By volume, cranberry juice makes up 12.5 percent of Bee’s punch and 25 percent of Flo’s punch. If 3 liters of Bee’s punch are mixed with 6 liters of Flo’s punch, approximately what percent of the mixture, by volume, is cranberry juice?
Indicate \(\underline{all}\) such ratios.
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Solution
Estimate to solve this one.The answer must be between 12.5 percent and 25 percent because you are mixing two punches with these percentages—eliminate all choices except choices (B) and (C).Because there is more of Flo’s punch, the answer must be closer to 25 percent than to 12.5 percent, so choice (C) is correct.
36a = 25b
ab ≠ 0
| Quantity A | Quantity B |
| 5⁄6 | a⁄b |
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Solution
Divide both sides by b, then divide both sides by 36 to find:\(\frac{a}{b}=\frac{25}{36}\). Although \(\frac{5}{6}\times \frac{5}{6}=\frac{25}{36}\), this does not mean that \(\frac{25}{36}\) reduces to 5⁄6. Use the Bowtie to compare the two fractions: 36 times 5 equals 180, while 6 times 25 equals 150. 180 is bigger, so Quantity A is greater.
Keri, Neill, and Rich use toilet paper in their apartment in a ratio of 3:2:2. Rich buys two cases of toilet paper online for everyone’s use at 28 rolls per case, an average of $3.50/roll and a $14 delivery charge. If they each contribute to the cost of the toilet paper in direct proportion to the amount they use, how much mustKeri contribute?
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Solution
90
The three roommates spend a total of $210 on toilet paper. If you make your ratio box you will see that your ratio total is 7 and your actual total is 210, so your multiplier is 30.Keri therefore needs to contribute $90.00 to the cost of the toilet paper.
b is a multiple of positive integer a.
| Quantity A | Quantity B |
| The ratio of a to b | 1⁄2 |
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Solution
Because you have variables, make your set-up on your scratch paper. Plug in different values for a and b. First, try 2 and 4:2⁄4 reduce to 1⁄2, making the quantities equal.Eliminate choices (A) and (B). Next, try 2 and 6.2⁄6 reduces to 1⁄3, which is less than 1⁄2. Quantity B is now greater, so eliminate choice (C).You’re left with choice (D).
James can swim 750 yards in 10 minutes. If he swims at the same constant rate, how many minutes will it take him to swim 4.2 times this distance?
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Solution
42
To swim 4.2 times the original distance, James needs to swim for 4.2 times the original time. Multiply the original time of 10 minutes by 4.2 to get 42, the correct answer.
a=1⁄6 and 6⁄7=5⁄b, what is the value of a + b ?
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Solution
Cross-multiply to find the value of b,\(\frac{35}{6}\).Then substitute in the values of a and b:\(a+b=\frac{1}{6}+\frac{35}{6}=\frac{36}{6}=6\)
If a certain vitamin pill has 400 milligrams of magnesium, then how many grams of magnesium are in a bottle of 500 vitamin pills?
(1 gram = 1,000 milligram)
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Solution
First, find that 400 milligrams × 500 pills = 200,000 milligrams total.Then, convert to grams by dividing by 1,000 to find the answer: 200 grams. When doing multiple conversions, be sure to label carefully and watch for arithmetic errors.
CHARITABLE ANNUAL DONATIONS TO CHARITY GROUP X
What is the approximate ratio of Company F’s average annual donation to charity group X for the period 1980–1990 to that for the period 1980–2000?
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Solution
Read the chart carefully and then ballpark.Company F’s annual average donation was about 150 for 1980–1990 and about 100 for 1980–2000.Reduce 150:100 to 3:2.
A certain recipe calls for 2 cups of sugar and 31⁄2 cups of flour. What is the ratio of sugar to flour in this recipe?
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Solution
A ratio is a part-to-part relationship, but it can be expressed and manipulated just like a fraction—in this case,\(\frac{3}{3\frac{1}{2}}\). None of the answers have a fractional value in the denominator, so you need to find a multiplier that will get rid of the fraction. In this case, just doubling the entire ratio will do the trick: \(\frac{3}{3\frac{1}{2}}\times \frac{2}{2}=\frac{4}{7}\).
Evangeline must spend 3⁄8 of her weekly salary on rent and 1⁄6 of her remaining salary on food. Which of the following could be percentages of her weekly salary that Evangeline devotes to entertainment, while still enabling her to place 4⁄9 of her salary into a savings account?
Indicate \(\underline{all}\) such percentages
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Solution
A,B, and C
The problem has percents in the answers and no real values in the question, so it’s a Hidden Plug In; the hidden variable in this case is Evangeline’s weekly salary.To find an easy number, try multiplying the denominators in the problem: 8 × 6 × 9 = 432, so Evangeline makes $432 per week. Now work the problems in bite-sized pieces. She spends 3⁄8 of $432, or $162, on rent, leaving her with $432 − $162 = $270. Of that, she spends 1⁄6, or $45, on food, leaving her with $270 − $45 = $225. If she wants to put 4⁄9 of her weekly salary, or $192, into a savings account, then she can only spend $225 − $192 = $33 on entertainment. Finally, use your on-screen calculator to determine that 33 is just over 7.6% of 432, so choices (A), (B), and (C) will all work.