Math word questions

One cup of butter is enough for 8 of Roger's cookies, and one cup of sugar is enough for 5 of Roger's cookies. If he used 15 more cups of sugar than butter, how many cookies did he make?

200 cookies

An amusement park has 12 different major rides. A coupon gives its holder access to any 3 of these major rides for free. How many sets of three rides are possible?

220 Steps: Combination question so nCr. 12c3 order doesn't matter, so: FCR / r factorial 12x11x10 / 3! 12x11x10 / 3x2x1 = 220 NOTE that this is answer the answer for 12c9.

How many factors does 21,600 have?

1. prime factorization = 2^5 * 3^3 * 5^2 2. 5, 3, 2 3. 6, 4, 3 4. 6 * 4 * 3 = 72

A classical concert will consist of two overtures in the first half and one symphony in the second half. A symphony and two overtures determine a program. The conductor can choose the program from 10 overtures and from 6 symphonies. How many programs are possible?

10 overtures AND 6 symphonies. 10 overtures --> 10C2 (bc order doesn't matter) 10*9/2 = 45 possibilities 6 symphonies --> 6 possibilities "and" means multiply so: 45 * 6 = 270

What is the sum of all integers from 45 to 155 inclusive?

11,100 The sum of an even spaced set of numbers = N (a1 + an) / 2 = 111(45+155)/2 = 111(200)/2 = 11,100

Medium: If x is the greatest common divisor of 90 and 18 and y is the least common multiple of 51 and 34, then x + y =

120

For adult males, heights are normally distributed with a mean of 175 cm and a standard deviation of 10 cm. What percent of adult males have a height less than 185?

185 is one SD above the mean. This would include everyone between the mean (M) and one SD above the mean (M+S) as well as everyone below M. M to (M+S) --> 34% Below M --> 50% answer: 84%

A parabola with a vertex (2,1) has a y-intercept of 9. What is the x-coordinate of the other point on the parabola with a y-coordinate of 9.

4 (0,9) and (4,9) The x-coordinate of the vertex tells us the line of symmetry is x=2. The y-intercept is 2 to the left of the line of symmetry so the other y-coordinate is 2 to the right of the line of symmetry.

If P and Q are integers, and (4P +Q) is odd, what must be true?

4P must be even --> even + Q = odd Q must be odd OR: Try four cases: where 1 for odd numbers and 2 for even numbers The cases that give odd answers have Q odd. We can't draw a conclusion about P.

What is the smaller positive integer that, when divided by 12, has a remainder of 5?

5 NOT 17

A certain school district has specified 5 different novels and 4 different non-fiction books from which teachers can can choose to assemble a summer reading list. Each summer reading list must have exactly 3 novels and 2 non-fiction books. How many different summer reading lists are possible?

60 Combinations! nCr for both novels and non-fiction books. Order doesn't matter, so we need to divide the FCP by r! (r factorial) 5C3 = 5x4x3 / 3! = 5x4x3 / 3x2x1 (simplify) = 10 4C2 = 4x3 / 2! = 4x3 / 2x1 (simplify) = 6 Since any collection of novels can go with any collection of non-fiction books, we can multiply these two together: 10 x 6 = 60 Can also use Pascal's triangle.

If K is the least positive integer that is divisible by every integer from 1 to 8 inclusive, then K =

840 K has to include the prime factors for each number 1-8 1 2 3 4 = 2 x 2 5 6 = 2 x 3 7 8 = 2 x 2 x 2 K = 1 x 2 x 3 x 2 x 5 x 7 x 2

Medium: If x is an integer, is y an integer? (DATA SUFFICIENCY) 1. (4x+4y)/2 = 6 2. (3x+6y)/3 = 5

A. Statement I ALONE is sufficient to answer the question, but statement 2 alone is NOT sufficient.

Consider the positive integers from 1-100. If n is a number in that set, then for how many numbers n is it true that |x-30| > 20

Absolute value question. New origin = 30 The distance between x and 30 is greater than 20. (x > 50) OR (x < 10) Answer has to include all numbers from 1-9 AND 51-100. 9 + 50 = 59

Suppose we start with 5 liters of a 30% HCl solution. How much water must we add to create a 20% solution?

Amount of HCl = (5)*(.3) = 1.5 liters HCl 5-1.5 = 3.5 liters water new solution: 0.20 = 1.5 liters / (amount of new solution) 1.5 / .2 = 7.5 liters 7.5 - 5 = 2.5 liters added

A bathroom floor is a rectangle, 2 meters by 3 meters. It is to be tiled with square tiles, 4 centimeters on a side. How many tiles will it take to fill the floor? (1 meter = 100 cm)

Area of floor = 6 m^2 * (100cm/1m)^2 = 60,000cm^2 The area of the tile is 4x4. So we want to divide the area of the bathroom by the area of the tiles. (two 4s) 60,000/4/4 = 15,000/4 = 3750

14, 23, 31, 40, 49, 59... Find the 41st term of this sequence.

Arithmetic sequence where 9 is being added to each term. a1 = 14 and d=9 an = 14 + (n-1)*9 n(41)=14+(40)*9 =14+360 =374 is the 41st term on the list

If K, (K+200), (K+150), and 15K are all multiples of P, then P could equal which of the following? A. 20 B. 25 C. 75 D. 100 E. 150

B. 25 Rule: If K and X are multiples of r, then (K+X),(K-X), and K*X must also be multiples of P. NOT K/X so in this case, 200 and 350 are both multiples of P and we can subtract them (350-200 = 150) to get a new multiple of P. 200-150 = 50 (another multiple of P) The only number on this list that is a FACTOR of 50 (the smallest multiple we have) is 25.

If n is an integer greater than 50, then the expression (n^2 - 2n)(n+1)(n-1) MUST be divisible by which of the following? I. 8 II. 12 III. 18 A. I only B. II only C. I and II only D. II and III only E. I, II, and III only

C. Factor an n out of the first term --> n (n-2) notice that we have a product of four consecutive integers (n-2)(n-1)n(n+1) There are four terms --> we know there must be two odd and two even integers 8 There must be a multiple of 4 (n=4) and another even number. Anything divisible by a multiple of 4 and another even number must be divisible by 8 12 There must be a multiple of 4 and 3 (3 is less than four). 12 is divisible by 4 and 3. 18 18 is 2 * 3 * 3. We don't know if we'll have two multiples of 3 in this set. Could be change, but can't guarantee.

Harriet has six novels she wants to read, one of which is Emma by Jane Austen. She plans to create a reading list of four of these novels for an upcoming trip, and different orders count as different lists. How many reading lists are possible if Emma must be on the list?

Clue: must include one item --> Count what we DON'T want. Total possible reading lists = (6)(5)(4)(3) = 360 Q (possible arrangements if Emma is NOT included on the list) --> (5)(4)(3)(2) = 120 n! - Q = R 360 - 120 = 240 240 possible lists if Emma must be on the list.

A librarian has 5 identical copies of book A, two identical copies of book B, and a single copy of book C. In how many distinct orders can be arrange these 8 books on a shelf?

Counting with identical items. N = n! / (a!)(b!) N = 8! / (5!)(2!) 8*7*6*5*4*3*2 2*5*4*3*2*1 Cancel to get: 8*7*6 / 2 = 168

An amusement park has 12 different major rides. A coupon gives its holders access to any 3 of these ride for free. How many sets of three rides are possible?

Order doesn't matter! Start with FCP and divide by 3! (12*11*10)/3! (12*11*10)/(6) = 220

Suppose P, Q, R, and S are integers. If P is even and (P*Q + R*S) is an odd integer, then which of the following must be true? I. Q is odd II. R is odd III. S is odd A. I only B. I and II only C. I and III only D. II and III only E. I, II, and III

D: II and III only P is even --> P and Q always even even + R*S = odd R*S must be odd R and S both have to be odd

In a certain game, in Phase 1 you flip one coin as many as three times. If you flip three tails, you lose. As soon as you get your first head, you advance to Phase 2. In Phase 2, you roll a six-sided die once. If you roll a 6, you win. For any other roll, you lose. What is the probability of winning?

Die roll is independent of the coin roll so the Phases are independent of each other. P(win) = P(winning phase 1 AND rolling a 6 in phase 2) Two phases are independent so we can multiply these: P(p1 win) * P(p2 win) P(p2 win) = 1/6 Flipped AT LEAST one heads coin will win p1 so finding the compliment (rolling all three tails) can help. P(win p1) = 1 - P(TTT) P(TTT) = (1/2)^3 = (1/8) P(win p1) = 1-1/8 = 7/8 P(win it all) = (7/8)*(1/6) = 7/48

In a certain summer school program, there are five periods in the day. Each student takes English, Math, History, Science, and Science Lab. In how many orders can a student schedule be arranged, given that Science Lab must immediately follow the Science Class?

Draw out five options. S can be in the first four slots, not the 5th, because SL needs to follow it. S=4 SL then must be directly after it. SL =1 The non-restrictive classes: English can then go in 3 slots. E=3 M=2 H=1 N = 4*1*3*2*1 =24 24 possible schedules

For a formal dinner, guests have the choice of one of 4 salads, one of 5 appetizers, one of 12 entrees, and one of 4 desserts. How many different meals are possible?

FCP key word "and" means multiply. N = 4*5*12*4 = 960

Suppose we have six different books that we will place on a shelf. In how many different orders can we place these six books?

FCP N = 6*5*4*3*2*1 = 720 We have 6 options for the first book, then 5 for the second book, and so on.

A small division of a company, with 25 employees, will choose a three-person steering committee consisting of a facilitator, a union rep, and a secretary. How many different possible steering committees could be chosen?

FCP key word "and" means multiply Facilitator --> 25 options Union rep --> 24 options Secretary --> 23 options 25*24*23 = 13,800

What is the sum of all the multiples of 20 from 160 to 840 inclusive?

First need to figure out how many terms there are. 160 = 20 * 8 840 = 20 * 42 Inclusive counting so 35 total terms. The number of pairs = 35/2 = 17.5 Sum = (N/2)(a1+an) = 17.5(160+840) =17.5*1,000 = 17,500

From a set of ten different items, Lisa is going to select three as a gift for someone. How many different three items can she pick?

Order doesn't matter! Start with FCP: 10*9*8 but divide by 3! = 3*2 10*9*8 / 3*2 = 120

For any number x, which of the following must be greater than x? I. x+2 II. 2x III. x^x

I and III only. 2x --> if x is negative x^2 --> if x is between 1 and 0

Which of the following could be true of at least some of the terms in the sequence defined by bn = (2n-1)(2n+3) I. divisible by 2 II. divisible by 3 III. divisible by 5

II and III. Start by plugging 1 into b(1). You get 5. Plug in 2 and we get 21 (divisible by 3). (2n-1) is just all odd integers. 2n+3 is always odd too. (odd)*(odd) = always odd. NOT divisible by 2.

A director needs to select 3 certified Safety Monitors for a committee. The director believed there were only 6 such certified Safety Monitors and calculated N1, the number of possible committees. Two new employees completed training and became certified Safety Monitors. If the new number of possible committees, drawing from all 8 Safety Monitors, is N2, what is the ratio of N2 to N1?

N1 = 6C3 (order doesn't matter) N1 = (6*5*4)/3! = 5*4 = 20 N2 = 8C3 (order doesn't matter) N2 = (8*7*6)/3! = 8*7 = 56 N2/N1 = 56/20 = 14/5

Cars P&Q are approaching each other on the same highway. Car P is moving at 49 MPH and car Q is moving at 61 MPH. At 2:00 PM, they are approaching each other and 121 miles apart. Eventually they pass each other. At what clock time are they moving away from each other and 44 miles apart?

Opposite directions --> ADD speeds D = (Rgap)*T Rgap = 49+61 = 110 121+44 = (110)*(T) 165/110 = T 15/10 = 3/2 = 1.5 hours ANSWER = 3:30 PM

Contract negotiations opened on the morning of March 20th, continued every day without a break, and ended late in the evening of May 10th. For how many calendar days were contract negotiations in session?

Inclusive counting example In March, from the 20th to the 31th: 31-21+1 = 12 days April: 30 days May: 10 days 12+30+10 = 52 days

Let S be the set of all positive integers that, when divided by 8, have a remainder of 5. What is the 76th term in this set?

Know this means it is an arithmetic sequence. When divided by one number, we get the same remainder every time. an = 5 +(n-1)*8 a(76) = 5 + (75)*8 = 5 + 150*4 = 5 + 600 = 605 is the 76th term in this set

The children Al, Betty, Check, Dahlia, Ed, Fran, and George will sit in seven adjacent chairs. Dahlia must be in the middle chair, and George must be next to Dahlia. In how many orders can they be arranged?

Most restrictive case: Dahlia must be in the middle chair. D= 1 Next restrictive case: George must sit next to Dahlia. G=2 For the unrestricted children: Al can sit in 5 places. 5 B = 4 C = 3 D = 2 F = 1 N = 1*2*5*4*3*2*1 = 240 So there are 240 different arrangements that satisfy these restrictions.

If P is an odd integer and (P^2 + Q*R) is an even integer, then which of the following must be true? A. either Q or R is an odd integer B. either Q or R is an even integer C. both Q and R are odd integers D. both D and R are even integers E. nothing can be concluded

P^2 is odd, so Q*R must be odd. But we don't know that Q and R are even integers. eg. Q = 2 and R = 5/2 E: nothing can be concluded

A committee of three will be selected from a group of eight employees, including Alice and Bob. What is the probability that the chosen committee of three includes Alice and not Bob?

Probability / Counting question. The number of 3 groups we can make from 8 employees = 8C3 = (8)(7)(6) / 3! = 56 So the 56 will be the denominator of the probability. For the numerator, count the "success" options with A but not B. A _ _ if Alice is on the committee and it doesn't include Bob, then there are 6 other options for the 2 remaining spots. 6C2 = (6)(5)/2 = 15 Probability = 15/56

Supper we start with unlimited supplies of a 20% HSO solution and of a 50% HSO solution. We combine X liters of the first and Y liters of the second to produce 7 liters of a 40% HSO solution. What does X equal? Mixture question!

Question about adding two different starting solutions. We need two different equations: 1. Total equation. X+Y = 7 liters 2. Amount of solute equation. final solution equation: .4 = HSO / 7 HSO in final solution = 2.8 liters EQUATION = (.2X) + (.5Y) = 2.8 liters --> solve for Y to get one variable --> Y = 7-X (.2X) + (.5*(7-X)) = 2.8 2X + 35 - 5x = 28 -3X = -7 X = 7/3

A sequence is defined by s(n) = (s (subscript n-1) -1) * (s (subscript n-2) and s(1) = 2 and s(2) = 3 Find the value of b(6)

Recursive seuqnce! Can't jump to find b(6), must find the values before it. Think of this as the nth value is the value before it minus 1 times the value of the term 2 before it. The third term is the value of the term before it minus 1, times the term two before it.

What is the sum of all the multiples of 5 that are greater than 100 and less than 200.

SUM5 = (N/2)(a1+an) 19 terms total (not inclusive of 100 and 200) Starts with 105 and ends with 195. 9.5 = the number of pairs SUM = (9.5)(300) =2700+150 = 2,850

Six children will sit in a row of six chairs, but Jackie and Marilyn cannot be seated next to each other. How many arrangements are possible?

See the word "not" think of counting the arrangements without the restriction. total # of arrangements = 6! = 720 # of arrangement NOT obeying the restriction --> there's 10 different options if Jackie and Marilyn are sitting next two each other. That leaves four chairs. 10*4! = 10*24 = 240 n! - Q = R 720 - 240 = 480

At a certain school of 200 students, the students can study French, Spanish, both, or neither. Just as many students study neither as both. One quarter of those who study spanish also study french. The total number who study french is 10 fewer than those who study Spanish only.

Sketch diagram with sections A + B + C + D = total French + Both + Spanish + Neither = Total B=D (1/4)(spanish studiers) = middle group (1/4)(B+C) = B B+C = 4B C = 3B (A+B) = C-10 (A+B)=3B-10 A=2B-10 Now everything is in terms of B. Can add to equation (2B-10) + B + 3B + B = 200 7B=210 B=30

If 7K is a positive integer and if Sqrt(K) > K, then is K an integer? (A) Yes (B) No (C) Can't be determined

Sqrt sign (radical) actually appears in the question so we know that we only want the positive square root, not the negative. (irrelevant here but still) (B) No. This is the rule if 0 < K < 1. Above 1, taking the square root will make a smaller number. BUT between 0 and 1, taking the square root of a number will make a bigger number.

Solve for k: 3/ [1-[8/(7+k)]] = 15

Substitute A for the entire denominator A = 1/5 Set 1/5 = 1 - [8/(7+k)] Set the second part equal to B B = 4/5 k = 3

Let t be a sequence of the form a(n) = a1 + d*(n-1). If a(3) = 17 and a(19) = 65, find a(10).

The first formula is used for arithmetic sequences.

N = 135 is the lowest number in a set of 41 consecutive multiples of 5. What is the difference between the lowest and highest numbers in the set?

Think in terms of a sequence 1st = 135 2nd = 135 + 5 3rd = 135 + 5*2 4th = 135 + 5*3 ... nth = 135 + 5*(n-1) 41st = 135 + 5*40 Difference between the highest and the lowest = (135 + 5*40) - 135 = 5*40

A librarian has a set of ten book, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf. How many different arrangements of the ten books are possible?

Think of the four Lincoln books as one big book. In how many ways can we put these seven books in order on a shelf? 7! That method of counting gives us a position for the Lincoln books among the other six books, but we can still put the four Lincoln books in any order. There are 4! orders for the Lincoln books. Total arrangements = N = (4!)*(7!)

There are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students are in both chorus and Italian, 45 students in both chorus and baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how much students are in none of the three activities?

Use the three-way venn diagram. Draw it! 130

A swimming pool has the capacity of a rectangular solid (5m)*(8m) on the surface and (2m) deep. The pool is filled with grains of sand. If each grain of sand is (1mm)^3, and 1m=1000mm, how many grains of sand are in the pool?

Volume of the pool = 5*8*2 = 80m^3 Question is how many cubic millimeters are in 80 cubic meters. (80m^3)*(1000mm/1m)^3 (80)* (10^3)^3 80*(10^9) =80,000,000,000 (billion) or 8*(10^10)

A parabola has an x-intercept at (-4,0). If the vertex is at (2,5), find the other x-intercept.

We know the vertex, so we know the line of symmetry is x=2. The point (-4,0) is six units to the left of the line of symmetry. Its reflection has to be six units to the right of the line of symmetry (8,0).

What is the units digit of 57^123?

only the 7^x matters. look for pattern: 7^1 = 7 7^2 = __9 (units digit of 9) 7^3 = __3 (9*7 = 63 so units digit of 3) 7^4 = __1 (3*7 = 21 so units digit of 1) 7^4 = _7 --> starts over! period of 4 so for 7 to the power of any multiple of 4, the units digit is 1 and we can use that to find 123. 120 is a multiple of 4 so 7^120= __1 7^121= __7 7^122= __9 7^123= __3

Suppose we roll one fair six-sided die eight times. What is the probability that we roll at least one six?

probability. notice "at least one" --> use compliment rule. P(at least one six) = 1 - (probability of 0 sixes) The probability of not rolling a six in one roll is (5/6). The probability of not getting a single six on eight rolls is (5/6)^8 P(at least one six) = 1 - (5/6)^8

A car and a truck are moving in the same direction on the same highway. The truck is moving at 50 MPH and the car is traveling at a constant speed. At 3:00 PM, the car is 30 miles behind the truck and at 4:30 PM, the car overtakes and passes the truck. What is the speed of the car?

same direction --> SUBTRACT speeds D = (Rgap)*T 30 = (Rgap)*(1.5) 20 MPH = rate the gap is shrinking Rcar - Rtruck = R gap Rcar - 50 = 20 Rcar = 70 MPH

Solve for x: (x^2+1)^2 - 15(x^2+1) = -50

see that it's a quadratic if you move the 50 over substitute u for the expression x = +/-2 x = +/-3

bn = [b(subscript n-1) - 1] ^2 + 3 and b1 = 1. Find the value of b4.

think Recursive sequence! So can't just sump to b(5). need to plug in "seed" value of b1 to get b2 and use b2 to get b3 answer: 39

What is the remainder when 43^86 is divided by 5?

units digit question!! 3^1 = units digit 3 3^2 = 3*3 = units digit 9 3^3 = 3*9 = units digit _7 3^4 = 3*7 = units digit _1 then just repeats (repeating pattern 4) 43^84 must be _1 43^85 = _3 43^86 = _9

Point J (5,2) and K (-2,-5) are two vertices of an isosceles triangle. If L is the third vertex and has a y-coordinate of 4, what is the x-coordinate of L?

x=-4 (-4,4) J and K are reflections over the line y=-x so any point on that line would be equidistant from both J and K.

If y=5+x and y=12-x, and if y^2=x^2+K equals which of the following? A) 17 B) 25 C) 60 D) 119

y= 5+x --> y-x=5 y=12-x --> y+x=12 K = y^2 - x^2 = (y+x)(y-x) = 5*12 = 60