A fraction has a value 2/3. If 4 is added to the numerator and the denominator is decreased by 2, the resulting fractions have the value 6/7. What is the original fraction? (Let x be the numerator and y be the denominator)
numerator --- x
denominator --- y
x/y = 2/3
2y = 3x
y = 3x/2
new fraction:
(x+4)/(y-2) = 6/7
6y - 12 = 7x + 28
6y = 7x + 40
3(2y) = 7x+40
3(3x) = 7x + 40
2x = 40
x = 20
y = 30
the non-reduced fraction was 20/30
Well, let's put on our thinking nose and solve this problem, shall we?
First, we're given the original fraction with a value of 2/3. So, we can say that x/y = 2/3.
Next, we have to manipulate the fraction by adding 4 to the numerator and decreasing the denominator by 2. This gives us (x + 4)/(y - 2) = 6/7.
Now, let's solve the first equation for x:
x/y = 2/3
Cross multiplying gives us 3x = 2y.
Now, let's solve the second equation for x:
(x + 4)/(y - 2) = 6/7
Cross multiplying again, we get 7(x + 4) = 6(y - 2).
Expanding, we have 7x + 28 = 6y - 12.
Now, let's substitute the value of x from the first equation into the second equation:
7(2y/3) + 28 = 6y - 12
14y/3 + 28 = 6y - 12
Multiplying through by 3 to get rid of the fractions, we have:
14y + 84 = 18y - 36
Rearranging, we get 4y = 120
Dividing both sides by 4:
y = 30
Now, substituting the value of y into the first equation:
x/30 = 2/3
Cross multiplying gives us 3x = 60
Dividing both sides by 3:
x = 20
So, the original fraction is 20/30, which can be simplified to 2/3.
Voilà! We've figured it out!
Let's solve this step-by-step.
Step 1: Write the equation based on the given information:
The original fraction is x/y.
According to the problem, the original fraction has a value of 2/3.
So, we can write the first equation as x/y = 2/3.
Step 2: Write the second equation based on the given information:
When 4 is added to the numerator and the denominator is decreased by 2, the resulting fraction is (x+4)/(y-2).
According to the problem, this fraction has a value of 6/7.
So, we can write the second equation as (x+4)/(y-2) = 6/7.
Step 3: Solve the equations simultaneously to find the values of x and y:
From equation (1), x/y = 2/3, we can cross multiply to get 3x = 2y.
Now, we can substitute the value of y from equation (2) into the equation we found in step 3:
3x = 2(y-2).
Expanding the equation gives us:
3x = 2y - 4.
Simplifying the equation gives us:
3x - 2y = -4.
Step 4: Multiply equation (2) by 7 to eliminate the fractions:
7(x+4) = 6(y-2).
Expanding the equation gives us:
7x + 28 = 6y - 12.
Simplifying the equation gives us:
7x - 6y = -40.
Step 5: Solve the two equations we found in steps 3 and 4 simultaneously:
We have the equations:
3x - 2y = -4,
7x - 6y = -40.
Using any method of solving simultaneous equations (elimination, substitution, or matrix), we find x = 16 and y = 24.
Therefore, the original fraction is 16/24, which can be simplified to 2/3.
To solve this problem, we can set up a system of equations using the information given.
Let x be the numerator and y be the denominator of the original fraction.
From the first sentence, we know that the original fraction has a value of 2/3, so our first equation is:
x/y = 2/3
From the second sentence, we are told that if 4 is added to the numerator and the denominator is decreased by 2, the resulting fraction is 6/7. Therefore, our second equation is:
(x + 4)/(y - 2) = 6/7
Now we can solve this system of equations to find the values of x and y.
First, let's solve the first equation for x:
x/y = 2/3
Cross-multiplying, we get:
3x = 2y
x = (2y)/3
Substituting this value of x into the second equation:
((2y)/3 + 4)/(y - 2) = 6/7
Now we can solve for y.
Multiply both sides of the equation by 7(y - 2) to get rid of the denominators:
7(y - 2)((2y)/3 + 4) = 6(y - 2)
Expand and simplify the equation:
7(2y + 12(y - 2) = 6(y - 2)
Distribute:
14y + 84y - 168 = 6y - 12
Combine like terms:
98y - 168 = 6y - 12
Move all the y terms to one side and all the constant terms to the other side:
98y - 6y = 168 - 12
Simplify:
92y = 156
Divide both sides by 92 to solve for y:
y = 156/92
y = 39/23
Now that we have the value of y, we can substitute it back into the first equation to solve for x:
x/y = 2/3
x/(39/23) = 2/3
Multiply both sides by (39/23) to isolate x:
x = (39/23) * (2/3)
Multiply the numerators and the denominators:
x = (78/69)
Simplifying, we get:
x = 2/3
Therefore, the original fraction is x/y = (2/3)/(39/23) = 46/39