ASTB math practice

An island is defended by a battery of coastal guns placed at the easternmost point of the island and having a maximum range of 5 miles. A ship, sailing due north at 24 mph along a course that will bring it within 4 miles of these guns, is approaching the position where it will be 5 miles from these guns. Assuming that the ship will maintain a straight course and the same speed, for approximately how long will the ship be within range of the coastal guns? a. 15 minutes b. 20 minutes c. 25 minutes d. 30 minutes

a. 15 minutes The time for the ship to travel from point A to point C must be determined. Let x = the distance from A to B; 2x = the distance from A to C. 4²+x²=5² 16+x²=25 x²=9 x=3 miles 2x=2(3)=6 miles Let t = time required to travel from A to C (in minutes). 24/60=6/t t=15 minutes *24/60=0.4 *6/15=0.4

A cash box contains a certain number of coins, of which 63 are dimes and 33 are nickels. The rest of the coins in the box are quarters. If the probability of selecting a quarter from this bank is one in five, how many quarters does the bank contain? a. 24 b. 27 c. 30 d. 35

a. 24 Let x = the number of quarters in the cash box (the numerator of the formula's fraction), and let x + 96 = the total number of coins (the fraction's denominator). Then solve for x: 1/5=x/x+96 5x=x+96 -x -x 4x=96 /4 /4 x=24

Angle E is 40° smaller than its complement. The number of degrees in angle E is a. 25° b. 45° c. 60° d. 90°

a. 25° Let x equal the number of degrees in angle E. x+x+40=90 2x=90-40 2x=50 x=25

Successive discounts of 20 percent and 10 percent are equivalent to a single discount of a. 28 percent. b. 29 percent. c. 30 percent. d. 31 percent.

a. 28 percent. 100 × .20 = 20; 100 - 20 = 80; 80 × .10 = 8; 80 - 8 = 72; 100 - 72 = 28.

A farmer who is 6 feet tall wants to determine the height of his barn. He notices that his shadow is 14 feet long and that his barn casts a shadow 70 feet long. How high is the barn? a. 30 feet b. 35 feet c. 40 feet d. 45 feet

a. 30 feet 6 : 14 = x : 70 14x=70 x 6 14x=420 /14 /14 x = 30

If 3n - 2 = 27, what does n equal? a. 5 b. 7 c. 8 d. 12

a. 5 3^n-²=27 3³=27 n-2=3 +2 +2 n =5

A tank that holds 500 gallons of water can be filled by one pipe in 25 minutes and emptied by another in 50 minutes. How long would it take to fill the tank if both pipes are open? a. 50 minutes b. 52 minutes c. 55 minutes d. 56 minutes

a. 50 minutes The filling rate is 500/25=20 gallons/minute. The emptying rate is 500/50=10 gallons/minute. The difference, 20-10 =10, is how many gallons are added per minute with both pipes open, so it takes 500/10=50 minutes to fill the tank.

In a 4-hour examination of 420 questions, there are 60 mathematics problems. If twice as much time should be allowed for each mathematics problem as for each of the other questions, how many minutes should be spent on the mathematics problems? a. 60 minutes b. 65 minutes c. 70 minutes d. 75 minutes

a. 60 minutes Let x = minutes to be spent on each math problem. X×60 + X/2 × 360 =240 60X+180X=240 240X=240 x=1

What is the perimeter of a right triangle whose legs are 3 and 4 feet? a. 10 feet b. 12 feet c. 14 feet d. 16 feet

b. 12 feet Using the Pythagorean theorem, the hypotenuse of the right triangle = 5: 3′ + 4′ + 5′ = 12

A naval detachment has enough rations to feed fifteen people for 8 days. If five more people join the detachment, for how many fewer days will the rations last? a. 1 b. 2 c. 3 d. 4

b. 2 Let x = the number of ration days for 20 persons. 15×8=20xx 20x=120 x=120/20 x=6 ration days for 20 persons. Therefore 8 days - 6 days = 2 days fewer.

A family drove from New York to San Francisco, a distance of 3,000 miles. They drove 1/8th of the distance the first day and 1/7th of the remaining distance the second day. How many miles were left to be driven? a. 2,200 miles b. 2,250 miles c. 2,400 miles d. 2,500 miles

b. 2,250 miles 1/8 of 3000=375 3000-375=2625 1/7 of 2625=375 2625-375=2250 2250 miles still to be driven

Two ships are 2,000 miles apart and sailing toward each other. One sails at the rate of 80 miles per day and the other at the rate of 100 miles per day. How far apart will they be at the end of 9 days? a. 364 miles b. 380 miles c. 440 miles d. 500 miles

b. 380 miles 80 × 9 days = 720 miles; 100 × 9 days = 900 miles; 720 + 900 = 1620 miles; 2000 - 1620 = 380 miles.

If the sum of the edges of a cube is 24 inches, what is the volume of the cube? a. 4 cubic inches b. 8 cubic inches c. 16 cubic inches d. 64 cubic inches

b. 8 cubic inches A cube has 12 edges, so each edge is 24/12=2 Volume=length x width x height V=2×2×2 v=8

A closed rectangular box with a square base is 3 inches high. If the volume of the box is 48 square inches, what is the box's surface area in square inches? a. 66 b. 80 c. 81 d. 90

b. 80 First, determine the dimensions of the square base. The box's height is given as 3. Accordingly, knowing the box's volume to be 48: 48=3w 48/3=w 16=w Because the base is square, each side must be 4 inches (4 × 4 = 16). Now you can calculate the total surface area: 2w+2wh+2w=2×16+2×12+2×12 =32+24+24 =80

A class of 216 recruits consists of three racial and ethnic groups. If 1/3 are black, 1/4 are Hispanic, and the remaining recruits are white, how many of the recruits in the class are white? a. 94 b. 90 c. 75 d. 68

b. 90 1/3×216=72 blacks 1/4×216=54 hispanics 72+54=126 216-126=90 whites

A CD system originally priced at $1,000 is first discounted by 20%, then later by another 10%. If a 5% tax is added to the purchase price, how much would a customer buying the system at its lowest price pay for it to the nearest dollar, including tax? a. $720 b. $750 c. $756 d. $800

c. $756 After the first 20% discount, the price was $800 ($1,000 minus 20% of $1,000). After the second discount, which is calculated based on the $800 price, the price of the CD system is $720 ($820 minus 10% of $800). A 5% tax on $720 is $36. Thus, the customer has paid $720 + $36 = $756.

1,000,000 may be represented as a. 10⁴ b. 10⁵ c. 10⁶ d. 10⁷

c. 10⁶ 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000, or 10 raised to the 6th power.

If x2 varies directly as y and if x = 2 when y = 10, what is the value of y when x = 8? a. 32 b. 130 c. 160 d. 168

c. 160 4 : 10 = 64 : y 4y=64×10 y=640/4 y=160

A bridge crosses a river that is 1,400 feet wide. One bank of the river holds 1/5 of the bridge, while the other holds 1/10 of it. How long is the bridge? a. 1,700 feet b. 1,800 feet c. 2,000 feet d. 2,100 feet

c. 2,000 feet 1/5x+1/10x+1400=x 2x+x+14000=10x -3x -3x 14000=7x 2000=x

What is the square root of 4 raised to the fifth power? a. 8 b. 16 c. 32 d. 64

c. 32 √4=2 2⁵=2×2×2×2×2 =32

The third root of 64 is a. 2 b. 3 c. 4 d. 8

c. 4 4 × 4 × 4 = 64.

In the figure shown below, what is the measure of angle x? (a circle with a triangle drawn inside it, top of triangle [point O] in center of circle with bottom two points [point A to left, point B to right with x inside point B angle] touching bottom of circle. Gap between triangle and circle is 80°) a. 35° b. 45° c. 50° d. 70°

c. 50° Arc AB = 80° therefore, AOB = 80°. The two radii are equal. Angle x = (180°-80°) = 1/2(100°) = 50°

If a driver completes a trip of 240 miles at the rate of 30 mph, at what rate would the driver have to travel on the return trip in order to average 40 mph for the round trip? a. 50 mph b. 55 mph c. 60 mph d. 65 mph

c. 60 mph Traveling at 30 mph for 8 hours, the driver travels 240 miles; the round trip was 480 miles. Coming back, 480 miles at 40 mph would take 12 hours: 480/40=12. If the first 240 miles took 8 hours, the return 240 miles must be covered in 4 hours, which equals a speed of 60 mph.

If x is less than 0 and y is greater than 0, then a. x is greater than y. b. y is greater than x. c. xy is less than 0. d. xy is greater than 0.

c. xy is less than 0. When a negative number and a positive number are multiplied, the product is negative.

If (x - y)2 = 60 and x2 + y2 = 40, then xy = a. -40 b. -20 c. -12 d. -10

d. -10 (x-y)2 = x²-2xy+y² 60=40-2xy -40 -40 -2xy=20 /-2 /-2 xy = -10

A field can be plowed by 12 machines in 7 hours. If 4 machines are broken and cannot be used, how many hours will it take to plow the field? a. 7 1/2 hours b. 8 1/2 hours c. 9 1/2 hours d. 10 1/2 hours

d. 10 1/2 hours Let x equal the number of hours to plow with 8 machines. So, 12×7=8xx 8x=84 x=84/8 x=10 1/2 hours

If a = 4b and 8b = 30c, then a = a. 6c b. 9c c. 12c d. 15c

d. 15c a = 4b ×2 ×2 2a = 8b = 30c 2a = 30c /2 /2 a = 15c.

Two trains running in the same direction on the same track travel at the rates of 40 and 45 mph, respectively. If the slower train starts out an hour earlier, how long will it take the faster train to catch up with it? a. 5 hours b. 6 hours c. 7 hours d. 8 hours

d. 8 hours The slower train is 40 miles ahead in one hour. The difference in rate is 5 mph, and 40/5=8

How much pure acid must be added to 10 ounces of a 55% acid solution in order to produce a 75% acid solution? a. 5 ounces b. 6 ounces c. 7 ounces d. 8 ounces

d. 8 ounces Pure Acid -No. of Ounces : x -Parts Pure Acid : 100 -No. of Ounces of Pure Acid : 100x 40% Acid Solution -No. of Ounces : 10 -Parts Pure Acid : 55 -No. of Ounces of Pure Acid : 550 60% Acid Solution -No. of Ounces : 10 + x -Parts Pure Acid : 75 -No. of Ounces of Pure Acid : 75(10 + x) 100x+550=75(10+x) 100x+550=750+75x -75x-550-550-75x 25x=200 x=8

Jane received grades of 93, 82, and 72 on three tests. What grade must she receive on the next test so that her average for these four tests is 85? a. 88 b. 89 c. 90 d. 93

d. 93 Jane's current test grades total 247: 93 + 82 + 72 = 247. To average 85, Jane's grades must total 340: 85 × 4 = 340. To raise her average grade, Jane must earn 93 on the next test: 340 - 247 = 93.