Quantitative Reasoning - Armed Services Vocational Aptitude Battery ASVAB

1. Two jets leave from the same airport at the same time, flying in opposite directions. One jet travels at 400 mph, and the other travels at 500 mph. How many miles apart will they be after 2.5 hours?
   Answer: 2,250 miles Because the jets are flying in opposite directions, their speeds combine. The combined speed is $400 \text{ mph} + 500 \text{ mph} = 900$ mph. To find the distance apart, multiply the combined speed by the time: $900 \text{ mph} \times 2.5 \text{ hours} = 2,250$ miles.
2. A recipe calls for 2 cups of flour for every 3 eggs. If a baker wants to use 12 eggs, how many cups of flour will be needed?
   Answer: 8 cups The ratio of flour to eggs is 2:3. The baker is using 12 eggs, which is $12 \div 3 = 4$ times the amount in the recipe. Therefore, the amount of flour needed is also 4 times the original amount: $2 \text{ cups} \times 4 = 8$ cups.
3. A restaurant bill is $72. A customer wants to leave an 18% tip. The customer pays with a $100 bill. How much change should they receive?
   Answer: $15.04 First, calculate the tip amount: $18\%$ of ($72). $\$72 \times 0.18 = \$12.96$. Next, add the tip to the bill to find the total cost: $\$72 + \$12.96 = \$84.96$. Finally, subtract the total cost from the amount paid to find the change: $\$100 - \$84.96 = \$15.04$.
4. A soldier can run 2 miles in 16 minutes. At this same rate, how long would it take him to run 5 miles?
   Answer: 40 minutes First, find the soldier's rate per mile. If he runs 2 miles in 16 minutes, his rate is $16 \div 2 = 8$ minutes per mile. To find the time for 5 miles, multiply this rate by the distance: $8 \text{ minutes/mile} \times 5 \text{ miles} = 40$ minutes.
5. A car's fuel tank holds 18 gallons and was completely full. After driving 288 miles, the tank is now empty. How many miles per gallon did the car get?
   Answer: 16 To find the miles per gallon (MPG), divide the total miles driven by the number of gallons used. $288 \text{ miles} \div 18 \text{ gallons} = 16$ MPG.
6. During a physical fitness test, a soldier completed 50 push-ups. This was 125% of the minimum requirement. What is the minimum number of push-ups required?
   Answer: 40 Let 'x' be the minimum requirement. The problem states that $1.25 \times x = 50$. To solve for x, divide both sides by 1.25: $x = 50 \div 1.25$. Since $1.25 = \frac{5}{4}$, the equation is $x = 50 \div \frac{5}{4} = 50 \times \frac{4}{5}$. This simplifies to $10 \times 4 = 40$.
7. A piece of equipment depreciates in value by 10% each year. If the equipment was purchased for $3,000, what is its value after two years?
   Answer: $2,430 After the first year, the value depreciates by 10% of $3,000, which is $300. The new value is $\$3,000 - \$300 = \$2,700$. After the second year, the value depreciates by 10% of the new value: 10% of $2,700 is $270. The final value is $\$2,700 - \$270 = \$2,430$.
8. An electronics store offers a 15% discount on a television that is regularly priced at $600. How much is the sale price of the television?
   Answer: $510 First, calculate the discount amount: 15% of $600. $\$600 \times 0.15 = \$90$. Then subtract the discount from the original price: $\$600 - \$90 = \$510$.
9. A box contains red and blue marbles. The ratio of red marbles to blue marbles is 4:7. If there are 28 red marbles, what is the total number of marbles in the box?
   Answer: 77 The ratio of red to blue is 4:7. If there are 28 red marbles, this corresponds to the '4' part of the ratio. The multiplier is $28 \div 4 = 7$. To find the number of blue marbles, multiply the '7' part of the ratio by this multiplier: $7 \times 7 = 49$ blue marbles. The total number of marbles is the sum of red and blue: $28 + 49 = 77$.
10. A military unit has 320 members. The ratio of officers to enlisted personnel is 1:15. How many officers are in the unit?
    Answer: 20 The ratio 1:15 means there are $1+15=16$ parts in total. To find the value of one part, divide the total number of members by the total number of parts: $320 \div 16 = 20$. Since officers represent 1 part, there are $1 \times 20 = 20$ officers.
11. A company produces defective items at a rate of 3 for every 500 produced. If the company produces 4,000 items in a day, how many can be expected to be defective?
    Answer: 24 Set up a proportion: $\frac{3 \text{ defective}}{500 \text{ total}} = \frac{x \text{ defective}}{4000 \text{ total}}$. To solve for x, you can see that 4000 is 8 times 500 (since $4000 \div 500 = 8$). Therefore, the number of defective items will also be 8 times greater: $3 \times 8 = 24$.
12. A map has a scale where 1 inch represents 40 miles. If two cities are 5.5 inches apart on the map, what is the actual distance between them in miles?
    Answer: 220 miles To find the actual distance, multiply the map distance by the scale factor. $5.5 \text{ inches} \times 40 \text{ miles/inch}$. This is $5 \times 40 + 0.5 \times 40 = 200 + 20 = 220$ miles.
13. A concrete mixture requires cement, sand, and gravel in a ratio of 1:3:5 by volume. If a builder uses 15 cubic feet of sand, how many cubic feet of gravel are needed?
    Answer: 25 The ratio of sand to gravel is 3:5. Set up a proportion: $\frac{3}{5} = \frac{15}{x}$, where x is the amount of gravel. Cross-multiply to solve for x: $3x = 5 \times 15$, so $3x = 75$. Divide by 3: $x = 25$. Therefore, 25 cubic feet of gravel are needed.
14. Two workers can assemble a machine in 10 hours. If one worker can do it alone in 15 hours, how long would it take the second worker to do it alone?
    Answer: 30 hours This is a work-rate problem. Let the rate of the two workers together be $\frac{1}{10}$ machine/hour. The rate of the first worker is $\frac{1}{15}$ machine/hour. Let the rate of the second worker be $\frac{1}{T}$. The equation is $\frac{1}{15} + \frac{1}{T} = \frac{1}{10}$. To solve for T, first find a common denominator (30): $\frac{2}{30} + \frac{1}{T} = \frac{3}{30}$. So, $\frac{1}{T} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30}$. Therefore, T = 30 hours.
15. An investment of $2,000 earns simple interest at a rate of 6% per year. How much total interest will have been earned after 5 years?
    Answer: $600 Simple interest is calculated using the formula I = Prt (Interest = Principal x Rate x Time). The interest earned in one year is $\$2,000 \times 0.06 = \$120$. Over 5 years, the total interest is $\$120 \times 5 = \$600$.
16. A train travels 210 miles in 3.5 hours. What is its average speed in miles per hour (mph)?
    Answer: 60 mph Speed is calculated by dividing distance by time (Speed = Distance / Time). So, $210 \text{ miles} \div 3.5 \text{ hours}$. To simplify, you can treat 3.5 as $\frac{7}{2}$. The calculation becomes $210 \div \frac{7}{2} = 210 \times \frac{2}{7}$. Since $210 \div 7 = 30$, the result is $30 \times 2 = 60$ mph.
17. A pump fills a tank at a rate of 40 gallons per minute. The tank has a capacity of 4,800 gallons. How many hours will it take to fill the tank?
    Answer: 2 hours First, find the total time in minutes by dividing the tank's capacity by the pump's rate: $4,800 \text{ gallons} \div 40 \text{ gallons/minute} = 120$ minutes. Then, convert minutes to hours by dividing by 60: $120 \text{ minutes} \div 60 \text{ minutes/hour} = 2$ hours.
18. A salesperson earns a 5% commission on all sales. If they sell a vehicle for $24,000, what is the amount of their commission?
    Answer: $1,200 To find the commission, calculate 5% of $24,000. Convert 5% to a decimal, which is 0.05. Then multiply: $\$24,000 \times 0.05 = \$1,200$.
19. On a 60-question test, a student answered 51 questions correctly. What percentage of the questions did the student answer correctly?
    Answer: 85% To find the percentage, divide the number of correct answers by the total number of questions and multiply by 100. $\frac{51}{60} \times 100$. Simplify the fraction $\frac{51}{60}$ by dividing both numerator and denominator by 3, which gives $\frac{17}{20}$. Then, $\frac{17}{20} = 0.85$, and $0.85 \times 100 = 85$%.
20. A recruit's monthly salary was $1,800. After a year, he received an 8% raise. What is his new monthly salary?
    Answer: $1,944 First, calculate the amount of the raise, which is 8% of $1,800. $\$1,800 \times 0.08 = \$144$. Then, add the raise to the original salary: $\$1,800 + \$144 = \$1,944$.

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