Mixture and Comparison Problems
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SSAT Middle Level: Quantitative › Mixture and Comparison Problems
A paint store mixes white paint with blue paint to create a custom color. White paint costs $$25 per gallon and blue paint costs $$35 per gallon. If 12 gallons of the mixture costs $$28 per gallon, what percentage of the mixture by volume is white paint?
White paint makes up 80% of the mixture by volume
White paint makes up 75% of the mixture by volume
White paint makes up 65% of the mixture by volume
White paint makes up 70% of the mixture by volume
White paint makes up 60% of the mixture by volume
Explanation
When you encounter a mixture problem involving costs, you're dealing with a weighted average situation. The key insight is that the final price per gallon ($28) falls between the two component prices ($25 and $35), and its position tells you the proportions.
Let's say white paint makes up $$x$$ fraction of the mixture, so blue paint makes up $$(1-x)$$. The weighted average formula gives us:
$$25x + 35(1-x) = 28$$
Solving this equation:
$$25x + 35 - 35x = 28$$
$$-10x = -7$$
$$x = 0.7 = 70%$$
We can verify: If 70% is white and 30% is blue, then $$0.7(25) + 0.3(35) = 17.5 + 10.5 = 28$$ ✓
Answer A (60%) would give us $$0.6(25) + 0.4(35) = 29$$, which is too expensive. Answer B (65%) yields $$0.65(25) + 0.35(35) = 28.5$$, still too high. Answer D (75%) produces $$0.75(25) + 0.25(35) = 27.25$$, which is too cheap. Each wrong answer represents a common calculation error or misapplication of the weighted average formula.
The correct answer is C: white paint makes up 70% of the mixture.
Study tip: In mixture problems, always check that your answer makes intuitive sense. Since $28 is closer to $25 than to $35, you'd expect more white paint than blue paint in the mixture, which 70% white confirms.
A tea merchant blends two types of tea. Type X costs $$12 per pound and Type Y costs $$18 per pound. If he wants to create 25 pounds of a blend that costs $$15 per pound, and then sell it at a 20% markup, what will be his profit per pound?
The profit per pound will be $$3.00
The profit per pound will be $$4.20
The profit per pound will be $$3.60
The profit per pound will be $$2.40
The profit per pound will be $$1.80
Explanation
This problem combines two key concepts: mixture problems and profit calculations. When you see a blending scenario with different costs per unit, you need to set up equations based on both quantity and total cost.
First, let's find how much of each tea type is needed. Let $$x$$ = pounds of Type X tea. Then $$(25-x)$$ = pounds of Type Y tea. The total cost equation is: $$12x + 18(25-x) = 15(25)$$. Solving: $$12x + 450 - 18x = 375$$, so $$-6x = -75$$ and $$x = 12.5$$ pounds. This means 12.5 pounds of Type X and 12.5 pounds of Type Y.
The blend costs $$15 per pound to make. With a 20% markup, the selling price is $$15 × 1.20 = 18$$ per pound. Therefore, profit per pound = selling price - cost = $$18 - 15 = 3.00$$.
Choice A ($$1.80) represents only the markup amount without adding it to the base cost. Choice B ($$2.40) likely comes from incorrectly calculating 20% of $$12 instead of $$15. Choice D ($$3.60) might result from taking 20% of $$18 (the selling price) instead of understanding that profit is the difference between selling price and cost.
Remember that profit per unit equals selling price minus cost price. When dealing with percentage markups, always apply the percentage to the cost price to find the selling price, then subtract the original cost to find profit.