
A: To solve the equation, you can cross-multiply to get rid of the fractions. This gives you the equation $(x + 8) \cdot 7 = (x + 5) \cdot 10$. Expanding both sides of the equation, you get $7x + 56 = 10x + 50$. Then, you can subtract 7x from both sides to get $56 = 3x + 50$. Finally, you can subtract 50 from both sides to get $6 = 3x$.
Q: How do I find the value of x?
A: To find the value of x, you can divide both sides of the equation by 3. This gives you $x = 2$.
Q: What is the significance of the variable x in this problem?
A: The variable x represents the denominator of the original fraction. When the numerator and the denominator are increased by 5, the new fraction becomes $\frac{x + 8}{x + 5}$.
Q: How do I simplify the fraction $\frac{5}{2}$?
A: To simplify the fraction, you can divide both the numerator and the denominator by their greatest common divisor, which is 1. Therefore, the simplified fraction is still $\frac{5}{2}$.
Q: What is the real-world application of this problem?
A: This problem can be applied to real-world situations where fractions are used to represent ratios or proportions. For example, in cooking, a recipe may call for a certain ratio of ingredients, which can be represented as a fraction.
Q: What are some common mistakes to avoid when working with fractions?
A: One common mistake is to forget to simplify the fraction after solving the problem. Another common mistake is to forget to check if the fraction is in its simplest form before solving the problem.
Q: How do I check if a fraction is in its simplest form?
A: To check if a fraction is in its simplest form, you can divide both the numerator and the denominator by their greatest common divisor. If the result is a whole number, then the fraction is not in its simplest form.
Q: What is the final answer to this problem?
A: The final answer is $\frac{5}{2}$.
Additional Resources
Conclusion
In this Q&A article, we have discussed the problem of finding the original fraction when the numerator is 3 more than the denominator and the fraction becomes $\frac{10}{7}$ when the numerator and the denominator are increased by 5. We have also provided answers to frequently asked questions and discussed the significance of the variable x in this problem. Additionally, we have provided tips and tricks for working with fractions and common mistakes to avoid.
Final Answer
The final answer is: