A grocer sells apples for $1.35 each, pears for $1.45 each and oranges for $1.60 each. She also offers a discount of 70c for anyone who buys two pieces of fruit, and a discount of $1.60 for anyone who buys three pieces of fruit. One day she sells 50 pieces of each type of fruit, for a total of $202. How many customers did she have, assuming each customer bought 1, 2 or 3 pieces of fruit?
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Let's break down the problem and find the solution step by step:
1. We know the prices of each fruit: apples cost $1.35, pears cost $1.45, and oranges cost $1.60.
2. The grocer offers two types of discounts:
a) A discount of 70c if two pieces of fruit are purchased together.
b) A discount of $1.60 if three pieces of fruit are purchased together.
3. From the problem statement, we gather that 50 pieces of each fruit were sold in total. Let's denote the number of customers who bought one piece of fruit as 'x', the number of customers who bought two pieces as 'y', and the number of customers who bought three pieces as 'z'.
4. Given the total quantity sold, we can form three equations:
a) x + y + z = 50 (Total number of customers)
b) x + 2y + 3z = 50 (Total number of fruit pieces)
5. To calculate the amount the grocer earned, we multiply the number of pieces by the respective fruit prices and deduct the applicable discounts:
Amount earned = (number of apples * price per apple) + (number of pears * price per pear) + (number of oranges * price per orange)
= x * $1.35 + y * $1.45 + z * $1.60
6. We know the total earnings from fruit sales were $202:
x * $1.35 + y * $1.45 + z * $1.60 = $202
Now, we have a system of two equations with three variables. To solve this, we can use an elimination method or substitution method.
Let's solve it using the substitution method:
1) Rearrange the equation x + y + z = 50 to get x = 50 - y - z.
2) Substitute the value of x in the second equation:
(50 - y - z) * $1.35 + y * $1.45 + z * $1.60 = $202
3) Simplify the equation:
$67.50 - $1.35y - $1.35z + $1.45y + $1.60z = $202
4) Combine like terms:
$0.10y + $0.25z = $134.50
5) Multiply through by 10 to eliminate decimal points:
y + 2.5z = 1345
6) Use the restriction y + z = 50 from the first equation to substitute y:
50 - z + 2.5z = 1345
7) Simplify the equation:
1.5z = 1295
8) Solve for z:
z = 1295 / 1.5
z = 863.33
However, z represents the number of customers who bought three pieces of fruit, which should be a whole number. Therefore, the solution is not possible.
Hence, there seems to be an error or inconsistency in the given information or problem statement.