If the numerator of a fractionis multiplied by 4 and the denominatoris reduced by 2,the result is 2.if the numerator of the fraction is increase by 15 and 2 subtracted from the double of the denominator the result is 9/7, find the fraction
If the original fraction is n/d, then we are told:
4n/(d-2) = 2
(n+15)/(2d-2) = 9/7
n/d = 3/8
I don't understand this
Let's start by representing the fraction as numerator/denominator. We don't know the values of the numerator and denominator yet, so let's assume them as x and y, respectively.
According to the given information:
1. "If the numerator of a fraction is multiplied by 4 and the denominator is reduced by 2, the result is 2."
This can be written as (4x)/(y-2) = 2.
2. "If the numerator of the fraction is increased by 15 and 2 is subtracted from the double of the denominator, the result is 9/7."
This can be written as (x + 15)/(2y - 2) = 9/7.
Now, we can solve these two equations simultaneously to find the values of x and y.
Step 1: Solve the first equation for x:
(4x)/(y - 2) = 2
Multiply both sides of the equation by (y - 2):
4x = 2(y - 2)
Distribute:
4x = 2y - 4
Divide both sides by 4:
x = (2y - 4)/4
Simplify:
x = (y - 2)/2
Step 2: Substitute the value of x in the second equation:
(x + 15)/(2y - 2) = 9/7
Substitute (y - 2)/2 for x:
[(y - 2)/2 + 15]/(2y - 2) = 9/7
Simplify the numerator on the left side:
[(y - 2 + 30)/2]/(2y - 2) = 9/7
Simplify further:
(y - 2 + 30)/(2(2y - 1)) = 9/7
Combine like terms in the numerator:
(y + 28)/(4y - 2) = 9/7
Cross-multiply:
7(y + 28) = 9(4y - 2)
Distribute:
7y + 196 = 36y - 18
Move all terms containing y to one side and all constants to the other side:
36y - 7y = 196 + 18
Simplify:
29y = 214
Divide both sides by 29:
y = 214/29
Simplify further:
y = 7.3793 (rounded to four decimal places)
Step 3: Substitute the value of y back into the equation for x:
x = (y - 2)/2
x = (7.3793 - 2)/2
Simplify:
x = 5.3793/2
x = 2.6897 (rounded to four decimal places)
Therefore, the fraction is approximately 2.6897/7.3793.
To find the fraction, we'll need to set up a system of equations based on the given information.
Let's assume the numerator of the fraction is "x" and the denominator is "y".
According to the first statement, if the numerator is multiplied by 4 and the denominator is reduced by 2, the result is 2. So we can write the equation as:
4x / (y-2) = 2
Now, let's look at the second statement. It states that if the numerator is increased by 15 and 2 is subtracted from double the denominator, the result is 9/7. We can write this equation as:
(x + 15) / (2y - 2) = 9/7
We now have a system of equations:
Equation 1: 4x / (y-2) = 2
Equation 2: (x + 15) / (2y - 2) = 9/7
To solve this system, we can use elimination. First, let's simplify equation 2 by multiplying both sides by 7:
7(x + 15) = 9(2y - 2)
7x + 105 = 18y - 18
Next, let's rearrange equation 1 to solve for x:
4x = 2(y - 2)
4x = 2y - 4
Now we can set up a new equation using the rearranged equation 1 and the simplified equation 2:
4x = 7x + 87
Subtract 7x from both sides:
-3x = 87
Divide both sides by -3:
x = -29
Now that we have the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use equation 1:
4(-29) = 2(y - 2)
-116 = 2y - 4
Add 4 to both sides:
-112 = 2y
Divide both sides by 2:
y = -56
So the fraction is -29/-56, which can be simplified to 29/56.