PROBLEMA 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? 43 . 78 $ 100 275 . 1 100 075 . 0 20 . 0 = = = + + x x x x x 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? 150 x 1 60 1 100 1 1 rt minutes 100 minutes 40 and hour 1 = = | . | \ | + = = x PROBLEM What is the reciprocal of 3-i? ( ) ( ) i or i i i i i i i x i i 1 . 0 3 . 0 10 3 1 9 3 1 3 3 9 3 3 3 3 1 3 1 + + = + + = + ÷ + + = + + ÷ ÷ PROBLEM Find the sixth term of the arithmetic sequence with a 1 =11 and d=4. ( ) 31 4 ) 1 6 ( 11 ) 1 ( 6 6 1 = ÷ + = ÷ + = a a d n a a n PROBLEM Find the general term an, of the arithmetic sequence -8,-3,2... ( ) 13 5 8 5 5 5 ) 1 ( 8 ) 1 ( 1 ÷ = ÷ ÷ = ÷ + ÷ = ÷ + = n a n a n a d n a a n n n n 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? ( ) ( ) 5 6 . 0 12 . 0 6 . 3 28 . 0 2 . 4 16 . 0 15 24 . 0 15 28 . 0 16 . 0 = = = ÷ + = ÷ + x x x x x x PROBLEM Peanut and nut mixture contains 40% peanut. How much additional peanut mixture should be added to produce 8 lbs. of 50% peanut? ( ) ( ) ( ) lbs x x x x x x 33 . 1 8 . 0 6 . 0 4 4 . 0 2 . 3 8 50 . 0 1 8 40 . 0 = = = + ÷ = + ÷ + = x 15-x 15 16% 28% 24% + = 8-x x 8 40% 100% 50% 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? mph V V V t t d u 2 ) ( 67 . 1 4 5 4 5 67 . 1 hours 1.67 minutes 40 and hour 1 ÷ = = + + ÷ = + = PROBLEM The third term of a geometric progression is 32 and the fifth is 128. Find the first term and the common ratio. 1 3 1 1 3 1 3 1 1 32 ÷ ÷ ÷ = = = r a r a a r a a n n 59 . 1 70 . 12 128 1 1 6 1 1 6 1 6 = = = = ÷ ÷ r a r a r a a 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. min 45 min 30 1 18 1 15 1 1 18 1 1 1 2 2 2 2 1 = = = | | . | \ | + + = | | . | \ | + t t t t t t 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? 1 2 2 ÷ = = ÷ = D T D J J D ( ) 18 44 1 2 44 = = ÷ + + = + + D D D D T J D 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. 1 2 2 65 . 1 25 . 0 10 . 0 05 . 0 + = = ÷ = = + + N Q N D D N Q D N ( ) 5 1 4 4 65 . 1 1 25 . 0 2 10 . 0 05 . 0 = + = = = + + | . | \ | + Q Q N N N N PROBLEM The sum of two positive numbers is 35. There product is 304. What is the smallest number? ( ) ( )( ) 0 16 19 0 304 35 304 35 304 35 NUMBER LARGER 35 NUMBER SMALL 2 2 = ÷ ÷ = + ÷ = ÷ = ÷ = ÷ = x x x x x x x x x x 16 NUMBER SMALL 16 19 = = = x x 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. ( ) 95 . 24 56 . 21 80 . 0 08 . 0 80 . 0 PRICE BOOK PRICE SELLING 80 . 0 = = + = = x x x x x 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? ( ) ( ) ( ) ( ) 2 . 63 25 . 0 20 . 0 100 20 . 0 99 20 . 0 97 15 . 0 100 90 = + + + + = x x 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? 12 1 36 3 6 3 6 1 = = · = P 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? ( ) % 74 100 85 . 0 87 . 0 = = E x E PROBLEM How many ways can you position six person in circular table? ( ) ( ) ways C C n C 120 ! 1 6 ! 1 = ÷ = ÷ = PROBLEM How many permutation can you get in a letter word TENNESSEE? ways P P 3780 ! 4 ! 2 ! 2 ! 9 = = PROBLEM How many ways can you arrange the letters in a word “MONDAY” given that the first letter is a vowel. ways W W 240 1 2 3 4 5 2 = · · · · · = PROBLEM How many ways can you arrange the word “MONDAY” wherein the 4 letters are taken at a time? ways P 360 4 6 = 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 PROBLEM What is the median of the sequence 5 6 8 12 13 15? 10 2 12 8 = + = MEDIAN PROBLEM There are 7 paintings in a museum with 3 vacant slots. How many ways can you arrange the paintings in the given slot? ways P 210 3 7 = PROBLEM The 1 st and the 10 th term of a geometric progression is 1 and 4. Find the 17 th term. ( ) 9 1 1 10 1 1 4 1 4 = = = ÷ ÷ r r r a a n n 758 . 11 4 1 17 1 17 9 1 17 = | . | \ | = ÷ a a PROBLEM What is the sum of the series 9 1 3 1 1 + + ? 3 1 1 3 1 1 2 = = = r r a a r 2 3 3 1 1 1 1 1 = ÷ = ÷ = S r a S PROBLEM What is the sum of the first 50 terms of the series 325 160 85 10 + + + ? ( ) ( ) 92375 2 75 ) 1 50 ( ) 10 ( 2 50 2 ) 1 ( 2 75 10 85 1 1 2 = ÷ + = ÷ + = = ÷ = ÷ = S d n a n S a a d PROBLEM The special way in proving hypothesis and rank against the most basic tools in the mathematicians toolbox. - MATHEMATICAL INDUCTION PROBLEM What is i i 2 4 8 7 ÷ + ? i 10 23 5 3 + = 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? ( ) 33600 18000 150000 6 11 calculator of number - = = + ÷ x x x x PROBLEM If a man buys gasoline: 10L for 11.50 12L for 12.01 18L for 11.78 Find the average. ( ) ( ) ( ) 78 . 11 18 12 10 78 . 11 18 01 . 12 12 5 . 11 10 = + + + + = Average Average 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 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 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 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-five- centavo coins, and 4 ten-centavo coins to 10 boys. In how many ways can the boys profit if each have a coin? Solution: 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: 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 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 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 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 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 15 hours PROBLEM PROBLEM cos ( α – β ) – cos ( α + β ) = sin α sin β PROBLEM = PROBLEM A polar equation r = a represents circle with center at the origin PROBLEM A polar equation represents lemniscate PROBLEM A polar equation r = a csc θ represents Horizontal line PROBLEM A polar equation θ = a represents a Line through the origin 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. 3 3 6 4 75 . 1 75 . 1 2 1 = ÷ + = + = = x x x t t t hrs t T T PROBLEM According to Hooke’s law you can stretch a _____ to F=4.2x. if 14 7 s s F , Find the Value of x. 3 10 3 5 2 . 4 14 2 . 4 2 . 4 2 . 4 7 to x x = s s PROBLEM A 16-team… The bowling officials has P8000 budget for the prizes. If the prize given to the 16 th place is P275, find the prize given to the first place. 725 ) 275 ( 2 16 8000 ) ( 2 1 1 = + = + = S a a a n S n PROBLEM Factor (81-x 4 ) completely ) 3 )( 3 )( 9 ( ) 9 )( 9 ( 2 2 2 x x x x 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. 2000 8 . 0 1 400 = ÷ = S S PROBLEM Find the sum of a 20 term G.P. if the first term is 1 and r=2. 1048575 1 2 ) 1 2 ( 1 1 ) 1 ( 20 1 = ÷ ÷ = ÷ ÷ = S S r r a S n PROBLEM Find the 16 th term of the expansion (x-2) 20 5 1 16 1 16 20 1 16 20 508035072 ) 2 ( x x C = = ÷ ÷ ÷ ÷ PROBLEM There are 15 contestants in a contest. In how many ways the three winners are to be chosen? 15 C 3 = 455 ways 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: 52 C 3 = 22,100 Successful outcome: 13 C 3 = 286 P = successful outcome / total outcome = 286/22,100 = 0.0129 PROBLEM Answer: cot β PROBLEM 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. S = = 11 A = = 16.248 cm 2 A = r = = 2.954 cm PROBLEM Answer: tan 3β PROBLEM Answer: 2cosαcosβ PROBLEM Tickets of a concert are consecutively numbered. Manny sold tickets numbered 168-358. How many did he sell? Solution: a n = a n + (n-1)d 358 = 168 + (n-1)(1) a n = 359 tickets PROBLEM Probability of getting exactly 3 tails in 5 throws. P = 5C3(1/2) 3 (1/2) 5-3 P=5/16 PROBLEM A standard 52-deck card, how many 5-hand card will have 3aces and 2kings? C 1 = 4C3 = 4 C 1 = 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 = 2 n – 1 C = 2 5 – 1 C = 31