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PS · Problem Solving
705-805
percent
In a recurring manufacture–to–sale cycle, a toy manufacturer sells a type of toys to a retailer at a profit that varies between 50% and 60%, depending on the size of the retailer’s order (the retailer gets a proportional discount for bulk purchase of toys). The retailer marks up the price of these toys by 20% to 40% depending on the location of his different shops. The retailer then offers a discount, across all shops, of either 10% or $15 on the purchase of a toy to the customer. If the manufacturer incurs a cost of $100 to manufacture a toy, the profit earned by the retailer per toy must be between:
Answer Choices
Correct answer marked belowA
$12 and $45
B
$15 and $45
$12 and $49
Correct
D
$17 and $49
E
$17 and $45
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We start with manufacturer cost = $100.
Manufacturer's profit is 50% to 60%, so selling price to retailer is between and .
Retailer marks up by 20% to 40%, so marked price before discount is between:
- Minimum:
- Maximum:
Retailer then offers a discount of either 10% or $15.
To minimize final selling price (and thus retailer profit), we apply the larger discount: maximum discount is $15 (since 10% of minimum marked price 180 is $18, but discount choice is fixed per toy as either 10% or $15, we compare for each extreme).
- For minimum profit scenario: marked price = 180, discount = $15 (since 10% of 180 = 18 > 15, retailer would choose the smaller discount to minimize profit? Wait, we want retailer profit range. For minimum retailer profit, we want lowest final selling price minus highest cost to retailer. So we need to minimize selling price and maximize cost. Let's compute systematically.
Retailer's cost: between 150 and 160.
Retailer's selling price after discount:
- Minimum selling price occurs with smallest marked price and largest discount: marked price = 180, discount = $15 (since 15 > 10% of 180? Actually 10% of 180 = 18, so $15 is smaller discount than 10% here. To minimize selling price, we want largest discount. Largest discount is max(10% of marked price, 15). For marked price 180, 10% = 18, so largest discount = 18. But the problem says discount is either 10% or $15 across all shops, meaning each shop applies one of these two discounts uniformly. To find range of possible profits, we consider combinations that yield minimum and maximum profit.
Let's find minimum retailer profit:
Retailer profit = selling price after discount - cost to retailer.
To minimize profit, minimize selling price and maximize cost.
- Minimize selling price: smallest marked price = 180, and apply larger discount: discount = 10% of 180 = 18 (since 18 > 15), so selling price = 180 - 18 = 162.
- Maximize cost: retailer cost = 160.
So minimum profit = 162 - 160 = $2? That's not in options. But wait, we must also consider that discount is either 10% or $15, not necessarily the larger one for profit minimization. Actually, retailer chooses discount to maximize his profit? No, discount is fixed across shops, but we are to find possible profit range given all variations. So for each scenario (marked price, cost, discount type), profit varies. We need absolute min and max possible profit.
Let's compute all extremes:
Cost to retailer: C in [150, 160]
Marked price: M in [180, 224] (since M = C_retailer_in × (1+markup%), but careful: C_retailer_in is manufacturer selling price, which is between 150 and 160. Markup is applied to that cost. So M = C_retailer_in × (1.2 to 1.4). So minimum M = 150×1.2=180, maximum M=160×1.4=224.…
Manufacturer's profit is 50% to 60%, so selling price to retailer is between and .
Retailer marks up by 20% to 40%, so marked price before discount is between:
- Minimum:
- Maximum:
Retailer then offers a discount of either 10% or $15.
To minimize final selling price (and thus retailer profit), we apply the larger discount: maximum discount is $15 (since 10% of minimum marked price 180 is $18, but discount choice is fixed per toy as either 10% or $15, we compare for each extreme).
- For minimum profit scenario: marked price = 180, discount = $15 (since 10% of 180 = 18 > 15, retailer would choose the smaller discount to minimize profit? Wait, we want retailer profit range. For minimum retailer profit, we want lowest final selling price minus highest cost to retailer. So we need to minimize selling price and maximize cost. Let's compute systematically.
Retailer's cost: between 150 and 160.
Retailer's selling price after discount:
- Minimum selling price occurs with smallest marked price and largest discount: marked price = 180, discount = $15 (since 15 > 10% of 180? Actually 10% of 180 = 18, so $15 is smaller discount than 10% here. To minimize selling price, we want largest discount. Largest discount is max(10% of marked price, 15). For marked price 180, 10% = 18, so largest discount = 18. But the problem says discount is either 10% or $15 across all shops, meaning each shop applies one of these two discounts uniformly. To find range of possible profits, we consider combinations that yield minimum and maximum profit.
Let's find minimum retailer profit:
Retailer profit = selling price after discount - cost to retailer.
To minimize profit, minimize selling price and maximize cost.
- Minimize selling price: smallest marked price = 180, and apply larger discount: discount = 10% of 180 = 18 (since 18 > 15), so selling price = 180 - 18 = 162.
- Maximize cost: retailer cost = 160.
So minimum profit = 162 - 160 = $2? That's not in options. But wait, we must also consider that discount is either 10% or $15, not necessarily the larger one for profit minimization. Actually, retailer chooses discount to maximize his profit? No, discount is fixed across shops, but we are to find possible profit range given all variations. So for each scenario (marked price, cost, discount type), profit varies. We need absolute min and max possible profit.
Let's compute all extremes:
Cost to retailer: C in [150, 160]
Marked price: M in [180, 224] (since M = C_retailer_in × (1+markup%), but careful: C_retailer_in is manufacturer selling price, which is between 150 and 160. Markup is applied to that cost. So M = C_retailer_in × (1.2 to 1.4). So minimum M = 150×1.2=180, maximum M=160×1.4=224.…
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What it tests
Your ability to apply GMAT math concepts to solve a multi-step problem quickly and accurately. At the 705-805 level, accuracy and speed both matter.
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Overcomplicating the setup — most GMAT problems reward a direct, organized approach over brute calculation.
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705-805
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PS
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