## Mixtures (Weighted Averages)

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Question 1 |

At a charity fundraiser each male attendant donated $1,200 and each female attendant donated $800. If the average donation at the fundraiser was $900, what was the ratio of the number of male attendants to the number of female attendants?

A | 1:3 |

B | 1:2 |

C | 1:1 |

D | 2:1 |

E | 3:1 |

Question 1 Explanation:

Because the overall average donation is closer the average donation of a woman, there must be more women than men. Thus, we can eliminate C, D, and E. At this point you can use the teeter totter method or you can just do the calculation:
Let \(M\) be the number of males and \(F\) be the number of females.
\(\frac{1200M + 800F}{M+F} = 900\).
Multiply both sides by \(M + F\) to get \(1200M + 800F = 900M + 900F\).
Rearrange terms to get \(300M = 100F\).
Divide by \(F\) and \(300\) on both sides to get \(\frac{M}{F} = \frac{100}{300} = \frac{1}{3}\).
The correct answer is A.

Question 2 |

The math department at College University has only graduate students and undergraduate students. The average undergraduate student has been studying math for 12 years, while the average graduate student has been studying math for 18 years. If the average student in the math department of College University has been studying math for 14 years, and there are 120 students in the math department, how many of them are undergraduate students?

A | 40 |

B | 50 |

C | 60 |

D | 70 |

E | 80 |

Question 2 Explanation:

Because the overall average is closer to the average for undergraduates, there must be more undergraduates. Thus, we can eliminate answers A, B, and C automatically. Now you can use the teeter-totter method or do the direct calculation.
Let \(U\) be the number of undergraduates and \(G\) be the number of graduates. First, note that because there are \(120\) students, \(U + G = 120\), so \(G = 120 - U\). Now, \({12U + 18(120 - U)}{120} = 14\). Multiply both sides by \(120\) to get \(12U + 120(18) - 18U = 120(14)\). Leave things in factored for so you don't calculate until you have to: \(-6U = 120(14) - 120(18) = 120(-4)\). Divide by \(-6\) to get your final answer:
\(U = {120(-4)}{-6} = -20(-4) = 80
The correct answer is E.

Question 3 |

The average height of the men at a party is 72 inches, and the average height of the women at the party is 64 inches. If the average height of all the people at the party is 70 inches, how many people could be at the party

A | 110 |

B | 111 |

C | 112 |

D | 113 |

E | 114 |

Question 3 Explanation:

Let [latex]M\) represent the number of men and \(W\) represent the number of women.
\( \frac{72M + 64W}{M + W} = 70\). \(72M + 64W = 70M + 70W\). \(2M = 6W\). So the ratio of men to women must \(1 : 3\). For example, there could be one man and three women, or ten men and thirty women. If we use a multiplier in order to algebraically represent all possibilities we can see that the total number of people must be a multiple of 4. Using the divisibility rule for 4 we can quickly identify the correct answer: C.

Question 4 |

Naomi has 30 liters of a 5 percent sulfuric acid solution. She want to mix all of it with a 15 percent sulfuric acid solution to create a 12 percent solution. How many liters of the 15 percent solution does she need?

A | 30 |

B | 40 |

C | 50 |

D | 60 |

E | 70 |

Question 4 Explanation:

Clearly she needs more of the \(15\) percent solution as the overall average is closer to \(15\) than it is to \(5\). So we can definitely eliminate A, and it's also a safe bet to eliminate B. Let's check C: \(\frac{60(0.15) + 30(0.05)}{60 + 30}\). Looks ugly, right? It's not bad at all if we use benchmark percents. Fifteen percent is just ten percent plus five percent, and five percent is just half of ten percent. So, if you can move decimal places and divide by two, you're all set. \(\frac{9 + 1.5}{90}\). This is less than \(, so we need more [latex]15\) percent acid*. The answer must be E.
*How do we know without direct calculation that \(\frac{90 + 1.5}{90}\) is less than \(0.12\)? Work backwards: what is \(12\) percent of \(90\)? Ten percent is \(9\), one percent is \(0.9[latex], so twelve percent is [latex]9 + 2(0.9) = 10.8\)

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