Hiller
If the product of 8k and 4 is 16, then
give the product of 4k and 2.
A. 8
B. 6
C. 4
D. 2
E. NOTA
What is C?
8k(4) = 16, k = 0.5. 4k (2) = 4.
What is the derivative of 30x^3?
What is 90x^2?
Aaron has 10 identical golf balls that he is trying to paint. He has four different colors at his disposal:
Purple, Light Blue, Blue, and Ocher. How many different colorings of the balls are possible?
A. 120
B. 286
C. 720
D. 84
E. NOTA
What is B?
We can encode the different colorings in a list (x1, x2, x3, x4), where x1 is the number of balls colored Purple, x2 the number of balls colored Light Blue, x3 the number of balls colored Blue, and x4 the number of balls colored Ocher. Therefore, we have translated our problem to the following: Finding all non-negative integer solutions (x1, x2, x3, x4) such that
x1 + x2 + x3 + x4 = 10.
This is common stars-and-bars problem, which the answer is (10+4-1 4-1) = 286
In a stars-and-bars problem, (stars+bars bars) = (___choose___)
58 + 47
What is 105?
For k, a positive integer, kx - 20 =2k,
how many positive integers are in the
solution set for x?
A. 10
B. 9
C. 6
D. 4
E. NOTA
What is C?
kx - 20 = 2k , x = 2k+20/k = 2 + 20/k.
x will be an integer when k is a factor of 20. So, the factors are 1, 2, 4, 5, 10, 20.
So essentially solve for x, then see what condition must be met in order for x to be a positive integer
Evaluate: 6 + 8 ÷ 2 ( 1 + 2 )
A. 22/3
B. 18
C. 21
D. 30
E. NOTA
What is B?
6 + 8 ÷ 2 ( 1 + 2 )
= 6 + 4(3) = 6 + 12 = 18
Without the use of a calculator, what is 75^2?
What is 5625?
7 * (7 + 1) = 56. Append 25.
This trick works for squaring any number ending in 5. Multiply the first number by that number + 1. Then just put 25 at the end.
When 2024 in base-10 is expressed in base-7, what is the base-10 average of its digits?
A. 13/4
B. 8
C. 4
D. 7/2
E. NOTA
What is D?
2024 base 10 = 5 * 7^3 + 6 * 7^2 + 2 * 7 + 1 = 5621 base 7
The base-10 average of these digits is 5 + 6 + 2 + 1 =7/2
To convert a number from base-10 to any other base, you repeatedly divide the base-10 number by the new base (in this case, 7) and record the remainders. The sequence of remainders, read from bottom to top (last to first), forms the number in the new base.
The average is the sum of the digits divided by the count of the digits. Since the digits are themselves base-10 numbers, we use standard base-10 arithmetic for the average.
2024/7 is 289 remainder 1
289/7 is 41 remainder 2
41/7 is 5 remainder 6
5/7 is 0 remainder 5
32 x 5
What is 160?
Beavis is currently twice as old as his brother and he is 6 years older than his sister. How many years from now will Beavis’s age be 2/3 of the combined ages of his brother and his sister at that time?
A. 12
B. 16
C. 20
D. 24
E. NOTA
What is A?
B + x = 2/3 ( B/2 + B - 6 + 2x ) --------> B + X = B - 4 + 4/3x --------> x = 12
Find the sum of all prime numbers
between 1 and 60 that are each 1
greater than a multiple of 8?
A. 17
B. 41
C. 58
D. 98
E. NOTA
What is C?
The primes in that range which are
1 greater than a multiple of 8 are
17 and 41. The sum is 58.
Without the use of a calculator, what is 45% of 120?
What is 54?
50% of 120 is 60. 5% is 6. 60 - 6 = 54
Julie buys 3 math books, 2 biology book, 3 physics books, and 2 chemistry books. She wants to order
them in his bookshelf that has space for 10 books. In how many ways can Julie order his books in his
bookshelf if the books of the same topic must stay together?
A. 864
B. 1728
C. 2592
D. 3456
E. NOTA
What is D?
We consider 4 ways to arrange the subjects. In each subject there are 3!, 2!, 3!, 2! ways to
arrange them respectively. The calculation is 4! · 3! · 2! · 3! · 2! = 3456
25 x 12 / 5
What is 60?
How many ordered pairs, (x, y) of positive integers satisfy: x/12 + y/36 = 1
A. 9
B. 11
C. 12
D. 13
E. NOTA
What is B?
3x + y = 36, so (11,3), (10,6), ... (1,33) that makes 11.
* Clear the denominator by multiplying the entire equation by the least common multiple, 36.
Then isolate y, so it'll become y = 36 ( 12 - x )
For y to be positive, y >= 1 and solve for x, which would be 11 2/3 >= x.
Since x must be an integer, the possible values for x are the integers from 1 up to and including 11.
If x = −2 is a solution of the equation
x^4 − kx + 12 = 0, then find the value
of k.
A. -14
B. -2
C. 2
D. 10
E. NOTA
What is A?
x^4 − kx + 12 = 0 for x = −2
gives 16 + 2k + 12 = 0 and k = −14.
Find the sum of the last three digits of 995^3.
What is 20?
To find last three digits, do mod1000 = −5^2mod1000 = −125mod1000= 875. Sum is 20.
The reason we used mod1000 to solve is because when you divide a number by 1000 you get the last 3 digits. mod10 to find last digit, mod100 to find last 2 digits.
995 = -5 (mod1000) ------> this means that if you divide 995 by 1000, the result is some whole number * 1000 plus a remainder of -5.
995^3 = (-5)^3 (mod1000)
995 ^3 = -125 (mod1000)
-125 + 1000 = 875
If A = B (modM), then A = B + kM (mod M) for any integer k.
Consider the function f(x)=x^2-ax
Determine the value of a > 0 such that the range of f is the interval [−4, ∞)
A. 1
B. 2
C. 4
D. 16
E. NOTA
What is C?
Consider, the x-coordinate of the vertex of the parabola is a/2. Therefore, we desire f(a/2)= -4.
-4=(a/2)^2-a(a/2) ------> -4=a^2/4-a^2/2 ------> -4=-a^2/4 -------> a=+-4
Given a>0, the answer is a=4
** -b/2a = vertex of the x-coordinate
At 4:40, what is the degree measure of the larger angle formed by the hour and minute hands of a clock?
What is 260 degrees?
At 4:40pm the hour hand will be 2/3 of the way between 4 and 5 and the minute hand will be at the 8. From 8 clockwise to 4 is 240 degrees plus the 20 degrees past 4 makes a total of 260 degrees.
** The minute hand moves 360 degrees in 60 minutes, or 6 degrees per minute. So in 40 minutes it will be at 240 degrees
The hour hand moves 360 degrees in 12 hours, or 0.5 degrees per minute. So 4:40 is is 4 * 60 + 40 = 280 minutes. 280 * 0.5 = 140 degrees
Find the small angle: 240 - 140 = 100
Find the larger angle: 360 - 100 (small angle) = 260
The Snowman drove 180 miles from his home to a math competition. His average speed to the competition was 20mph less than his average speed returning along the same route. If the entire trip took 7.5 hours, what was the Snowman’s average speed to the competition, in mph?
A. 24
B. 40
C. 48
D. 60
E. NOTA
What is B?
180/R-20 + 180/R = 15/2 ----> 12/R-20 + 12/R = 1/2 (multiply by the common denominator)-------> 24R + 24(R - 20) = R^2 - 20R R^2 - 68R + 480 = 0 -------> (R - 60)(R - 8) = 0-----> R = 60 -----> 60 - 20 = 40
*R = speed
Whoville had W citizens at the end of
2020. By the end of 2021, the census
showed that the citizen population had
increased by 25%, and they had a total of
X citizens. By the end of 2022, the year's
increase (from X citizens) was again 25%,
and Whoville had Y citizens. Find the
change over the two years as a fraction of
the 2020 population: |Y − W|
divided by the original W.
A. 1.5
B. 3/4
C. 5/8
D. 9/16
E. NOTA
What is D?
At the end of 2020, we had W for
the population of Whos. By the end of
2021, we had 1.25W or
5/4W. This is X.
By the end of 2022, we have an increase
of 25% of this, so
(5/4W) * 5/4 = 25/16W. This is Y. |Y-W| = 25/16W-16/16W = 9/16W.
Divide by W to get 9/16
What is the maximum area of a right triangle with hypotenuse of length 8?
What is 16?
Maximum area is equal legs of length 4√2. Area is 16.
The maximum area for a right triangle with a fixed hypotenuse occurs when the triangle is also an isosceles right triangle. Just use pythagorean theorem
a^2 + a^2 = 64
2a^2 = 64
a^2 = 32
a = sqrt 32 = sqrt 16 * 2 = 4sqrt2
The result of the sum 2^2 + 3^2 + 4^2 + 5^2 + ... + 10^2 has what unit digit?
A. 5
B. 4
C. 3
D. 1
E. NOTA
What is B?
The units digits are 4+9+6+5+6+9+4+1+0 and the sum has a units digit of 4.
Find the median of the data set: 3, 18, 7, 44, 24, 27
What is 21?
18 + 24 / 2 = 42 / 2 = 21
A pentagon has angle measures in the ratio 10:11:11:12:16. What is the degree measure of the second largest angle?
A. 99
B. 108
C. 126
D. 144
E. NOTA
What is B?
Let the angles be equal to 10x, 11x, 11x, 12x, and 16x. Then, the sum of them must
be equal to 540 degrees. So, 60x = 540, and x = 9. The second largest angle is 12x,
which is 108.