AMCAT Math 1 — Questions and Answers

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:

45 km
33 km
50 km (Correct answer)
24 km

Correct answer: 50 km

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.

72 (Correct answer)
70
73
78

Correct answer: 72

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?

90 (Correct answer)
86
78
74

Correct answer: 90

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.

60
316
133
120 (Correct answer)

Correct answer: 120

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.

$755
$890
$450 (Correct answer)
$442

Correct answer: $450

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.

Question 6: A nine-member family has an average age of 22 years. Carlo is the youngest, and he is six years old. What was the family's average age soon before Carlo was born?

17
18 (Correct answer)
15
20

Correct answer: 18

The current total age of the 9-member family is 9 * 22 = 198 years. When Carlo was born 6 years ago, there were 8 family members, and each of them was 6 years younger. So, the sum of their ages 6 years ago was (198 - 6) - (8 * 6) = 192 - 48 = 144 years. The average age of the 8 members before Carlo was born was 144 / 8 = 18 years.

Question 7: The average of 7 numbers is 60. The average of the first three numbers is 50, while the average of the final three is 70. What is the remaining number?

60 (Correct answer)
58
75
63

Correct answer: 60

The total sum of the 7 numbers is 7 * 60 = 420. The sum of the first three numbers is 3 * 50 = 150, and the sum of the final three numbers is 3 * 70 = 210. To find the remaining (fourth) number, subtract the sum of the first three and final three numbers from the total sum: 420 - (150 + 210) = 420 - 360 = 60.

Question 8: When a specific sum is invested for ten years, the simple interest gained doubles. What rate of interest is being offered?

15%
45%
13%
20% (Correct answer)

Correct answer: 20%

Let the principal be P. The statement "the simple interest gained doubles" implies that the simple interest (SI) earned is twice the principal, so SI = 2P. Using the formula SI = (P * R * T) / 100, where T = 10 years, we have 2P = (P * R * 10) / 100. Dividing both sides by P and simplifying, we get 2 = R / 10, which means the rate of interest (R) is 20%.

Question 9: A guy rows a boat at 15 mph on calm water. Determine the speed of the river if it takes him 4 hours and 30 minutes to paddle a boat to a location 30 miles distant and back.

12 mph
10 mph
15 mph
5 mph (Correct answer)

Correct answer: 5 mph

Let the speed of the river be 'x' mph. The boat's speed downstream is (15 + x) mph, and upstream is (15 - x) mph. The total time for a 30-mile trip each way is 4.5 hours. So, 30/(15 + x) + 30/(15 - x) = 4.5. Solving this equation, we get 30(15 - x) + 30(15 + x) = 4.5(225 - x²), which simplifies to 900 = 4.5(225 - x²). This leads to 200 = 225 - x², so x² = 25, and thus x = 5 mph.

Question 10: If 13x = 371293, find the value of 22x-1.

128
620
512 (Correct answer)
64

Correct answer: 512

The problem as stated, 13x = 371293 and finding 22x-1, leads to x = 28561 and 22x-1 = 628341, which is not among the options. However, the options (128, 512, 64) are powers of 2. If we assume the expression was intended to be 2^(2x-1) and the first equation was a typo for 13x = 65 (implying x=5), then 2^(2*5-1) = 2^9 = 512. This is the most plausible interpretation given the answer choices, suggesting a significant typo in the original problem statement.

Question 11: Dion and Jane can finish a job in 72 hours; Jane and Lisa in 120 hours; and Dion and Lisa in 90 hours. What time can Dion do it on his own?

120 hours (Correct answer)
150 hours
90 hours
118 hours

Correct answer: 120 hours

Let D, J, and L be the time Dion, Jane, and Lisa take individually. Their combined work rates are 1/D + 1/J = 1/72, 1/J + 1/L = 1/120, and 1/D + 1/L = 1/90. Adding these equations gives 2(1/D + 1/J + 1/L) = 1/72 + 1/120 + 1/90 = 5/360 + 3/360 + 4/360 = 12/360 = 1/30. Therefore, the combined rate of all three is 1/D + 1/J + 1/L = 1/60. To find Dion's rate, subtract Jane and Lisa's combined rate: 1/D = 1/60 - 1/120 = 2/120 - 1/120 = 1/120. Thus, Dion can do the job alone in 120 hours.

AMCAT Math 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: