(a) To find the fraction of cupcakes Jacob had remaining after giving some to his neighbor, we need to subtract the fraction he gave away from 1.
Jacob gave away 1/7 of the cupcakes, so the fraction he has left is 1 - 1/7.
To subtract fractions, we need a common denominator. In this case, the common denominator is 7.
So, we have (7/7) - (1/7) = 6/7.
Therefore, Jacob had 6/7 of the cupcakes remaining after giving some to his neighbor.
(b) To determine how many cupcakes Jacob gave to his neighbor, we can start by subtracting the number of cupcakes he gave to his aunt and the number of cupcakes he had left from the total number of cupcakes he baked.
Jacob baked a certain number of cupcakes, but the exact quantity is not mentioned in the question. Let's represent this unknown quantity as "x."
According to the information provided, Jacob gave 1/7 of the cupcakes to his neighbor. This can be expressed as (1/7)x. Additionally, he gave 27 cupcakes to his aunt. Therefore, the total number of cupcakes given away is (1/7)x + 27.
The remaining cupcakes that Jacob had can be represented as 15.
So, we can set up an equation to solve for x:
x - [(1/7)x + 27] = 15
To simplify the equation, we can distribute the negative sign:
x - (1/7)x - 27 = 15
Next, we combine like terms:
(6/7)x - 27 = 15
To isolate x, we add 27 to both sides of the equation:
(6/7)x = 42
Finally, we multiply both sides by the reciprocal of (6/7), which is (7/6):
x = (42)(7/6) = 49
Therefore, Jacob baked a total of 49 cupcakes.
To find out how many cupcakes he gave to his neighbor, we substitute this value into the expression (1/7)x:
(1/7)(49) = 7
Thus, Jacob gave 7 cupcakes to his neighbor.