If the numerator and denominator of a fraction are each increased by 3, the fraction is equivalent to two-thirds. If the numerator and denominator are each increased by 11, the fraction is equivalent to four-fifths. What is the fraction?
To solve this problem, we'll set up a system of equations using the given information. Let's assume that the fraction is represented by "a/b," where a is the numerator and b is the denominator.
According to the first condition, if the numerator and denominator are each increased by 3, the new fraction is equivalent to two-thirds:
(a + 3) / (b + 3) = 2/3
Now, let's solve the equation above for a.
Cross-multiplying, we get:
3(a + 3) = 2(b + 3)
Expanding both sides, we have:
3a + 9 = 2b + 6
Rearranging the terms, we find:
3a - 2b = -3 [Equation 1]
Next, let's use the second condition. If the numerator and denominator are each increased by 11, the new fraction is equivalent to four-fifths:
(a + 11) / (b + 11) = 4/5
Again, solving for a by cross-multiplying, we get:
5(a + 11) = 4(b + 11)
Expanding both sides, we have:
5a + 55 = 4b + 44
Rearranging the terms, we find:
5a - 4b = -11 [Equation 2]
Now, we have a system of equations (Equation 1 and Equation 2) to solve simultaneously:
3a - 2b = -3 [Equation 1]
5a - 4b = -11 [Equation 2]
We can solve this system using any suitable method, such as substitution or elimination. Let's use the method of elimination.
Multiplying Equation 1 by 2 and Equation 2 by 3, we have:
6a - 4b = -6 [Equation 3]
15a - 12b = -33 [Equation 4]
Subtracting Equation 3 from Equation 4, we find:
9a - 8b = -27
Simplifying further, we get:
9a = 8b - 27 [Equation 5]
Now, let's substitute Equation 5 into either Equation 1 or Equation 2 to find the value of b.
Using Equation 1, we have:
3a - 2b = -3
Substituting 8b - 27 for 9a, we obtain:
3(8b - 27) - 2b = -3
Expanding and simplifying, we get:
24b - 81 - 2b = -3
Combining like terms, we have:
22b - 81 = -3
Adding 81 to both sides, we find:
22b = 78
Dividing both sides by 22, we obtain:
b = 3
Now that we have the value of b, we can substitute it back into Equation 1 to find a:
3a - 2(3) = -3
Simplifying, we get:
3a - 6 = -3
Adding 6 to both sides, we have:
3a = 3
Dividing both sides by 3, we obtain:
a = 1
Therefore, the fraction is 1/3.