+2666 We have: \({b+7 \over b + 4} = {c \over 9}\)
Cross multiply: \(9(b+7) = c(b + 4)\)
Distribute both sides: \(9b + 63 = bc + 4c \)
Subtract 9b from both sides: \(bc + 4c - 9b = 63\)
Factor out b: \(b(c-9) + 4c = 63\)
Subtract 36 from both sides (Simon's Favorite Factoring Trick): \(b(c-9) + 4c - 36 = 27\)
Factor out c: \(b(c-9) + 4(c-9) = 27\)
Simplify the left-hand side: \((b+4)(c-9) = 27\)
The (positive and negative) factor pairs of 27 are (1,27), (-1,-27), (-3,-9) and (3,9).
Each of those cases has 2 pairs of integers that work, which makes for \(4 \times 2 = \color{brown}\boxed{8}\) integers.
+2666 We have: \({b+7 \over b + 4} = {c \over 9}\)
Cross multiply: \(9(b+7) = c(b + 4)\)
Distribute both sides: \(9b + 63 = bc + 4c \)
Subtract 9b from both sides: \(bc + 4c - 9b = 63\)
Factor out b: \(b(c-9) + 4c = 63\)
Subtract 36 from both sides (Simon's Favorite Factoring Trick): \(b(c-9) + 4c - 36 = 27\)
Factor out c: \(b(c-9) + 4(c-9) = 27\)
Simplify the left-hand side: \((b+4)(c-9) = 27\)
The (positive and negative) factor pairs of 27 are (1,27), (-1,-27), (-3,-9) and (3,9).
Each of those cases has 2 pairs of integers that work, which makes for \(4 \times 2 = \color{brown}\boxed{8}\) integers.
