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ASVAB - Arithmetic Reasoning-
Mathematics Knowledge
John bought a camera on sale that normally costs $160. If the price was reduced 20% during the sale, what was the sale price of the camera? (a) $ (b) $ (c) $ (d) $140 - ANS (c) $ This question asks you to determine the sale price of a camera that normally sells at $160 and is discounted 20%. To solve, determine what 20% of $160 equals. Rewrite 20% as a decimal. 20% = 0.20. So 20% of $160 = 0.20 x $160 = $32. The sale price of the camera would be $160 - $32 = $128, choice (c) A subway car passes 3 stations every 10 minutes. At this rate, how many stations will it pass in one hour? (a) 15 (b) 18 (c) 20 (d) 30 - ANS (b) 18 First, set up the rate as a proportion, where (x) is the number of stations. 3 stations/10 minutes = (x) stations/1 hour Then, convert the units. 3 stations/10 minutes = (x) stations/60 minutes Cross multiply and solve for (x). 180 = 10(x) 18 = (x) On a certain map, 3/4 inch represents one mile. What distance, in miles, is presented by 1 3/4 inches?
(a) 1 1/ (b) 2 1/ (c) 2 1/ (d) 5 1/4 - ANS (b) 2 1/ In this question, the ratio is implied: for every 3/4 inch of map there is 1 real mile, so the ratio of inches to the miles they represent is always 3/4 to 1. Therefore, you can set up the proportion: number of inches/ number of miles = 3/4 / 1 = 3/ Now 1 3/4 inches = 7/4 inches. Set up a proportion: 7/4 inches 7/4 inches / number of miles = 3/ Cross-multiply: 7/4(4) = 3 (number of miles) 7= 3(number of miles) 7/3 = number of miles or 2 1/3 = number of miles A certain box contains baseballs and golf balls. If the ratio of baseballs to golf balls is 2:3 and there are 30 baseballs in the box, how many golf balls are in the box? (a) 18 (b) 20 (c) 36 (d) 45 - ANS (d) 45 You can express the ratio of baseballs to golf balls as 2/3. Since you know the number of baseballs, you can set up a proportion: 2/3 = 30/ (x) where (x) is the number of golf balls. To solve, cross-multiply to get 2(x) = 90, or x = 45. Four people shared a taxi to the airport. The fare was $36.00, and they gave the driver a tip equal to 25% of the fare. If they equally shared the cost of the fare and tip, how much did each person pay? (a) $9. (b) $10. (c) $10.
(a) 90 (b) 94 (c) 95 (d) 96 - ANS (d) 96 Be careful with a question like this one. You're given the percent decrease (25%) and the new number (72), and you're asked to reconstruct the original number. Don't just take 25% of 72 and add it on. That 25% is based not on the new number, 72, but on the original number - the number you're looking for. The best way to do a problem like this is to set up an equation: (original number) - (25% of original number) = new number (x) - 0.25(x) = 72 0.75x = 72 x = 96 Alternatively, you can use the answer choices to determine the correct answer. The original number of jelly beans has to be reducible by 25%, or 1/4. That means the original number of jelly beans has to be a multiple of 4 (or else you'd be reducing by pieces of jelly beans). Only the correct answer, 96, is a multiple of 4. A student finishes the first half of an exam in 2/3 the time it takes him to finish the second half. If the entire exam takes him an hour, how many minutes does he spend on the first half of the exam? (a) 20 (b) s (c) 27 (d) 36 - ANS (b) 24 The time it takes to complete the entire exam is the sum of the time spent on the first half of the exam and the time spent on the second half. The time spent on the first half is 2/3 of the time spent on he second half. If (S) represents the time spent on the second half, then the total time spent is 2/3(S) + (S) or 5/3 (S). You know this total time is one hour, or 60 minutes. Set up a simple equation and solve for (S). 5/3(S) = 60 3/5 x 5/3(S) = 3/5 x 60 (S) = 36 So the second half takes 36 minutes. The first half takes 2/3 of this, or 24 minutes. You could also find the first half by subtracting 36 minutes from the total time, 60 minutes.
A 25 ounce solution is 20% alcohol. If 50 ounces of water are added to it, what percent of the new solution is alcohol? (a) 6 2/3 % (b) 7 1/2 % (c) 10% (d) 13 1/3% - ANS (a) 6 2/3% You're asked what percent of the new solution is alcohol. The (part) is the number of ounces of alcohol; the (whole) is the total number of ounces of the new solution. There were 25 ounces originally. Then 50 ounces were added, so there are 75 ounces of new solution. How many ounces are alcohol? 20% of the original 25 - ounces solution was alcohol. 20% is 1/5, so 1/5 of 25, or 5 ounces are alcohol. Now you can find the percent of alcohol in the new solution: % alcohol = alcohol/total solution x 100% = 5/75 x 100% = 20/3% = 6 2/3% Marty has exactly 5 blue pens, 6 black pens, and 4 red pens in his backpack. If he pulls out one pen at random from his backpack, what is the probability that the pen is either red or black? (a) 2/ (b) 3/ (c) 2/ (d) 1/3 - ANS (a) 2/ To find probability, determine the number of desired outcomes and divide that by the number of possible outcomes. The probability formula looks like this: Probability = # of desired outcomes/# of possible outcomes In this case, Marty is pulling one pen at random from his knapsack, and you want to determine the probability that the pen is either red or black. There are 5 blue pens, 6 black pens, and 4 red pens in the knapsack. Let's return to the probability formula: Probability = # of desired outcomes/ # of possible outcomes = number of red + black pens/number of red+ black + blue pens = 4 + 6/ 4+6+5 = 10/15=2/ From 1980 through 1990, the population of Country X increased by 100%. From 1990 to 2000, he population increased by 50%. What was the continued increase for the period 1980 - 2000? (a) 150%
(d) 300 - ANS (a) 225 The calculations aren't too bad on this one. The most important thing to keep in mind is that you're solving for 75% of (x) and not for (x) itself. First, you are told that 50% of (x) is 150. That means that half of (x) is 150, and that (x) is 300. So 75% of (x) = 0.75 x 300 = 225. The total fare for two adults and three children on an excursion boat is $14. If each child's fare is one half of each adult's fare, what is the adult fare? (a) $2. (b) $3. (c) $3. (d) $4.00 - ANS (d) $4. This question where Backsolving (plugging in an answer choice to see if it's correct) can save you a lot of time. Let's start with choice (b) and see if it works. If (b) is correct, an adult's ticket would cost $3.00, and a child's ticket would cost $1.50. The total fare you're asked for is for two adults and three children. If an adult's fare was $3.00, that total fare would be 2($3.00) + 3($1.50) = $6.00 + $4.50 = $10.50. That's too low since the question states that the total fare is $14.00. Now see what happens if an adult fare was more expensive. If (d) was correct, an adult's ticket would cost $4.00 and a child's ticket would cost $2.00. The total fare would equal 2($4.00) + 3($2.00) = $8.00 + $6.00 = $14.00. That's the total fare you're looking for, so (d) is correct. What is the prime factorization of 140? (a) 2 x 70 (b) 2 x 3 x 5 x 7 (c) 2 x 2 x 5 x 7 (d) 2 x 2 x 2 x 5 x 7 - ANS (c) 2 x 2 x 5 x 7 To find the prime factorization of a number, find one prime that will go into the number (here 2 is a good place to start). Express the number as that prime multiplied by some other number. 140 = 2 x 70 Then keep breaking down the larger factor until you are left with only prime numbers. 140 = 2 x 2 x 35 140 = 2 x 2 x 5 x 7
A painter charges $12 an hour while his son charges $6 an hour. If the father and son worked the same amount of time together on a job, how many hours did each of them work if their combined charge for their labor was $108? (a) 6 (b) 9 (c) 12 (d) 18 - ANS (a) 6 When the painter and his son work together, they charge the sum of their hourly rates, $12 + $6, or $18 per hour. Their bill equals the product of this combined rate and the number of hours they worked, Therefore $108 must equal $18 per hour times the number of hours they worked. Divide $108 by $18 per hour to find the number of hours. $108 / $18 = 6. 4! = __. (a) 4 (b) 16 (c) 24 (d) 256 - ANS (c) 24 The exclamation mark indicates a factorial. A factorial is an integer multiplied by every smaller integer, down to the number 1, like this: 4! = 4 x 3 x 2 x 1 = 24 At garage (A), it cost $8.75 to park a car for the first hour and $1.25 for each additional hour. At garage (B), it costs $5.50 to park a car for the first hour and $2. for each additional hour. What is the difference between the cost of parking a car for 5 hours at garage (A) and parking it for the same length of time at garage (B)? (a) $2. (b) $1. (c) $1. (d) $1.25 - ANS (b) $1. Compute the cost of parking a car for 5 hours at each garage. Since the two garages have a split-rate system of charging, the cost for the first hour is different from the cost of each remaining hour. The first hour at garage (A) costs $8. The next 4 hours cost 4 x $1.25 = $5.
(a) 2, (b) 2, (c) 20, (d) 20,200 - ANS (d) 20, If you change each digit 5 into a 7 in the number 258,546, the new number would be 278,746. The difference between these two numbers would be 278,746 - 258,546 = 20,200. Micheal bought 2 1/4 pounds of lumber at $4.00 per pound. If a 7% sales tax was added, how much did Micheal pay? (a) $9. (b) $9. (c) $10. (d) $11.77 - ANS (a) $9. Since 1 pound of lumber costs $4.00, 2 1/4 pounds of lumber cost 2.25 x $4.00 = $9.00. Then add 7% sales tax to $9.00. Find 7% of $9.00 by multiplying 0.07 x $9. = $0.63. Add $0.63 to $9.00 to get $9.63, choice (a). The ratio of 3 1/4 t0 5 1/4 is equivalent to the ratio of __. (a) 3 to 5 (b) 4 to 7 (c) 8 to 13 (d) 13 to 21 - ANS (d) 13 to 21 The question asks which of five ratios is equivalent to the ratio of 3 1/4 to 5 1/4. Since the ratios in the answer choices are expressed in whole numbers, turn this ratio into whole numbers. Start by turning the ratio into improper fractions: 3 1/4 : 2 1/ = 13/4: 21/ Multiply both sides of the ratio by 4. = 13: A cat is fed 3/8 of a pound of cat food every day. For how many days will 72 pounds of this cat food feed the cat? (a) 160 (b) 172 (c) 180 (d) 192 - ANS (d) 192
Set up the proportion. 3/8 lb / 1 day = 72 lbs / (x) days Cross multiply. 3/8 (x) = 72 (x) = 72 x 8/ (x) = 192 After spending 5/12 of his salary, a man has $420 left. What is his salary? (a) $ (b) $ (c) $ (d) $1,008 - ANS (c) $ You can save valuable time by estimating on this one. Pay special attention to how much you have left and how much you've already spent. If a man spent 5/12 of his salary and was left with $420, that means that he had 7/12 left, and if the man's salary is (x) dollars, then 7/12(x) = $420 That means that $420 is a little more than half of his salary. So his salary would be little less than 2($420) = $840. Choice (c), $720 is a little less than $840. So (c) works perfectly, and it's the correct answer here. A stock decreases in value by 20%. By what percent must the stock price increase to reach its former value? (a) 15% (b) 20% (c) 25% (d) 40% - ANS (c) 25% They key to this question is that while the value of the stock decreases and increases by the same amount, it doesn't decrease and increase by the same percent. When the stock first decreases, the amount of changes is part of a larger whole. If the stock were to increase to its former value, that same amount of change would be a larger percent of a smaller whole. Pick a number for the original value of the stock, such as $100. (Since it's easy to take percents of 100, it's usually best to choose 100.) The 20% decrease represents $20, so the stock decreases to a value of $80. Now in order for the stock to reach the value of $100 again, there must be a $20 increase. What percent of $80 is $20? It's $20/$80 x 100%, or 1/4 x 100%, or 25%.
(c) 35 (d) 36 - ANS A) 31 x + x + 8 = 70, solve for x. x = 31 A sales manager buys antacid in bottles the gross. If he goes through 3 bottles antacid everyday, how long will the gross last? (a) 144 days (b) 3 days (c) 20 days (d) 48 days - ANS D) 144 days 144 bottles in a gross, and 144 / 3 (bottles per day) = 48 days Jenny's test grades are 93,89, 96, and 98. If she wishes to raise her average to 95, what does she need to score on her next test? (a) 100 (b) 99 (c) 97 (d) 95 - ANS B) 99 x = the unknown score. (93 + 89 + 96 + 98 + x) / 5 or (376 + x) / 5. Now solve for x. x= 99 How many square feet of carpeting are needed to carpet a 12 - foot × 12 - foot room? (a) 24 (b) 120 (c) 48 (d) 144 - ANS D) 144 To determine sq. footage, multiply length by width. 12 x 12 = 144 Joe received an hourly wage of $8.15. His boss gave him a 7% raise. How much does Joe make per hour now? (a) $0. (b) $8. (c) $8. (d) $13.85 - ANS C) $8. To calculate the new wage first multiply $8.15 x 0.07 = $0.57. Then add that to the original wage, $8.15 + $0.57 = $8.
Alice leaves her house, driving east at 45 miles per hour (mph). Thirty minutes later her husband Dave notices she forgot her cell phone and sets off after her. How fast must Dave travel in order to catch up with Alice 3 hours after he leaves? (a) 49 mph (b) 50.5 mph (c) 52.5 mph (d) 54 mph - ANS C) 52.5 mph Distance = speed x time. First, find how far Alice had traveled 31/2 hours at 45 mph, or 157.5 mph. Dave has three hours to cover this distance. Now find his speed. 157.5 mi / 3 hr = 52.5 mph. Miriam bought five cases of motor oil on sale. A case of motor oil normally costs $24.00, but she was able to purchase the oil for $22.50 a case. How much money did Miriam save on her entire purchase? (a) $7. (b) $1. (c) $8. (d) $22.50 - ANS A) $7. Subtract the sale price from the original price, $24.00 - $22.50 = $1.50. Now, multiply that number by the amunt of cases to get the final answer. $1.50 x 5 = $7.50. A security guard walks the equivalent of six city blocks when he makes a circuit around the building. If he walks at a pace of eight city blocks every 30 minutes, how long will it take him to complete a circuit around the building, assuming he doesn't run into any thieves? (a) 20.00 minutes (b) 3.75 minutes (c) 22.50 minutes (d) 7.5 minutes - ANS C) 22.50 minutes Divide 30 by 8 to determine how long it takes to walk one city block = 3.75 minutes. Then, multiply by 6, the number it takes to complete the circuit. The population of Grand Island, Nebraska, grew by 600,000 people between 1995 and 2005, one-fifth more than the town council predicted. The town originally predicted the city;s population to grow by (a) 400, (b) 500, (c) 300, (d) 100,000 - ANS B) 500, Let x = the original number predicted. An ¹/₅ would make the population growth ⁶/₅ or 120%, of x. Express the equation as 1.2x = 600,000, solve for x. x = 500,000.
MIchael needs 55 gallons if paint to paint an apartment building. He would like to purchase the paint for the least amount money possible. Which of the following should he buy? (a) two 25 - gallon buckets at $550 each (b) eleven 5 - gallon buckets at $108 each (c) six at 10 - gallon buckets at $215 each (d) fifty-five 1 - gallon bucket at $23 each - ANS B) Eleven 5 - gallon buckets at $ each Choice (a) doesn't have enough paint (2 ×25 gal = 50 gal) so it's wrong. Choice (b): 11 × $108 = $1, Choice (c): 6 × $215 = $1, Choice (d): 55 × $23 = $ As a member of FEMA, you're required to set up a contingency plan to supply meals to residents of a town devastated by a tornado. A breakfast ration weights 12 ounces and the lunch and dinner rations weight 18 ounces each. Assuming a food truck can carry 3 tons and that each resident will receive 3 meals per day, how many residents can you feed from one truck during a 10 - day period? (a) 150 residents (b) 200 residents (c) 250 residents (d) 300 residents - ANS B) 200 Residents First find how many ounces each truck can hold. 2,000lb. are in a ton, so each truck can carry three times that 6,000lb. There are 16 ounces to a pound so one truck can carry 96,000 ounces ( 16oz. × 6,000lb.). Now find the daily total for each persons three meals 12oz + 18oz + 18oz = 48oz. Now 96,000 × 48 = 2,000 daily rations. This rations need to last 10 days (2,000 ÷ 10 = 200) So 200 residents can be fed by one truck for 10 days. A train headed south for Wichita left the station at the same time a train headed north for Des Moines left the same station. The train headed to Wichita traveled 55 miles per hour. The train headed for Des Moines traveled at 70 miles per hour. How many miles apart are the trains at the end of three hours? (a) 210 miles (b) 165 miles (c) 125 miles (d) 375 miles - ANS D) 375 miles D. Distance = speed × time, take the distance traveled by the Wichita train (55mph × 3hr = 165 miles) then add that to the distance traveled by the Des Moines train (70mph × 3hr = 210 miles). The trains are 375 miles apart in 3 hours. Another option: add the two trains mph then multiply by 3 hours (55 + 70) = 125 × 3 = 375 miles
A carpenter needs to cut four sections, each 3 feet, 8 inches long, from a piece of molding. If the board is only sold by the foot, what's the shortest length of board she can buy? (a) 15 feet (b) 14 feet (c) 16 feet (d) 12 feet - ANS A) 15 feet Each section is 3²/₃ feet, times that by four and you get 14 ²/₃ so 15 is the shortest board possible. Kiya had only one coupon for 10% off one frozen turkey breast. The turkey breast costs $8.50 each, and Kiya wanted to buy two. How much did she pay? (a) $16. (b) $17. (c) $15. (d) $7.65 - ANS A) $16. First find the discounted turkey's price ($8.50 × .10 = $0.85, $8.50 - $0.85 = $7.35) then add another $8.50 because she needs two turkey breast. $7.35 + $8.50 = $16. Your car uses gasoline at a rate of 21 miles per gallon. If gasoline costs $2.82 per gallon and you drive for 7 hours at a speed of 48 miles per hour, how much will you pay for gasoline for the trip? (a) $38. (b) $45. (c) $47. (d) $59.27 - ANS B) $45. First determine the miles traveled d = st (7hr × 48mph = 336 miles), then divide by the rate per gallon, 21. So now you know you need 16 gallons at $2.82 per gallon to pay for the trip. 16 × $2.82 = $45. If you roll two six-sided dice, what's the probability of not rolling a five on either die? (a) ¹/₃₆ (b) ¹/₆ (c) ⁴/₃₆ (d) ²⁵/₃₆ - ANS D) 25/ The probability of rolling a 5 is a 1 out of 6 chance, so the probability of NOT rolling a 5 is 5 out of 6. With two die the probability is ⁵/₆ × ⁵/₆ = ²⁵/₃₆ Jack loaned Bob $1,500 at an annual interest rate of 7%. After one year, how much will Bob owe Jack?
An aircraft flies over Boondock Air Force Base at 10:20 a.m. At 10:32 a.m., the plane passes over Sea Side Naval Air Station, 120 miles away. How fast is the aircraft traveling? (a) 400 mph (b) 500 mph (c) 600 mph (d) 700 mph - ANS C) 600 mph The plane traveled 120 miles in 12 minutes or ¹/₅ of an hour. d=st, 120 = ¹/₅x or 120 × 5 = 600 mph What is the reciprocal of 1/2? - ANS 2/ Because 1/2 times 2/1 = 1 Solve for x: If the scale on a road map is 1 inch = 250 miles, how many inches would represent 1,250 miles? - ANS x = 5, 5 inches = 1,250 miles. 1 inch = 250 miles, x = 1250 miles. The problem can be expressed as two ratios set equal to each other, known as proportion: 1/250 = x/ I = Prt Where I = the amount of interest, P = the initial amount invested, r = interest rate, and t = the length of time the money is invested. This is called the _ _ formula. - ANS Simple interest What is the square root of 54? - ANS 7.3 (to the nearest tenth) You know 7x7 = 49, 8x8= 64 Closer to 49, so try 7.3 x 7.3 = 53.29 ... 7.4 x 7.4 = 54. 53.29 is closer Arcs of a circle and angles are measured in degrees and in minutes or even seconds. 1 degree =? - ANS 60 seconds 60 mins .. 1 min = 60 sec.
Complementary angles are two angles that equal _ degrees when added together. - ANS 90 Supplementary angles are two angles that equal _ degrees when added together - ANS 180 Quadrilaterals are shapes with four sides whose angles total _ degrees. - ANS 360 A circle always is _ degrees - ANS 360 Which of the following fractions is the largest? 2/3 , 5/8, 11/16 or 3/4? - ANS 3/ Find a common denominator for the fractions - which is 48 2/3 > 32/48, 5/8 > 30/48, 11/16 > 33/48, 3/4 > 36/ The largest numerator is the largest fractions, which is 3/4. Express 403,000,000,000,000 in scientific notation. - ANS 4.03 x 10^ Solve: A baker has S lbs of sugar to use in baking. After she uses 50 lbs to make donuts, how much sugar does she have left? A) s + 50 B) 50 - 2 C) s - 50 D) s / 50 - ANS C) S - 50 Six pizzas are pepperoni, seven are hamburger, four are cheese, and three are "with everything". What's the probability that a randomly selected pizza is pepperoni?
- ANS 1/4. There are 6 + 7 + 4 + 3 = 20 pizzas total. So the probability is 6/20, which is 1/4. an _ is a positive, negative, or zero whole number. - ANS Integer _ are whole numbers - ANS Factors _ is limiting a number to fewer decimal places - ANS Rounding Sum also means: - ANS Addition Difference also means: - ANS Subtraction
