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What are the 2 formulas for weighted averages? |
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Explain exactly what a weighted average is? |
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"Of the shoe stores in City X, 30 carry Brand A shoes, 40 carry Brand B shoes, and 25 carry Brand C shoes. If each store carries at least one of the brands, 32 of the stores carry two of the three shoe brands, and none of the stores carry all three, how many shoe stores are there in City X?"
What is the 3-overlapping sets formula and how does it apply to this question? |
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Definition
Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither
G1+G2+G3 is the sum of all the stores that carry these brands,only we've double-counted some of them. Since 32 of the stores carry two brands, we've double-counted exactly 32, so we subtract 32, the "sum of 2-group overlaps."
"2*(all three)". If one of the stores in City X carried all three shoe brands, that store is being counted three times--once in the 30 A's, once in the 40 B's, and once in the 25 C's. But it's still only one store. Thus, if it's represented three times, we need to subtract it twice. In this case, the question tells us that the "all three" term is equal to zero.
"Neither" equals 0 because every shoe store in City X carries at least one of the brands.
So, to solve this example...
Total = 30 + 40 + 25 - 32 - 2*0 + 0 = 63
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"A class has 40 students, 25 of whom are boys and 12 of whom play basketball. If 8 of the boys play basketball, how many girls do not play basketball?"
What is the overlapping sets formula? Explain how it works in relation to this question. |
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Definition
Total = Group1 + Group2 - Both + Neither
In the example above, 40 is Total, 25 is Group1, 12 is Group2, and 8 is Both. (Group1 and Group2 are interchangeable.)
Basically, G1+G2 is the sum of all of the people who do one or the other, but that sum double-counts the number who do both. That's why we subtract both.
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Three rugs have a combined area of 200 square meters. By overlapping the rugs to cover floor area of 140 square meters, the area that is covered by exactly two layers of rug is 24 square meters. What is the area that is covered with three layers of rug ?
A. 18 square meters
B. 20 square meters
C. 24 square meters
D. 28 square meters
E. 30 square meters |
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Definition
ANSWER = A
This is a triple overlapping set.
Here is how I did it (thanks for Horacio Quiroga of MGMAT for teaching me this technique)
S is the sum of just the single carpets, D is sum of double overlaps, and T is sum of the triple overlaps
S + 2D + 3T = 200 (accounts for the total area of carpets and overlaps)
S + D + T = 140 (excludes double counting)
Subtracting the equations gets me:
D + 2T = 60
I know that the total "double overlap" is 24 so I substitute in
2T = 60 - 24
T = 18
So far this technique has gotten me through all triple overlapping sets |
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To save time when working with proportions, you should cancel factors out of proportions before cross-multiplying. How can you cancel factors with proportions? |
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Definition
You can cancel factors either vertically within a fraction or horizontally across an equals sign in a proportion.
NEVER cancel factors diagonally across an equals sign. |
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When should you use an Unknown Multiplier for complex ratio problems?
When CAN you use it? |
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Definition
You should use an unknown multiplier when neither quantity in the ratio is already equal to a number or a variable expression.
You CAN use it only ONCE per problem. Every other ratio in the problem must be set up with a proportion. |
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| If each term in a ratio is increased by an absolute amount , what happens to the ratio that results? Is it the same or different? |
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Definition
If each term in a ratio is increased by an absolute amount
(rather than, say, being multiplied or divided by a number), it has unpredictable effects on the ratios, and the ratio does not stay the same
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