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.plz tell me step by step
done. see related questions below.
OK, let's see whether I can clarify it a bit.
If the original fraction is n/d, then we are told:
numerator of a fraction is multiplied by 4: change n to 4n
denominator is reduced by 2: change d to d-2
4n/(d-2) = 2
Similar changes for the 2nd half, giving us
(n+15)/(2d-2) = 9/7
Now just find n and d:
4n/(d-2) = 2
(n+15)/(2d-2) = 9/7
4n = 2(d-2)
7(n+15) = 9(2d-2)
4n = 2d-4
7n + 105 = 18d - 18
4n-2d = -4
7n-18d = -123
36n-18d = -36
7n-18d = -123
29n = 87
n=3
Now use that in 4n-2d = -4 to get
d = 8
So the original fraction is 3/8
check:
4*3/(8-2) = 12/6 = 2
(3+15)/(2*8-2) = 18/14 = 9/7
Let's represent the fraction as x/y.
According to the given information, when the numerator of the 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
Simplifying this equation, we get:
4x = 2(y-2)
Next, we are given that when 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
To solve this equation, we can cross-multiply:
7(x + 15) = 9(2y - 2)
Expanding both sides:
7x + 105 = 18y - 18
Now let's simplify further:
7x = 18y - 18 - 105
7x = 18y - 123
We now have two equations:
4x = 2(y-2)
7x = 18y - 123
To find the values of x and y, we can solve these two equations simultaneously. Let's first solve the first equation for x:
4x = 2(y-2)
Divide both sides by 2:
2x = y - 2
x = (y-2)/2
Substitute this value of x in the second equation:
7[(y-2)/2] = 18y - 123
Now, let's simplify:
7(y-2) = 2(18y - 123)
7y - 14 = 36y - 246
Now, bring all the y terms to one side and the numbers to the other side:
36y - 7y = 246 - 14
29y = 232
Divide both sides by 29:
y = 232/29
Simplifying this fraction, we get:
y = 8
Now, substitute the value of y into the expression for x:
x = (y-2)/2
x = (8-2)/2
x = 6/2
x = 3
Therefore, the fraction is 3/8.
Let's begin by assigning variables to the numerator and denominator of the fraction. We'll call the numerator 'x' and the denominator 'y'.
Step 1: Translate the given information into equations:
From the first statement:
4x / (y - 2) = 2
From the second statement:
(x + 15) / (2y - 2) = 9/7
Step 2: Simplify the equations:
For the first equation, we can cross multiply to eliminate the denominator:
4x = 2(y - 2)
For the second equation, we can multiply both sides by (2y - 2) to eliminate the denominator:
7(x + 15) = 9(2y - 2)
Step 3: Simplify further:
For the first equation:
4x = 2y - 4
For the second equation:
7x + 105 = 18y - 18
Step 4: Rearrange the equations to solve for 'x' in terms of 'y':
From the first equation:
4x = 2y - 4
Divide both sides by 4:
x = (2y - 4) / 4
x = (y - 2) / 2
From the second equation:
7x + 105 = 18y - 18
Subtract 105 from both sides:
7x = 18y - 18 - 105
7x = 18y - 123
Divide both sides by 7:
x = (18y - 123) / 7
Step 5: Set the two expressions for 'x' equal to each other:
(y - 2) / 2 = (18y - 123) / 7
Step 6: Cross multiply and simplify:
7(y - 2) = 2(18y - 123)
7y - 14 = 36y - 246
Combine like terms:
0 = 36y - 7y - 246 + 14
0 = 29y - 232
Step 7: Solve for 'y':
29y = 232
Divide both sides by 29:
y = 8
Step 8: Substitute the value of 'y' back into one of the equations to find 'x':
x = (8 - 2) / 2
x = 6 / 2
x = 3
Therefore, the fraction is 3/8.