When nine is added to both the numerator and the denominator of an original fraction, the new fraction equals six sevenths. When seven is subtracted from both the numerator and the denominator of the original fraction, a third fraction is two thirds. Find the original fraction.
Use algebraic equations to justify your answer.
Just put the words into symbols. If the original fraction is n/d, then
(n+9)/(d+9) = 6/7
(n-7)/(d-7) = 2/3
Now just solve for n and d.
I'm not really sure how I'm supposed to solve for n or d.
as usual, first just clear fractions:
(n+9)(7) = 6(d+9)
(n-7)(3) = 2(d-7)
7n+63 = 6d+54
3n-21 = 2d-14
7n-6d = -9
3n-2d = 7
Now can you solve for n and d?
15/19
isnt it 1/2
To solve this problem, we can start by assigning variables to the numerator and denominator of the original fraction. Let's say that the original fraction is represented as x/y, where x is the numerator and y is the denominator.
According to the problem, when nine is added to both the numerator and denominator, the new fraction becomes (x + 9)/(y + 9), which equals six sevenths or 6/7.
This can be stated as: (x + 9)/(y + 9) = 6/7.
Similarly, when seven is subtracted from both the numerator and denominator of the original fraction, the new fraction becomes (x - 7)/(y - 7), which equals two thirds or 2/3.
This can be stated as: (x - 7)/(y - 7) = 2/3.
Now we have a system of two equations with two unknowns. To solve this system, we can use a method called substitution.
First, let's solve one equation for one variable. We'll solve the second equation for x in terms of y:
(x - 7)/(y - 7) = 2/3
Cross-multiplying, we get:
3(x - 7) = 2(y - 7)
3x - 21 = 2y - 14
3x = 2y - 14 + 21
3x = 2y + 7
x = (2y + 7)/3
Now, substitute this expression for x into the first equation:
(x + 9)/(y + 9) = 6/7
[(2y + 7)/3 + 9]/(y + 9) = 6/7
[(2y + 7 + 27)/3]/(y + 9) = 6/7
(2y + 34)/(3y + 27) = 6/7
Cross-multiplying, we get:
7(2y + 34) = 6(3y + 27)
14y + 238 = 18y + 162
14y - 18y = 162 - 238
-4y = -76
y = -76 / -4
y = 19
Now substitute the value of y back into the expression for x:
x = (2y + 7)/3
x = (2(19) + 7)/3
x = (38+7)/3
x = 45/3
x = 15
Therefore, the original fraction is 15/19.