AMCAT Math

Question 1

John Clark and Mae begin their journeys at the same time, traveling at speeds of 40 and 50 kilometers per hour, respectively. If John takes 15 minutes longer to complete the route than Mae, then the total distance traveled is:

Accepted Answer

50 km

Answer

Let the distance be D. John's time is D/40 and Mae's time is D/50. Since John takes 15 minutes (or 1/4 hour) longer, we set up the equation: D/40 - D/50 = 1/4. Finding a common denominator (200), we get (5D - 4D)/200 = 1/4, which simplifies to D/200 = 1/4. Solving for D, we find D = 200/4 = 50 km.

Question 2

The total of the two integers equals 184. If one-third of one surpasses one-seventh of another by 8, identify the smaller number.

Accepted Answer

72

Answer

Let the two integers be x and y. We have two equations: x + y = 184 and (1/3)x - (1/7)y = 8. Multiplying the second equation by 21 gives 7x - 3y = 168. Substituting y = 184 - x from the first equation into the modified second equation yields 7x - 3(184 - x) = 168, which simplifies to 10x - 552 = 168. Solving for x, we get 10x = 720, so x = 72. Then, y = 184 - 72 = 112. The smaller number is 72.

Question 3

The ratio between the two integers is 4:5, and their L.C.M is 200. Then what is the sum of the two numbers?

Accepted Answer

90

Answer

Let the two integers be 4x and 5x, based on their ratio. The Least Common Multiple (L.C.M.) of 4x and 5x is 20x, because 4 and 5 are coprime. Given that the L.C.M. is 200, we set 20x = 200, which means x = 10. The two numbers are therefore 4 * 10 = 40 and 5 * 10 = 50. Their sum is 40 + 50 = 90.

Question 4

Find the area of the triangle with sides of 10 cm, 24 cm, and 26 cm.

Accepted Answer

120

Answer

First, check if the triangle is a right-angled triangle using the Pythagorean theorem (a² + b² = c²). Here, 10² + 24² = 100 + 576 = 676, and 26² = 676. Since 10² + 24² = 26², it is a right-angled triangle. The area of a right-angled triangle is (1/2) * base * height, so (1/2) * 10 cm * 24 cm = 120 cm².

Question 5

Ken borrowed $20,000 from his buddy Richard at 15% yearly simple interest and lent it to Mike at the same rate compounded annually. Find Ken's gain after two years.

Accepted Answer

$450

Answer

Ken pays simple interest to Richard and collects compound interest from Mike. The simple interest Ken pays is $20,000 * 0.15 * 2 = $6,000. The amount Mike pays Ken after two years compounded annually is $20,000 * (1 + 0.15)² = $20,000 * 1.3225 = $26,450, meaning Mike pays $6,450 in interest. Ken's gain is the difference between the interest he received and the interest he paid: $6,450 - $6,000 = $450.

(AMCAT) Aspiring Minds Computer Adaptive Test Practice Test

(AMCAT) Aspiring Minds Computer Adaptive Test Practice Test

AMCAT Math