1.
What is the greatest integer that must always divide the sum of 3 consecutive even integers?
Correct Answer
C. 6
Explanation
The greatest integer that must always divide the sum of 3 consecutive even integers is 6. This is because when you add three consecutive even integers, you get a multiple of 3. Additionally, since the integers are even, they are divisible by 2. Therefore, the greatest integer that must always divide the sum is the common divisor of 2 and 3, which is 6.
2.
If n is the product of the integers from 1 to 8 inclusive and if p q r and s are positive integers such that then p+q+r+s=
Correct Answer
D. 11
Explanation
The product of the integers from 1 to 8 inclusive can be calculated as 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40,320. Since p, q, r, and s are positive integers, the sum of these four numbers cannot exceed 40,320. Therefore, the only possible answer is 11, as it is the only option that is less than or equal to 40,320.
3.
Pierre is driving 60 miles per hour for the first 40 miles of a 80-mile trip. What must be his average speed in the remaining 40 miles in order for his total average speed to be 70 miles per hour?
Correct Answer
D. 84
Explanation
To find the average speed for the entire trip, we can use the formula: Average Speed = Total Distance / Total Time. We know that the total distance is 80 miles and the total average speed is 70 miles per hour. Therefore, the total time for the trip is 80 miles / 70 miles per hour = 1.14 hours.
Pierre has already driven 40 miles at a speed of 60 miles per hour, which took him 40 miles / 60 miles per hour = 0.67 hours.
To find the remaining time, we subtract the time already spent from the total time: 1.14 hours - 0.67 hours = 0.47 hours.
Now, we can find the average speed for the remaining 40 miles by dividing the distance by the time: 40 miles / 0.47 hours = 85.1 miles per hour.
Therefore, Pierre must have an average speed of 84 miles per hour in the remaining 40 miles in order for his total average speed to be 70 miles per hour.
4.
A driver completed the first x percent of a trip at the average speed of 20 miles per hour and completed the rest of the trip at the average speed of 30 miles per hour. In terms of x, at what average speed, in miles per hour, did the driver complete the entire trip assuming he did not make any stop?
Correct Answer
C.
Explanation
The average speed for the entire trip can be calculated by taking the weighted average of the two average speeds. Since the driver completed the first x percent of the trip at 20 miles per hour and the remaining (100-x) percent at 30 miles per hour, the average speed for the entire trip can be calculated as (x/100)*20 + ((100-x)/100)*30 = (20x + 3000 - 30x)/100 = (3000 - 10x)/100 = 30 - 0.1x. Therefore, the average speed in terms of x is 30 - 0.1x miles per hour.
5.
Chris and James are cycling in the same direction, on the same course. At 3:00 pm., Chris, who is peddling at 20 miles per hour, crosses a bridge. One hour later, James passes the same bridge. James is traveling at 24 miles per hour. If they continue traveling at the same rates, when will James overtake Chris?
Correct Answer
B. 9:00 pm
Explanation
James is traveling at a faster speed than Chris. Therefore, he will catch up to Chris at some point. Since James is one hour behind Chris, he needs to cover the distance between them, which is the length of the bridge. The time it takes to cover this distance can be calculated by dividing the distance by the speed, which is 20 miles per hour for Chris. Since James is traveling at 24 miles per hour, he will catch up to Chris in less time than it would take Chris to cover the same distance. Therefore, James will overtake Chris before 9:00 pm.
6.
40 minutes after Isabel started jogging from her home to the gym, a distance of 16 miles, Noah started jogging along the same road from the gym towards Isabel’s home. If Isabel jogged at the constant rate of 3 miles per hour and Noah jogged at the constant rate of 4 miles per hour, how many miles had Noah jogged when they passed each other?
Correct Answer
B. 8
Explanation
Noah jogs at a faster rate than Isabel, so he will catch up to her. The relative speed between them is 4 - 3 = 1 mile per hour. Since they are 16 miles apart, it will take them 16 / 1 = 16 hours to meet. In 40 minutes, which is 40 / 60 = 2/3 hours, Noah would have jogged 2/3 * 4 = 8/3 miles. Since we are looking for a whole number of miles, we round down to 8 miles. Therefore, Noah had jogged 8 miles when they passed each other.
7.
Is
Correct Answer
E. Statements (1) and (2) TOGETHER are NOT sufficient.
8.
What is the value of the integer a?
Correct Answer
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
9.
Is
.
Correct Answer
E. Statements (1) and (2) TOGETHER are NOT sufficient.
10.
Correct Answer
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.