PROBLEM
A small group has $100 to spend for lunch. The group decided to
give a tip of 20%(before tax). The sales tax is 7 ½ %. How much
should they spend?
x + 0.20 x + 0.075 x = 100
1.275 x = 100
x = $78.43
PROBLEM
An un-experienced worker unloads a truck within 1 hour and 40
minutes. Together with a trainee they can work for 1 hour. In how
many hours can the trainee spend working alone?
1 hour and 40 minutes = 100 minutes
rt = 1
1
 1
+ 60 = 1

 100 x 
PROBLEM
What is the reciprocal of 3-i?
1
3−i
(3 + i ) =
1
3+i
x
3 − i (3 + i ) 9 + 3i − 3i + 1
3+i
=
9 +1
3+i
=
or 0.3 + 0.1i
10
PROBLEM
Find the sixth term of the arithmetic sequence with a1=11 and d=4.
a n = a1 + (n − 1)d
a6 = 11 + (6 − 1)(4)
x = 150
a6 = 31
PROBLEM
Find the general term an, of the arithmetic sequence -8,-3,2...
a n = a1 + (n − 1)d
a n = −8 + (n − 1)(5)
x=5
a n = 5n − 5 − 8
a n = 5n − 13
PROBLEM
A solution of system of linear inequalities is bounded if
•
IT IS ENCLOSED WITHIN A CIRCLE
PROBLEM
A mixture containing 16% of a drug is combined to a 28% of a drug
to obtain a 15mL of 24% of a mixture. How many 16% of a drug
should be added?
16%
+
28%
0.16 x + 0.28(15 − x ) = 0.24(15 )
0.16 x + 4.2 − 0.28 x = 3.6
0.12 x = 0.6
=
PROBLEM
Peanut and nut mixture contains 40% peanut. How much additional
peanut mixture should be added to produce 8 lbs. of 50% peanut?
40%
15-x
0.40(8 − x ) + 1(x ) = 0.50(8)
3.2 − 0.4 x + x = 4
0.6 x = 0.8
24%
15
100%
8-x
x = 1.33lbs
x
+
x
=
50%
8
PROBLEM
A small boat travelling 5 miles upstream, later back down stream for
1 hour and 40 minutes. The stream current is 4 mph. What is the
velocity when it is in still water?
PROBLEM
Together John and Michael can paint a wall for 18 minutes. John
alone can finish the wall 15 minutes more than Michaels painting.
Find the time of each.
1 hour and 40 minutes = 1.67 hours
1 1
 + 18 = 1
 t1 t 2 
 1
1

+ 18 = 1
 t 2 + 15 t 2 
t u + t d = 1.67
5
5
+
= 1.67
V −4 V +4
t 2 = 30 min
V = (−)2mph
t1 = 45 min
PROBLEM
The third term of a geometric progression is 32 and the fifth is 128.
Find the first term and the common ratio.
a n = a1 r
n −1
a3 = a1 r 3−1
32 = a1 r
3−1
a 6 = a1 r 6−1
PROBLEM
Daniel is twice as old as Jimmy. Terry is one year younger than
Daniel. If the sum of their age is 44. How old is Daniel?
128 = a1 r 6−1
D = 2J → J =
a1 = 12.70
T = D −1
r = 1.59
D
2
D + J + T = 44
D
D + + (D − 1) = 44
2
D = 18
PROBLEM
A Bank contains $1.65 Nickel, Dimes and Quarter. There are twice
as many as Nickel as Dimes as and one more Quarter than Nickel.
There are how many Quarter.
0.05 N + 0.10 D + 0.25Q = 1.65
N = 2D → D =
N
2
Q = N +1
x(35 − x ) = 304
35 x − x 2 = 304
x 2 − 35 x + 304 = 0
(x − 19 )(x − 16 ) = 0
0.80 x = SELLING PRICE
x = BOOK PRICE
N
0.05 N + 0.10  + 0.25(N + 1) = 1.65
2
N =4
0.80 x + 0.08(0.80 x ) = 21.56
Q = 4 +1
Q=5
PROBLEM
The sum of two positive numbers is 35. There product is 304. What is
the smallest number?
x = SMALL NUMBER
35 − x = LARGER NUMBER
PROBLEM
A woman pays $21.56 in buying a book which is marked 20% off.
The sales tax is 8%. How much is the book.
x = 19
x = 16
SMALL NUMBER = 16
x = 24.95
PROBLEM
A boy got 100 an average in Homework and Test grade which are
97, 99, 100. Homework counts 15%. Each Test grade counts 20%
and 25% Final grade. What is the score he should get in Final Exam
to have a grade of 90?
90 = 100(0.15) + 97(0.20 ) + 99(0.20 ) + 100(0.20 ) + 0.25 x
x = 63.2
PROBLEM
A matrix is in reduced form containing 0’s in a row is below any ____
non zero element.
•
1
PROBLEM
Two balls are drawn in a bag with one red ball, two black balls and
three white balls. What is the probability if the first ball is red and the
second is white given if that the first drawn ball is return into the bag?
P=
P=
1 3 3
 =
6 6 36
1
12
PROBLEM
How many ways can you position six person in circular table?
C = (n − 1)!
C = (6 − 1)!
C = 120 ways
PROBLEM
How many permutation can you get in a letter word TENNESSEE?
P=
PROBLEM
A student is given a 87% chance to get 98% in the exam given that
he was able to solve all problems. But he has also 15% chance of
not solving the problem. What is the chance of him getting 95% in the
exam?
E = 0.87(0.85 )x100
E = 74%
9!
2!2!4!
P = 3780 ways
PROBLEM
How many ways can you arrange the letters in a word “MONDAY”
given that the first letter is a vowel.
W = 2  5  4  3  2 1
W = 240 ways
PROBLEM
How many ways can you arrange the word “MONDAY” wherein the 4
letters are taken at a time?
PROBLEM
There are 7 paintings in a museum with 3 vacant slots. How many
ways can you arrange the paintings in the given slot?
6 P4 = 360 ways
PROBLEM
What is the mode of the sequence 4 21 11 7 4 8 6 9?
•
4. IT IS UNIMODAL
PROBLEM
What is the mode of the sequence 15 6 7 9 3 8 11 15 3 4
9?
•
15 and 9. BIMODAL
7 P3 = 210 ways
PROBLEM
The 1st and the 10th term of a geometric progression is 1 and 4. Find
the 17th term.
a n = a1 r n −1
4 = (1)r 10−1
r=4
1
1
a17 = 1 4 9 


a17 = 11.758
17−1
9
PROBLEM
PROBLEM
What is the median of the sequence 5 6 8 12 13 15?
8 + 12
MEDIAN =
= 10
2
1
3
1
9
What is the sum of the series 1 + + ?
r=
a2
a1
1
r= 3
1
1
r=
3
a1
1
=
1− r 1− 1
3
3
S=
2
S=
PROBLEM
What is the sum of the first 50 terms of the series 10 + 85 + 160 + 325 ?
d = a2 − a1 = 85 − 10 = 75
n(2a1 + (n − 1)d ) 50(2(10) + (50 − 1)75 )
=
2
2
S = 92375
S=
PROBLEM
The special way in proving hypothesis and rank against the most
basic tools in the mathematicians toolbox.
•
MATHEMATICAL INDUCTION
PROBLEM
What is
7 + 8i
?
4 − 2i
3 23
= + i
5 10
PROBLEM
A calculator manufacturer wants to earn $18000. They produce the
calculator for $6 and sell it for $11. If the overhead runs $150000.
What is the number of calculator to be produced?
x - number of calculator
11x − (6 x + 150000 ) = 18000
x = 33600
PROBLEM
If a man buys gasoline:
10L for 11.50
12L for 12.01
18L for 11.78
average.
Average =
10(11.5) + 12(12.01) + 18(11.78)
10 + 12 + 18
Average = 11.78
Find the
PROBLEM
If we partition the division so that the rectangles width is not equal
and the height is taken from sublevel, the sum of areas of the
rectangles are called:
•
REIMANN SUM
PROBLEM
How ways are there to select 3 juniors and 4 seniors from chorus
with 10 freshmen, 15 sophomores, 18 juniors and 20 seniors to sing
in an all-day chorus?
Solution:
(18C3) x (20C4)
ANSWER: 3,953,520 ways
PROBLEM
The mean score of the students is 65.3 and the standard deviation of
20.15. Find the two values at which must lie at least 75% of the data.
Solution:
k=2
65.3 – 2(20.15) = 25
65.3 + 2(20.15) = 105.6
ANSWER:
75% of the data lie between 25 and 105.6
PROBLEM
How many ways can 5 keys be arranged in a keychain?
Solution:
(n-1)!
(5-1)!
ANSWER: 24 ways
PROBLEM
A bracelet has 7 charms. How many arrangements of charms can be
made?
Solution:
(n-1)!
(7-1)!
ANSWER: 720 ways
PROBLEM
How many distinguishable permutations can be made from the letters
of MISSISSIPPI?
Solution:
ANSWER: 34,650
PROBLEM
A man tosses a 1 five-peso coin, 2 one-peso coins, 3 twenty-fivecentavo coins, and 4 ten-centavo coins to 10 boys. In how many
ways can the boys profit if each have a coin?
Solution:
PROBLEM
A box contains 3 red marbles and 5 black marbles. A red marble is
drawn and was not replaced back to the box. What is the probability
that a black marble will drawn?
ANSWER:
QUESTION
What occurs when the new coordinate axes have the same direction
as and are parallel to the original coordinate axes?
ANSWER: Translation of coordinate axes
PROBLEM
An office…
Let y = number of offices rented
ANSWER: 12,600
PROBLEM
A box contains 3 red marbles and 5 black marbles. A red marble is
drawn and replace back to the box before the second drawer. What
is the probability that a black marble will drawn?
ANSWER:
rent x offices = 68,400
(1600 + 100y) (40 – y) = 68,400
(y – 22) (y – 2)=0
y=2
rent = 1600 + 100(2)
ANSWER: rent = 1800
PROBLEM
Kelly sells corn dogs in a state fair. Booth rental and equipment total
$200 per day. Eacg corn dogs is $35 cents to make and sell for $2
each. How many corn dogs should she sell if she wants $460 profit?
Letx=corn dogs to sell each day
Revenue=2x
PROBLEM
Two men will meet at an intersection. One travels southward while
the other travels north. If one of the driver drives 30 mph and the
other drives 40mph, when will they be 35 miles apart?
rt=35
(30+40)t=35
t=0.5 or 30 minutes
Ans. x≥400
PROBLEM
Linda has $16,000 to invest. Part of her money is envested with a
bond of 5% and 6 %. How much should she invest in 6 % if she
wants an annual profit of $937.50?
Let x=invested in 6 %
16000-x = invested in 5%
0.0625x + 0.05(16000-x) = 937.50
X=11,000
PROBLEM
There are $8.75 nickels, dimes and quarters. There 5 more dimes
than nickel and 4 more quarters than dimes. How many dimes are
there?
0.05N+0.10D+0.25Q=8.75
5+N=D
Q=4+D
0.05(D-5)+0.10D+0.25(4+D)=8.75
D=20
PROBLEM
A working student earns $8 per week. How many hours should she
work per week to earn $120 to $200?
120≤ ≤200
PROBLEM
A polar equation r = a represents
→circle with center at the origin
PROBLEM
A polar equation
represents
→lemniscate
15
hours
PROBLEM
→
PROBLEM
cos ( α – β ) – cos ( α + β ) =
PROBLEM
A polar equation r = a csc θ represents
→Horizontal line
PROBLEM
A polar equation θ = a represents a
→ Line through the origin
→ sin α sin β
PROBLEM
=
→
PROBLEM
A polar equation r = a cos θ represents
→Circle tangent to the y-axis
PROBLEM
Melinda had a 6 mile hike or 1 hour 45 min. she first walked for 4mph
and then he finished her walk at 3mph. Find the distance that she
walked after finishing the 4mph walk.
PROBLEM
A 16-team… The bowling officials has P8000 budget for the prizes. If
the prize given to the 16th place is P275, find the prize given to the
first place.
tT = 1.75hrs
n
(a1 + a n )
2
16
8000 = (a1 + 275)
2
S = 725
S=
tT = t1 + t2
1.75 =
x 6− x
+
4
3
x=3
PROBLEM
According to Hooke’s law you can stretch a _____ to F=4.2x. if
7  F  14 , Find the Value of x.
7
4.2 x 14


4.2 4.2 4.2
x = 5 to 10
3
3
PROBLEM
Factor (81-x4) completely
(9 + x 2 )(9 − x 2 )
(9 + x 2 )(3 − x)(3 + x)
PROBLEM
The Government gives P500 taken from the lottery… If P500 is given
to an individual and that individual spends 80% of it, and the
receivers of that money also spends 80% of what they received, and
the cycle continues forever, Find the total amount of money used.
400
1 − 0 .8
S = 2000
S=
PROBLEM
Find the sum of a 20 term G.P. if the first term is 1 and r=2.
a1 (r n − 1)
r −1
1(2 20 − 1)
S=
2 −1
S = 1048575
S=
PROBLEM
Find the 16th term of the expansion (x-2)20
PROBLEM
A bridge deck consisting of 52 cards. If you were to pick 3 cards,
what is the probability that the cards are diamond?
Total outcome: 52C3 = 22,100
Successful outcome: 13C3 = 286
P = successful outcome / total outcome
= 286/22,100 = 0.0129
PROBLEM
= 20 C16−1 x 20−16−1 (2)16−1
Answer: cot β
= 508035072 x
5
PROBLEM
PROBLEM
There are 15 contestants in a contest. In how many ways the three
winners are to be chosen?
15C3 =
455 ways
Answer: sin α
PROBLEM
A circle with a radius, r, is inscribed in an irregular triangle having a
side of 5 cm, 7cm and 10 cm. Determine the radius of the circle.
PROBLEM
Tickets of a concert are consecutively numbered. Manny sold tickets
numbered 168-358. How many did he sell?
S=
Solution:
an = an + (n-1)d
358 = 168 + (n-1)(1)
= 11
= 16.248 cm2
A=
A=
r=
= 2.954 cm
an = 359 tickets
PROBLEM
Probability of getting exactly 3 tails in 5 throws.
PROBLEM
P = 5C3(1/2)3(1/2)5-3
Answer:
tan 3β
P=5/16
PROBLEM
Answer:
2cosαcosβ
PROBLEM
A standard 52-deck card, how many 5-hand card will have 3aces and
2kings?
C1 = 4C3 = 4
C1 = 4C3 = 6
C = (4)(6) = 24 hand
PROBLEM
In a single dice, find the probability of getting at least 3 two’s in 4
throws.
P1 = 4C3(1/6)3(5/6)4-3 = 5/324
P 2= 4C4(1/6)4(5/6)4-4 = 1/216
P = P1 + P2 = 13/648
PROBLEM
In throwing a 2-dice, find the probability of getting the sum of the two
dice greater than 10.
P1 = 2/36 ; for 5 and 6 output of the dice.
P2 = 1/36 ; for 6 and 6 output of the dice.
P = P1 + P2 = (2/36) + (1/36)
P= 1/12
PROBLEM
Find the combination of 5 object taken 1,2,3,4, and 5.
C = 2n – 1
C = 25 – 1
C = 31