If the numerator and denominator of a fraction are both increased by 5, the resulting fraction is equivalent to 2/3. However, if the numerator and denominator are both decreased by 5, the resulting fraction is equivalent to 3/7. What is the fraction?
(x+5)/(y+5) = 2/3
(x-5)/(y-5) = 3/7
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2 y + 10 = 3 x + 15
3 y - 15 = 7 x - 35
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3 x - 2 y = -5
7 x - 3 y = 20
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9 x - 6 y = -15
14 x -6 y = 40
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-5 x = -55
x = 11
then 33-2y = -5
2 y = 38
y = 19
11/19
Let's assume the fraction is expressed as numerator/denominator.
Let's also assume the original fraction is x/y.
According to the given conditions, if we increase both the numerator and denominator by 5, the resulting fraction is (x + 5)/(y + 5). This is equivalent to 2/3, so we can set up the equation:
(x + 5)/(y + 5) = 2/3
Now, if we cross-multiply, we get:
3(x + 5) = 2(y + 5)
Expanding and simplifying, we have:
3x + 15 = 2y + 10
Subtracting 2y from both sides, we get:
3x - 2y + 15 = 10
Subtracting 15 from both sides, we have:
3x - 2y = -5
This is our first equation.
Now, let's consider the second condition where we decrease both numerator and denominator by 5. The resulting fraction is (x - 5)/(y - 5), which is equivalent to 3/7:
(x - 5)/(y - 5) = 3/7
Cross-multiplying, we get:
7(x - 5) = 3(y - 5)
Expanding and simplifying, we have:
7x - 35 = 3y - 15
Adding 35 to both sides, we get:
7x - 3y = 20
This is our second equation.
Now, we have a system of equations:
3x - 2y = -5 (Equation 1)
7x - 3y = 20 (Equation 2)
To solve this system, we can use any method such as substitution or elimination.
Since the elimination method may be easier, let's eliminate the y variable.
Multiplying both sides of Equation 1 by 3 and Equation 2 by 2, we get:
9x - 6y = -15 (Equation 3)
14x - 6y = 40 (Equation 4)
Now we can subtract Equation 3 from Equation 4:
(14x - 6y) - (9x - 6y) = 40 - (-15)
Simplifying, we have:
14x - 6y - 9x + 6y = 40 + 15
5x = 55
Dividing both sides by 5, we get:
x = 11
Now, substitute the value of x into Equation 1:
3(11) - 2y = -5
33 - 2y = -5
Subtracting 33 from both sides, we have:
-2y = -38
Dividing both sides by -2, we get:
y = 19
Therefore, the fraction is 11/19.
To solve this problem, we can set up a system of equations. Let's denote the original fraction as x/y, where x is the numerator and y is the denominator.
According to the first condition, if the numerator and denominator are both increased by 5, we get (x+5)/(y+5) = 2/3.
According to the second condition, if the numerator and denominator are both decreased by 5, we get (x-5)/(y-5) = 3/7.
Now, we can solve this system of equations to find the values of x and y.
Let's start by solving the first equation:
(x+5)/(y+5) = 2/3
Cross-multiplying, we get:
3(x+5) = 2(y+5)
3x + 15 = 2y + 10
3x - 2y = -5 (Equation 1)
Next, let's solve the second equation:
(x-5)/(y-5) = 3/7
Cross-multiplying, we get:
7(x-5) = 3(y-5)
7x - 35 = 3y - 15
7x - 3y = 20 (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously.
Using any method (such as substitution or elimination), solving Equation 1 and Equation 2 will give us the values of x and y, which correspond to the numerator and denominator of the fraction.
Once we have the values of x and y, we can simplify the fraction x/y to find the solution to the problem.