Mixtures Quiz

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.

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.

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.

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 \(0.9[latex], so twelve percent is [latex]9 + 2(0.9) = 10.8\)