Solving Fractional Relationships: When Numerators and Denominators Increase

A fascinating aspect of algebra and fractions is understanding how changes in the numerator and denominator affect the fraction itself. In this article, we explore several scenarios where the numerator and denominator of a fraction are increased, leading us to find the original fraction. These types of problems offer a valuable exercise for enhancing mathematical comprehension and problem-solving skills.

Example 1

If the numerator of a fraction is increased by 400 and the denominator by 300, the resultant fraction is 15/14. Find the original fraction.

Solution:

[ frac{15}{14} frac{x 400}{y 300} ]

Solving for x and y, we get:

Step 1: Cross multiply to get:

[ 15(y 300) 14(x 400) ]

Step 2: Simplify the equation:

[ 15y 4500 14x 5600 ]

Step 3: Rearrange to isolate x in terms of y:

[ 15y - 14x 1100 ]

This equation can be solved for specific values of x and y. Using the method of substitution or trial and error, we find:

[ frac{x}{y} frac{11}{9} ]

Example 2

A similar problem is given where increasing the numerator and denominator by certain values results in a new fraction. Consider the following example:

A fraction is multiplied by 5a/6b and the result is 10/21. Find the original fraction.

Solution:

Step 1: Set up the equation:

[ frac{5a}{6b} frac{10}{21} ]

Step 2: Cross multiply to find the proportionality:

[ 35ab 120 ]

Step 3: Simplify to find the original fraction:

[ frac{a}{b} frac{12}{21} ] [ frac{a}{b} frac{12}{21} ]

Example 3

In another instance, we increase the numerator by 200 and the denominator by 280, resulting in a fraction of 3/14. Determine the original fraction.

Solution:

Step 1: Set up the equation:

[ frac{x 200}{y 280} frac{3}{14} ]

Step 2: Cross multiply to solve for x and y:

After solving, we find:

[ frac{x}{y} frac{41}{175} ]

Example 4

Another approach involves increasing the numerator by 400 and the denominator by 500. The new fraction is given as 15/14. Determine the original fraction.

Solution:

Step 1: Set up the equation:

[ frac{x 400}{y 500} frac{15}{14} ]

Step 2: Cross multiply to find the relationship:

[ 14(x 400) 15(y 500) ]

Step 3: Simplify to solve for x and y:

[ 14x 5600 15y 7500 ]

Step 4: Rearrange to isolate x in terms of y:

[ 14x - 15y 1900 ]

Conclusion

Understanding how changes in the numerator and denominator affect a fraction is crucial for solving complex algebraic problems. These examples demonstrate various strategies and approaches to find the original fraction when given certain conditions. Such exercises enhance mathematical skills and provide valuable practice for students and educators alike.