Mike enters the Vermilion game with a batting average of .274. After going 3-for-4 in the game, his average has gone up to .289. How many hits did he have for the season before the game started?
We recall that a batting average of .274 means h/b = 0.274, which is rounded from some long decimal.
So (1) h/b = 0.274, and (2) (h + 3)/(b + 4) = 0.289.
(1) h = 0.274b
(2) h + 3 = 0.289(b + 4) = 0.289b + 1.156
→ 0.274b + 3 = 0.289b + 1.156
→ 0.015b = 1.844
→ b = 1.844/0.015 = 122.933. Let’s assume 123.
Then h = 0.274 x 123 = 33.702. Let’s assume 34.
(2) h + 3 = 0.289(b + 4) = 0.289b + 1.156
→ 0.274b + 3 = 0.289b + 1.156
→ 0.015b = 1.844
→ b = 1.844/0.015 = 122.933. Let’s assume 123.
Then h = 0.274 x 123 = 33.702. Let’s assume 34.
Checking: 34/123 = 0.27642… So we’ve got rounding issues.
Since .276… is a bit too big, should we try b = 124? Then 34/124 = 0.27419….
Pretty good. And (34 + 3)/(124 + 4) = 37/128 = 0.28906…. Also good.
Pretty good. And (34 + 3)/(124 + 4) = 37/128 = 0.28906…. Also good.
If you start over using these 5-place decimals, you get 34 and 124 almost exactly.