Math problems

Let m = odd integer, n = even integer. List all the following that must be even. A) nm^2 B) n+m C) 3n - 2m D) n - m E) nm

A, C, E A number is even only when it has a factor or 2.

A contest has 8 competitors. Judges award prizes for gold, silver and bronze medals, with no ties. 1) How many ways can the judges award prizes? 2) How many different groups of 3 can get prizes?

#ways to order all 8 contestants = n! = 8! #ways to order first 3 prizes = n!/(n-k)! = 8!/(8-3)! = 8!/5! = 8x7x6 = 336 #ways that a group of 3 get prizes (order doesn't matter) = n!/(n-k)!k! = 336/3! = 336/6 = 56

The number of strawberries 100 people ate per year is normally distributed with a mean of 29 strawberries, and a standard deviation of 4 strawberries eaten by each person. How many of the surveyed people ate more than 25 strawberries during the course of the year? A) 16 B) 48 C) 50 D) 68 E)84

%ppl eating more than mean 29 strawberries = 50% 68% of people fall within +/-1SD of the mean, so 1SD=34%. %ppl eating more than 25 strawberries = mean + 1SD = 50+34 = 84%. Answer E.

f(x)=x-1 What happens when x is negative? Is (-x) -1 = -x-1? OR -(x-1) = -x+1?

(-x) -1 = -x-1

If (1/a)<(1/b)<0, then which of the following inequalities are true? A) a+b < ab B) |a| > |b| C) a < b D) (b/a) + (a/b) > 2

(1/a)<(1/b)<0 (1/a) and (1/b) must both be negative. Pick numbers: -1/2 < -1/4. If a=-2 and b=-4, then b<a. A) If a<0, b<0, then a+b<0 and ab>0, thus a+b<ab is TRUE. B) lal=l-2l=2, and lbl=4, so lbl>lal ; thus lal>lbl is false. C) a=-2 and b=-4, so b<a ; thus a<b is false. D) a^2+b^2 > 2ab, so (b/a)+(a/b)>2 is TRUE.

The lengths of two sides of a triangle are 22 and 11. A) 14 B) The length of the third side Is A or B bigger, C equal, D undetermined?

*The sum of the two shorter sides of a triangle cannot be less than the longest side, or else the triangle is not closed. Imagine 22 was the hypotenuse and base. Side x is greatest when the height is greatest, meaning a right triangle. 22^2 - 11^2 = 19^2, which is greater than 14. As the height shrinks, and the triangle becomes squatter/flatter. Side 11 is fixed, but Side X shrinks. The smallest that Side X can be is slightly > 22-11 = 11. Therefore 11< Side X < 19. Answer D.

n>0. The remainder when n is divided by 3 is 1, and the remainder when n+1 is divided by 2 is 1. A) The remainder when n-1 is divided by 6. B) 3 Is A or B bigger, C equal, D undetermined?

1) (n-1)/3 is an integer, so n could = 4, 7, 10, 13, 16, ... 2) (n+1-1)/2 = n/2, so n must be even. Revise 1) to include n = 4, 10, 16, ... A) What is remainder when (n-1)/6? If n=4, (4-1)/6=3/6=3. If n=10, (10-1)/6=9/6=1 3/6. So A) 3 and B) 3, therefore C is correct.

If [x^-2]([-y^2]x+1)=48, and if y^2=2, then what is a possible value for x^2? A. 34/70 B. 0.7655 C. 2/128 D. 0.9822 E. 12/63

1/x^2(-2x+1)=48 -2x+1=48x^2 0=48x^2 + 2x -1 = (6x+1)(8x-1) x=-1/6 or 1/8 x^2=1/36 or 1/64 C) 1/62 x 2/2 = 2/128

A line has end points -2 and 10. What is the midpoint coordinate of the line?

10 - (-2) = 12. Midpoint = 12/2 = 6 Midpoint coordinate = 10 - 6 = 4

What percent is 0.144?

14.4%

5 < sqrt(n) < 6 A) n^2 B) 500

25 < n < 36 625 < n^2 < 1296 A is larger.

A) (41)^2 - (21)^2 B) (41-21)^2 Is A or B bigger, C equal, D undetermined?

A) a^2 - b^2 = (a+b)(a-b) 41^2 - 21^2 - (41+21)(41-21)=62x20 B) (41-21)^2 = 20x20 A is bigger. No need to multiply out.

A) sqrt(2) B) cuberoot(3) Is A or B bigger, C equal, D undetermined?

A) sqrt(2) and B) cuberoot(3) Square both sides: A) 2 and B) 3^(2/3) Cube both sides: A) 8 and B) 3^2 =9 B is bigger!

The length of rectangle B is 10% less than the length of rectangle A. The width of rectangle B is 10% greater than the width of rectangle A. A) Area of A B) Area of B

A: l x w = area B: length of rectangle B is 10% less width of rectangle B is 10% greater B: 0.9l x 1.1w = 0.99area A is greater.

Alan is 3 times as old as Betsy. If Alan were 11 years older and Betsy were 1 year older, then Alan would be 5 times as old as Betsy. How old is Alan right now?

A=3B (A+11)=5(B+1) A = 9 years If having trouble writing an equation, try numbers: B=2, A=3B=6. A+11=17 B=3, A=9. A+11=20 B=4, A=12. A+11=23 B=5, A=15. A+11=26 B=6, A=18.. A+11=29 Which of these is 5(B+1)=A+11?

Given five letters, A B C D E, how many ways can you select 3 of five letters?

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE 10 ways Number of ways to order all five: n! = 5! Number of ways to pick and order 3 for 3 positions: n!/(n-k)! = 5!/2! = 5x4x3x2!/2! = 5x4x3 = 60 Number of ways to pick any 3 for 3 positions: n!/(n-k)! k! = 5!/2!3! = 5x4x3!/2x1x3! = 20/2 = 10

The distance from (b,2) to (-2,2) is 3. A) b B) 0

Always be careful when 0 is an option. Check -ve. On a graph, b is somewhere on the line y=2. b could be 1 or -5. Ans D.

A square is inscribed in a circle. If the area of the square is 4, what is the radius of the circle?

Area of square = 4, therefore side =2. The diagonal of the square = diameter of the circle. Also, it is a right triangle with side ratio 1:1:sqrt2, therefore 2:2:2sqrt2 and diameter is 2sqrt2, therefore radius = sqrt2.

Which of the following inequalities have at lease one positive and at least one negative solution for x? Indicate all posibilities. A) (5/3)x < x B) x^3 < x C) x-6 < x-7

B

Triangle ABC has points C=(0,0), B=(6,0), and A=(x,y). The area of ABC is 18. A) y B) 6 Is A or B bigger, C equal, D undetermined?

Base = Bx-Cx = 6-0=6 Area=1/2 x base x height = 18 Height = 36/6 = 6 = l Ay - x-axis I = lAy - 0l = l6l y could be -6 or 6. D undetermined.

X@Y = X^2 - Y A) (3 @ 2) @ -1 B) 50

Be careful with negative and subtraction! X@Y = X^2 - Y (3 @ 2) @ -1 3^2 - 2 = 9 - 2 = 7 7^2 - (-1) = 49 + 1 = 50 C.

How many 8" squares will it take to fill a rectangular floor measuring 18 feet x 24 feet?

Convert both lengths into inches. It is not enough to do 18 x 24 x 12 because area is length^2. 18x12 = 216" 24x12 = 288" Area = 216 x 288 = 62,208 # of 8" tiles needed = 62,208/64 = 972 tiles

x cannot equal y. x@y=(x+y)/(x-y) If q<0<p: A) p@q B) q@p Is A or B bigger, C equal, D undetermined?

D If lql = lpl, then both p@q and q@p = 0, meaning equal. The numerators are the same. Examine the denominators. Since q<0<p, A) p-q >0, and B) q-p <0. Not equal. A would be bigger than B.

n is an integer greater than 0. Option A) The number of different prime factors of 9n. Option B) The number of different prime factors of 8n. Is A or B bigger, C equal, D undetermined?

D) it depends what n is. If n=2: 9n=18 -> (1, 18; 2, 9; 3, 6) = 2,3 8n=16 -> (1, 16; 2, 8; 4, 4) = 2 If n=3: 9n=27 -> (1, 27; 3, 9) = 3 8n=24 -> (1, 24; 2, 12; 3, 8; 4, 6) = 2,3

If x is an integer, which of the following must be even? A) x^2 -x -1 B) x^2 -4x +6 C) x^2 -5x +5 D) x^2 +3x +8 E) x^2 + 2x +10

D) x^2 +3x +8

A triangle ABC has sides AB=4(sqrt2) and AC=7 and Angle A = 45'. What is the length of BC?

Don't assume that because an angle looks like 90' that it is, unless it is marked. Create at right angle by dropping a perpendicular line (BD) between side AC and angle B. Mini triangle ABD is a right 45:45:90 with hypotenuse 4(sqrt2). Without calculation, 1:1:sqrt2, therefore sides AD and BD are 4. Side DC = ADC-AD=7-4=3 Mini triangle BCD has sides 3:4:BC, therefore BC=5.

If n=14x22x39, which is not an integer? A) n/21 B) n/24 C) n/26 D) n/42 E) n/77

Don't multiply out. Find the factors 14: 2x7 22: 2x11 39: 3x13 Determine if the denominators have a common factor: A) n/21: n/7x3 OK B) n/24: n/2x12 NOPE! C) n/26: 2x13 OK D) n/42: 6x7 = 2x3x7 OK E) n/77: 7x11 OK

If it takes three days for 10 workers to build one house, how long will it take 15 workers to build four houses? 15, 10, 8, 6, 4 days?

Don't try to set up ratio or equations. Think logically. For one house, 10 workers need 3 days; 15 workers should be faster: 3x10/15=2 days per house. There are 4 houses, so 2x4=8 days.

If 3.2 = -n + 0.8m and 0.4m = -0.3n + 3.2, then what is the value of -n? a) 2 b) 1 c) 0 d) -1 e) -2

Find -n, NOT n. -n = -2. E, not A.

x>2 A) 3^(x+2) B) 4^x Is A or B bigger, C equal, D undetermined?

For small values of x, A > B. If x=3, then A) 3^5 =243, and B) 4^3 = 64. However as two integers are raised to increasing integer powers, the larger base number will eventually outgrow the smaller base number. If x=10, then A) 3^12 = 531,441, and B) 4^10 = 1,048,576.

k is an integer for which 1/(2^(1-k)) < 1/8. Option A) k Option B) -2 Is A or B bigger, C equal, D undetermined?

NOT 2^(1-k) < 8. Be careful with reciprocals. For 1/(2^(1-k)) to be less than 1/8, the denominator of 2^(1-k) must be greater than 8. Thus: 2^(1-k) > 8 = 2^3 1-k >3 -k > 2 k < -2 therefore B!

X is a non-negative integer. A) 2X B) 3X Is A or B bigger, C equal, D undetermined?

If X>0, then 3X>2X always. But 0 is a non-negative integer! If X=0, 2X=3X=0. Answer D.

A) The area of a rectangle with perimeter 24 B) 20 Is A or B bigger, C equal, D undetermined?

If sides = 11+11+1+1=24, Area = 11. If sides = 10+10+2+2=24, Area = 20. If sides = 8+8+4+4=24, Area = 32 If sides = 6+6+6+6=24, Area = 36. A Square is also a rectangle. Area can be stretched thin, or blown up.

3.7, 4.1, a, 8.5, 9.2, 2a The six numbers above are listed in increasing order. Which of the following values could be the range of the six numbers? Indicate all values. a) 4.0 b) 5.2 c) 7.3 d) 11.6 e) 12.9 f) 14.1

NOT which could be "in" the range, but what IS the range? Range = biggest - smallest = 2a - 3.7 The biggest value for 2a is when a<8.5. 2a<17. Range = 17-3.7 = 13.3. Therefore F) 14.1 is not possible. Anything less than E may be possible. The smallest values for a<4.1, but 2a cannot be less than 9.2. The smallest range possible is when 2a is a little bigger than 9.2. 9.2-3.7 =5.5. Therefore the possible ranges are 5.5 < Range < 13.3, which is C D E.

Germany's Gasoline tax = 20.4%, Income tax = 28.6%. The amount of Germany's gasoline tax revenue was approximately what percent less than the amount of its income tax revenue? a) 10% b) 20% c) 30% d) 40% e) 50%

Not 28.6-20.4 = 8.2% (change in values)/starting value = % 8.2/28.6 = 30% C.

From a box of 10 lightbulbs, you remove 4. How many different sets of 4 lightbulbs could you remove?

Number if all were ordered = 10! Number if first four were picked in order = 10!/(10-4)! = 10!/6! = 10 x 9 x 8 x 7 = 5040 But order doesn't matter = 10!/(10-6)!4! = 5040/(4x3x2x1) = 210 different sets

Let A, B, C, D be events for which P(A or B)=0.6, P(A)=0.2, P(C or D)=0.6, and P(C)=0.5. The events A and B are mutually exclusive, and the events C and D are independent. What is P(D)?

P(A)=0.2 P(B)=P(A or B) - P(A)=0.6-0.2=0.4 P(C)=0.5 P(CuD)=P(C)+P(D)-P(CnD)=0.6 0.5+P(D)-[P(C)xP(D)]=0.6 P(D)[1-0.5]=0.1 P(D)=0.1/0.5=0.2

2x/3 = 2y/5 = 2z/7 z is positive. A) x+y B) z Is A or B bigger, C equal, D undetermined?

Pick numbers. If z = 7, then 2z/7 = 2. Then 2x/3=2, thus x=3. Likewise y=5. A) x+y = 3+5 = 8 B) z = 7 Thus A is bigger.

A circle of area of 9Π cm2 has a square inscribed inside of it, so that each of the four corners of the square lie on the circle. What is the area of the square?

Remember Diameter = 2 radius = 6 Ans 18cm^2

S and T are positive and follow S=k/T, where k is a constant. If S increases by 50%, then the value of T decreases by what percent?

ST=k E.g. S=10, T=1, k=10 If S=15, k=10, then T=k/S=10/15=2/3 NOT 66.7%! By what PERCENT has T decreased? T'=1, T''=2/3. % decrease = (T'-T")/T'=1/3 33.3%!

Pat invested a total of $3000. Part of it was invested in an account that had 10% simple annual interest, and the remainder was in another account that paid 8% simple annual interest. If the interest earned at the end of the first year from these investments was $256, how much did Pat invest at 10% and how much at 8%?

Simultaneous equations: A+B=3000 0.1A + 0.08B = 256 --> A+0.8B=2560 B-0.8B = 0.2B = 3000-2560 = 440 --> B = 2200 A = 800

x<y<0 A) x/y B) y/x Is A or B bigger, C equal, D undetermined?

Since both x and y are negative, their ratios will be positive. Since x is more negative (i.e. bigger) than y, then A) x/y will always >1, whereas B) y/x will be a fraction <1. Check with negative fractions! Answer A

Two sides of a triangle T are 3 and 4. A) The third size B) 8 Is A or B bigger, C equal, D undetermined?

The largest side cannot be more than the sum of the two smaller sides, otherwise it would be incomplete. The largest that Side X can be < 3+4 = 7, which is less than 8. Answer B.

If 6 builders can build a house in 45 days, how long will it take 15 builders to construct the same house?

The number of days needed to complete building a house varies inversely as the number of builders employed. If t is time in days and n is the number of men, then t and n are related by t = (k/n), where k is the work that each man can perform. When t = 45 and n = 6, then k = 45×6 = 270, so it would take 270 days for 1 man to build a house. t = 270/15 = 18.

After 5 adults leave a party, there are three time as many children as adults. After a further 25 children leave the party, there are twice as many adults as children. A) The original number of adults B) 14 Is A or B bigger, C equal, D undetermined?

The wordy question is difficult to set up as an equation. You only need to compare the original number of adults to 14. 14-5=9 adults, so children=9x3=27. After 27-25=2 children left, so then 2x2=4 adults left, which does not equal 9. Pick either a smaller or bigger than 14. 13-5=8 adults, so children=8x3=24. After 24-25=-1 children left. Doesn't make sense, therefore adults > 14. (A)

Of the 180 people, 30% were women, and 25% were from minority groups. If 1/9 of the women are from minority groups, how many were neither women nor from minority groups? a) 75 b) 81 c) 87 d) 93 e) 99

Total minority = 0.25 x 180 = 45 Minority = Minor women + Minor men Minor women = 0.3 x 1/9 x 180 = 6 Minor men = Minority - Minor women = 45 - 6 = 39 Men = (180x0.7) - 39 = 126 - 39 = 87 C.

For all integers x, the function f is defined as: f(x)=(x-1) when x is even. f(x)=(x+1) when x is odd. If a and b are integers, and f(a)+f(b)=a+b, which of the following statements must be true? A) a=b B) a=-b C) a+b is odd D) Both a and b are even. E) Both a and b are odd.

Use all pieces of the question to help: f(a)+f(b)=a+b A) If a=b, whether or not a/b is even or odd, f(a)+f(a)=2a-2 OR 2a+2, which is not a+b=2a B) If a=-b and even, f(a)+f(-a)=(a-1)+(-a-1)=-2, which is not 2a. If a=-b and odd, it would be the same result -2. Rules out D and E. C) If a+b is odd, one must be even and the other must be odd. Then f(a=even)+f(b=odd)=(a-1)+(b+1)=a+b. The 1 + -1 = 0.

I expect to earn at least $1000 in interest after an initial investment of $20K. If I invest for one year when interest is compounded quarterly, what is the least annual interest rate to achieve the goal?

Value of investment = (Principal investment)(1+r/100n)^nt r = annual interest rate. n = compounded n times per year. t = years 21000 < 20000 (1+r/400)^4 21/20 < (1+r/400)^2 (sqrt(sqrt(1.05))) < 1+r/400 1.0122 - 1 < r/400 r > 4.91

If an integer is randomly selected from all positive 2-digit integers, what is the probability that there is: 1) a 4 in the tens place? 2) at least one 4 in the tens place or the units place?

_ _ Tens place possibilities: 1-9 Units place possibilities: 1-10 1) P(4 in the tens place) = 1/9 2) P(at least one four) = P(A) + P(B) - P(AnB) = 1/9 + 1/10 - (1/9x1/10) = (10 + 9 - 1)/90 = 18/90 = 1/5

How many 3-digit positive integers are odd and do not contain the digit 5?

_ _ _ First digit possibilities: 1-9 Second digit possibilites: 1-10 Third digit possibilities: 1-10 Third digit odd, not 5 possibilities: 1, 3, 7, 9 Probability (first digit is not 5) = 8/9 Probability (second digit is not 5) = 9/10 Probability (third digit is odd, not 5) = 4/10 Probability = 8/9 x 9/10 x 4/102 = 288/900 Possibilities = 288

Describe the domain, x and y intecepts of: a) y=2lx-1l b) y=100-900x c) y=5-(x+20)^2 d) y=(x+2)^1/2 e) y=x+lxl

a) y=2lx-1l is lxl shifted to the right by 1, and stretched vertically by a factor of 2. Domain: all real numbers. b) y=100-900x is a line with slope -900, y-cept=100, and x-cept=1/9. Domain: all real numbers c) y=5-(x+20)^2 is a parabola pointing down with vertex (-20,5), y-cept=-395, x-cept=-20+/-sqrt(5). Domain: all real numbers d) y=(x+2)^1/2 is a half parabola opening to the right with vertex (-2,0), x-cept=-2, y-cept=sqrt(2). Domain: numbers >= -2. e) y=x+lxl. For x>=0, y=2x. For x<0, y=0. Two lines, one is y=2x, and second is along x-axis for x<0. Domain: all real numbers.

If cube root(x) = 3, and x=square root (y), what is y?

cube root(x) = 3 --> x=3^3 = 27. 27=square root (y) --> y = 27^2 = 729.

Martha invited 4 friends to go to the movies. How many ways can they sit in a row of 5 seats, one person per seat? How many of those ways is Martha sitting in the middle?

n=5, pick and order all 5. 5! = 5x4x3x2x1 = 120 (Number of ways to select without order) x (Number of ways to order) = (Number of ways to select with order) Number of ways with Martha in the middle = (Number of ways to select with order) / (Number of ways to order) = 120/5 = 24

If x is an integer and if |x-2| > 1, then what are all of the possible values of x? A. x>3 B. x<-3 or x>1 C. x<1 or x>3 D. 1<x<3 E. x>1

|x-2| > 1 Solve: +(x-2)>1 x>1+2 Solve: -(x-2)<1 Don't forget to switch inequality sign! -x+2<1 -x<-1 (if multiplying both sides to remove -1, leave sign intact. If multiply by -1 to move negative to other sign, switch sign!) x<1 So x<1 and x>3, which is C. Also draw a graph!