Ratios, Rates, and Proportions - Quantitative Reasoning

1. A laboratory solution contains salt and water in the ratio $3:97$ by mass. If the solution has a total mass of $1,000$ grams, what is the mass of salt, in grams?
   Answer: 30 This question tests proportions in solutions. The ratio of salt to water is 3:97, total parts 100. For 1,000 grams total, salt is (3/100) × 1,000 = 30 grams. This calculates the mass fraction. Choice A is justified as correct. A distractor like 97 could be misreading the ratio. Adding masses directly might give 300.
2. A machine produces $180$ bolts in $12$ minutes at a constant rate. At the same rate, how many bolts does it produce in $50$ minutes?
   Answer: 750 This question tests rates of production. The machine produces 180 bolts in 12 minutes, giving a rate of 180 / 12 = 15 bolts per minute. For 50 minutes, the total is 15 × 50 = 750 bolts. This applies the constant rate proportionally over time. Choice A is justified as correct. A distractor like 720 might come from using hours instead of minutes. Incorrectly inverting the rate could lead to choices like 150.
3. A runner uses energy at a constant rate of $95$ calories per mile. Approximately how many calories does the runner use over $6.4$ miles?
   Answer: 608 This question tests rates of energy use. The runner uses 95 calories per mile, a constant rate. For 6.4 miles, total is 95 × 6.4 = 608 calories. This multiplies rate by distance. Choice A is justified as correct. A distractor like 152 might come from dividing instead. Using a rounded distance could lead to 589.
4. A store sells trail mix made by combining nuts and dried fruit in the ratio $5:3$ by weight. If a bag contains $24$ ounces of trail mix, how many ounces are dried fruit?
   Answer: 9 This question tests proportions in mixtures. The ratio of nuts to dried fruit is 5:3, total parts 8. For 24 ounces total, dried fruit is (3/8) × 24 = 9 ounces. This finds the portion correctly. Choice B is justified as the answer. A distractor like 15 could be the nuts portion instead. Reversing to 3:5 would give 7.2 ounces fruit.
5. A cyclist travels $18$ kilometers in $45$ minutes at a constant speed. What is the cyclist's speed, in kilometers per hour?
   Answer: 24 km/h This question tests rates for speed. The cyclist travels 18 km in 45 minutes, which is 0.75 hours, so speed is 18 / 0.75 = 24 km/h. This converts time and divides distance by time correctly. The proportional relationship gives the constant rate. Choice A is justified as the answer. A distractor like 40 might arise from using 45 minutes as 0.45 hours. Forgetting the conversion could lead to 0.4 km/h.
6. In a jar of marbles, the ratio of red marbles to blue marbles is $3:5$. If there are $40$ blue marbles, how many red marbles are there?
   Answer: 24 This question tests ratios of quantities. The ratio of red to blue marbles is 3:5, meaning for every 3 red, there are 5 blue. With 40 blue marbles corresponding to 5 parts, each part is 40 / 5 = 8 marbles. Thus, red marbles are 3 × 8 = 24. This justifies choice A as correct. A distractor like 15 might result from reversing the ratio to 5:3. Another error could be adding the parts incorrectly, leading to disproportionate calculations.
7. A recipe calls for the ratio of flour to sugar to be $5:2$ by weight. If $300$ grams of flour are used, how many grams of sugar are needed?
   Answer: 120 This question tests ratios in recipes. The ratio of flour to sugar is 5:2, indicating 5 parts flour to 2 parts sugar. With 300 grams of flour as 5 parts, each part is 300 / 5 = 60 grams. Sugar needed is then 2 × 60 = 120 grams. Choice A is justified as the correct amount. A distractor like 750 could arise from multiplying instead of dividing the parts. Reversing the ratio to 2:5 would lead to incorrect weights like 150 grams.
8. A rectangular photograph is enlarged so that each linear dimension is multiplied by $1.5$. If the original width is $8$ inches, what is the new width, in inches?
   Answer: 12 This question tests proportions in scaling. The enlargement multiplies each dimension by 1.5, a proportional increase. Original width 8 inches becomes 8 × 1.5 = 12 inches. This applies the scale factor directly. Choice C is justified as correct. A distractor like 9.5 might come from adding instead of multiplying. Using a different factor like 1.2 could lead to 9.6, close to distractors.
9. A map uses a scale of $1$ inch to represent $25$ miles. Two cities are $3.6$ inches apart on the map. What is the actual distance between the cities, in miles?
   Answer: 90 This question tests proportions in map scales. The scale is 1 inch represents 25 miles, setting up a direct proportion between map distance and actual distance. For 3.6 inches on the map, the actual distance is 3.6 × 25 = 90 miles. This applies the scale factor correctly to find the real-world measurement. Choice A is justified as the correct answer. A distractor like 144 could come from squaring the scale incorrectly. Reversing the proportion, such as dividing instead of multiplying, might lead to choices like 0.144.
10. A paint mixture is made by combining blue paint and white paint in the ratio $4:1$ by volume. If $3$ liters of white paint are used, what is the total volume of the mixture, in liters?
    Answer: 15 This question tests ratios in mixtures. The ratio of blue to white paint is 4:1, so total parts are 5. With 3 liters white as 1 part, blue is 4 × 3 = 12 liters, total 15 liters. This scales the ratio to find the whole. Choice B is justified as correct. A distractor like 12 might be the blue amount only. Reversing to 1:4 would give a smaller total like 3.75.
11. A solution is made by mixing acid and water in the ratio $1:9$ by volume. If the total volume of the solution is $500$ milliliters, what volume of acid is in the solution?
    Answer: 50 mL This question tests proportions in mixtures. The ratio of acid to water is 1:9, so acid is 1 part out of total 10 parts. With 500 mL total, acid is (1/10) × 500 = 50 mL. This calculation finds the component volume correctly. Choice A is justified as the answer. A distractor like 450 could be from taking the water portion instead. Reversing the ratio to 9:1 would yield 450 mL acid, a common error.
12. A tank is filled by a pipe at a constant rate of $12$ gallons per minute. How many gallons are added in $25$ seconds?
    Answer: 5 This question tests rates of flow. The pipe fills at 12 gallons per minute, a constant rate. In 25 seconds, or 25/60 = 5/12 minutes, gallons added are 12 × (5/12) = 5. This converts time and applies the rate. Choice A is justified as correct. A distractor like 300 could be from ignoring the time conversion. Using seconds directly might yield small values like 0.48.
13. A printer uses ink at a constant rate of $2.5$ milliliters per page. How many milliliters of ink are used to print $36$ pages?
    Answer: 90 mL This question tests rates of consumption. The printer uses 2.5 mL per page, establishing a direct proportion of ink to pages. For 36 pages, ink used is 2.5 × 36 = 90 mL. This multiplies the rate by the quantity accurately. Choice B is justified as correct. A distractor like 72 might result from using 2 mL per page instead. Dividing instead of multiplying could lead to small values like 14.4.
14. The ratio of students who study French to students who study Spanish in a school is $7:9$. If $144$ students study Spanish, how many students study French?
    Answer: 112 This question tests ratios of groups. The ratio of French to Spanish students is 7:9, meaning 7 French per 9 Spanish. With 144 Spanish as 9 parts, each part is 144 / 9 = 16 students. French students are 7 × 16 = 112. Choice A is justified as correct. A distractor like 81 could come from reversing to 9:7. Misadding parts might lead to incorrect multiples like 63.
15. A vehicle's fuel efficiency is $32$ miles per gallon. If the vehicle travels $240$ miles, how many gallons of fuel does it use?
    Answer: 7.5 This question tests rates in fuel efficiency. The vehicle gets 32 miles per gallon, inversely relating distance to fuel. For 240 miles, fuel used is 240 / 32 = 7.5 gallons. This divides distance by rate. Choice A is justified as correct. A distractor like 272 could be from multiplying instead. Reversing to gallons per mile might give 0.133.
16. A bill is split among three coworkers, Alex, Bri, and Chen, in the ratio $2:3:5$. If Chen pays $45$ dollars, how much is the total bill, in dollars?
    Answer: 90 This question tests ratios in splitting costs. The ratio among Alex, Bri, and Chen is 2:3:5, total parts 10. Chen's 45 dollars is 5 parts, so each part is 45 / 5 = 9 dollars, total bill 10 × 9 = 90 dollars. This scales the ratio to the total. Choice A is justified as correct. A distractor like 75 might come from incorrect part division. Reversing ratios could lead to 225.

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