If 2 is added to the numerator and denominator of a fraction, it becomes [tex]$\frac{9}{10}$[/tex] and if 3 is subtracted from the same numerator and denominator, it becomes [tex]$\frac{4}{5}$[/tex]. Find the fraction.
Answer (2)
Define the fraction as y x and set up the equations based on the given information: y + 2 x + 2 = 10 9 and y − 3 x − 3 = 5 4 .
Cross-multiply and simplify the equations to get 10 x − 9 y = − 2 and 5 x − 4 y = 3 .
Solve the system of equations using elimination to find x = 7 and y = 8 .
The fraction is 8 7 .
Explanation
Setting up the equations Let the fraction be y x . According to the problem, adding 2 to both the numerator and the denominator results in 10 9 , and subtracting 3 from both the numerator and the denominator results in 5 4 . This gives us two equations: y + 2 x + 2 = 10 9 y − 3 x − 3 = 5 4
Eliminating fractions To solve this system of equations, we first cross-multiply each equation to eliminate the fractions. This gives us: 10 ( x + 2 ) = 9 ( y + 2 ) 5 ( x − 3 ) = 4 ( y − 3 ) Expanding these equations, we get: 10 x + 20 = 9 y + 18 5 x − 15 = 4 y − 12
Rearranging the equations Now, let's rearrange the equations to bring them into a more standard form: 10 x − 9 y = − 2 5 x − 4 y = 3 To solve this system, we can use the method of substitution or elimination. Let's use elimination. Multiply the second equation by 2 to make the coefficients of x in both equations the same: 2 ( 5 x − 4 y ) = 2 ( 3 ) 10 x − 8 y = 6
Solving for y Now subtract the first equation from the modified second equation: ( 10 x − 8 y ) − ( 10 x − 9 y ) = 6 − ( − 2 ) 10 x − 8 y − 10 x + 9 y = 8 y = 8
Solving for x Now that we have the value of y , we can substitute it back into one of the equations to find the value of x . Let's use the equation 5 x − 4 y = 3 : 5 x − 4 ( 8 ) = 3 5 x − 32 = 3 5 x = 35 x = 7
Verifying the solution So, the fraction is y x = 8 7 . To verify the solution, we can plug these values back into the original equations: 8 + 2 7 + 2 = 10 9 8 − 3 7 − 3 = 5 4 Both equations hold true, so our solution is correct.
Final Answer Therefore, the fraction is 8 7 .
Examples
Fractions are used in everyday life, such as when cooking, measuring ingredients, or splitting a bill with friends. Understanding how to manipulate fractions and solve equations involving them is essential for many practical tasks. For example, if you want to scale a recipe that calls for 2 1 cup of flour by a factor of 4 3 , you need to multiply the fractions to find the new amount of flour needed. Similarly, if you are dividing a pizza into equal slices among a group of people, you are using fractions to determine the size of each slice. These skills are also crucial in more advanced fields like engineering, finance, and computer science.
The fraction in question is 8 7 . This was determined by setting up two equations based on the conditions given, solving the equations, and verifying that the values hold true in both scenarios provided in the problem.
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