In a certain positive fraction \frac{a}{b} , the numerator is 2 less than the denominator. If the numerator and denominator are interchanged, the fraction is increased by 1 \frac{1}{15} . Find a + b .
To solve this problem, we first need to set up equations based on the given information.
Let's assume that the denominator is represented by "b," and the numerator is represented by "a."
According to the problem, the numerator is 2 less than the denominator:
a = b - 2 (Equation 1)
When the numerator and denominator are interchanged, the fraction is increased by 1 1/15. Let's convert 1 1/15 into an improper fraction:
1 1/15 = (15 * 1 + 1) / 15 = 16 / 15
So, the new fraction, after interchanging the numerator and denominator, can be written as:
(b - 2)/b + 16/15
Now, we have the equation:
(b - 2)/b + 16/15 = a/b (Equation 2)
To find the sum, a + b, we need to eliminate the fractions by multiplying through by the denominators.
Multiply both sides of Equation 2 by "15b" to eliminate the fractions:
(15b)((b - 2)/b) + (15b)(16/15) = (15b)(a/b)
Simplifying this equation gives us:
15(b - 2) + 16b = 15a
Expanding and simplifying further:
15b - 30 + 16b = 15a
Combining like terms:
31b - 30 = 15a
We also know from Equation 1 that:
a = b - 2
Substitute this value of "a" into the equation above:
31b - 30 = 15(b - 2)
Expanding and simplifying:
31b - 30 = 15b - 30
Collecting terms:
31b - 15b = -30 + 30
16b = 0
Dividing by 16 on both sides:
b = 0
Substituting this value of "b" into Equation 1 to find "a":
a = b - 2
a = 0 - 2
a = -2
So, the value of "a" is -2, and the value of "b" is 0.
Finally, to find a + b, we simply calculate:
a + b = -2 + 0 = -2
Therefore, a + b equals -2.