Germaine spent $20 less than 1/3 of her money on vegetables. She spent $14 more than 1/2 of her remaining amount of money on meat. If she had $76 left, how much money did she have at first?
From the given information, Germaine spent $76 on vegetables and meat. This amount is $14 more than 1/2 * remaining amount of money on meat.
Therefore 1/2 * remaining amount of money = $76 - $14 = $<<76-14=62>>62
$1 * remaining amount of money = $62 * 2 = $<<62*2=124>>124
Germaine had $124 left after buying vegetables, meat, and other things.
1/3 * her money = $124 - $20 = $<<124-20=104>>104
Germaine had $104 * 3 = $<<104*3=312>>312 at first. Answer: \boxed{312}.
Let's break down the problem step by step:
Step 1: Calculate the amount of money Germaine spent on vegetables.
Let x be Germaine's initial amount of money.
According to the problem, Germaine spent $20 less than 1/3 of her money on vegetables. Therefore, she spent (1/3)x - $20 on vegetables.
Step 2: Calculate the remaining amount of money Germaine had after buying vegetables.
Since she spent (1/3)x - $20 on vegetables, the remaining amount of money is x - ((1/3)x - $20) = x - (1/3)x + $20 = (2/3)x + $20.
Step 3: Calculate the amount of money Germaine spent on meat.
Germaine spent $14 more than 1/2 of her remaining amount of money on meat. Hence, she spent (1/2)((2/3)x + $20) + $14 on meat.
Step 4: Calculate the remaining amount of money Germaine had after buying meat.
Since she spent (1/2)((2/3)x + $20) + $14 on meat, the remaining amount of money is (2/3)x + $20 - [(1/2)((2/3)x + $20) + $14] = (2/3)x + $20 - (1/3)x - $10 - $14 = (1/3)x - $4.
Step 5: Set up an equation using the given information.
According to the problem, Germaine had $76 left. Therefore, we have the equation (1/3)x - $4 = $76.
Step 6: Solve the equation for x.
To solve the equation, we can isolate (1/3)x: (1/3)x = $76 + $4 = $80.
Step 7: Solve for x.
Multiply both sides of the equation by 3: x = 3 * $80 = $240.
Therefore, Germaine had $240 at first.