Act questions types

take things from diffrent groups problem: How many outfits does sally have if she has 5 shrts, 5 pants and 2 hats?

Multiple options: 5*5*2 = 50

When using logs on calc

Remember every log has a base of 10 so, the buttons log(x) and in(x) will always assume a base of 10. You must use the 3rd function log(x)(y)

Good ellipse question

Remember that this ellipse equation given has 1. No (h,k) value so it's centered at the origin and 2. Is not in standard ellipse for, 1. Make it in standard ellipse form by diving by 36 on both sides 2 this yields: (x^2/9) + (y^2/4) 3. In this equation 9= horizontal radius = 3 and 4= verticals radius = 2 4. To be inscribed, the equation in our answer choices must have a radius less than 2 to be at least in side the ellipse

The ratio of x to z is 3 to 5, and the ratio of y to z is 1 to 5. What is the ratio of x to y? A. 5:3 B. 5:1 C. 3:1 D. 1:3 E. 1:1

Set numbers for each and create proportions then solve those portions

Ellipses Equation

(x-h)^2/(a^2) + (y-k)^2/(b^2) = 1 a = the x radius b = the y radius

Triangle properties

1) the sum of any two side lengths will ALWAYS be greater than the third side length 2) sum of the three angles = 180 3. If a^2 + b^2 = c^2 then it is a right triangle 4. If a^2 + b^2 < c^2 it is obtuse 5. If a^2 + b^2 > c^2 it is acute

Expected value

1. 12* 1/3 = 2. K * 2/3 = 2k/3 3. 4+ 2k/3 = 20 4. K=24

hard question

1. 2/a is really just another fraction in between these 2 fractions 2. We can easily see, by giving them like-numerators, what divided by 2 gives us our numbers 3. 2/22 < 2/a < 2/9 so the numbers can be between 22 and 16 4. There are 5 numbers between 22 and 16

Medium question

1. 510-105 = 405 meaning that from 8:00 to 1 he travels 405mi 2. 8:00 to 1 is 5 hours 3. 405km/5h =81km/h

A square and a rectangle have the same area. The length of the rectangle is 45 centimeters, and the width of the rectangle is 5 centimeters. What is the length, in centimeters, of a side of the square?

1. Area of R = 45*5 = 225 2. Area of S = s^2 3. 225=s^2 = s = 15

Harder one

1. Calculate x and add the probabilities together 1/10 (1) = 2/10 (2) = 3/10 (3) = 4/10 (4) = 5/10 (5) = 6/10 (6) = When we add up all the x values, we get 3.0

Multiplying matrixes

1. Determine the scaling of the matrixes ie: (heights x width) 2. If the 2 the two middle numbers match: (2x2) and (2x2); then they are "defined" and can be multiplied. If not they cannot be multiplied. 3. Bring down the outside numbers also 2 and 2, and that's the resulting dimensions of the product matrixes 4. Multiply the first 2# across by the first two numbers down in the second matrixes. Add those two products together to get the top right value of the end product matrixe 5. To find the determinate of a matrixes, cross multiply and subtract

A 12-centimeter-by-16-centimeter rectangle is inscribed in a circle as shown below. What is the area of the circle, in square centimeters? F. 5pi G. 14pi Н. 25pi J. 100pi

1. Draw a diagonal and treat like a right triangle and solve for the hypotenuse 2. Can't that in half to find radius 3. (Pi)(10)^2 = 100pi

The vector i represents 1 mile per hour east, and the vector j represents 1 mile per hour north. According to her GPS, at a particular instant, Tia is biking 30° west of north at 16 miles per hour. One of the following vectors represents Tia's velocity, in miles per hour, at that instant. Which one?

1. Draw a mini x-y plane and graph the vector northwest @ 30 2. Her velocity, 16mi/h is the resultant vector of 2 vectors, the northern and western components: -i + j 3. Sin(30)=(AB/16)=8 and cos(30)=(BC/16) = 8sqrt3 4. 8 = -i and 8sqrt3 = j so the resultant = -8 + 8sqrt3

Matrixes question

1. Find if matrixes is defined or undefined (h x l): [3x1] * [1*3] 2. The middle terms match so it is defined and thus can be multiplied. 3. The outside numbers creat the resultant = a [3x3] matrixes

Watch out for liscense plate probability

1. I'm these problems you must use hangman method. You must logically think through the possible outcome for a specific digit or code 2. Remember, to solve you must multiply the number of possible numbers for each feasible spot by each other 3. If the first number is 5 don't multiply by 5, multiply by one because there is only one number possibility for that spot

For whole numbers x and y, the list below has 4 as its mean, median, and mode. What is the value of x*y 2, 6, 4, 1, x, y

1. If the mode is 4 then 4 has to be x or y 2. If either x or t is 4 then: (2+6+4+1+4+x)/6=4. X=7 3. (4)(7)=28

What is the value of b in the solution to the system of equations below? 3a-b=18 a +3b =-4 F. -10 G. -3 H. 3 J. 6 K. cannot be determined with the given information

1. If you didn't recognize that it's just a system of equations with x,y for a,b you will fail

Due to inflation, a refrigerator that formerly sold for $450 now sells for 7% more. Which of the following calculations gives the current cost, in dollars, of the refrigerator? F. 450+7 G. 450+450(0.07) H. 450+450(0.7) J. 450+450(7) K. 450(0.07)

1. If you know percent increase you know that: X + X(0.05) is the same as X(1+0.05)

Asymptote question

1. In this case remember that when a asymptote is horizontal: y=# and when it's verticals: x=# 2. In the function f(x), if the highest power terms in the numerator and denominator are the same, then you take the ratios of both of this functions = 2/7. This only works for horizontal asymptotes 3. For verticals ones, you just set the denominator equal to zero

Another one like it

1. In this one we simple make x=a and y=b 2. Now just plug in but make sure that when you plug in for b, you only bulging the half the b is on

An integer from 10 through 99, inclusive, is to be chosen at random. What is the probability that the number chosen will have 0 as at least 1 digit?

1. Let's find how many numbers between 10 and 99 have 0 as at least 1 digit. 10,20,30,40,50,60,70,80,90 — 9 numbers 2. Let's find total numbers in this set = 99-10+1=90 3. 9/90= 1/10

What is log2 16

1. Logb y = (log y)/(log b) 2. X= (ln (16))/(ln (2))=4

The vector i represents 1 mile per hour east, and the vector j represents 1 mile per hour north. Maria is jogging south at 12 miles per hour. One of the following vectors represents Maria's velocity, in miles per hour. Which one?

1. Maria is traveling 12 mi/h south. j = Posotive y axis so -j is negative x axis 2. Plug in and we find she travels -12j

Median of a frequency table

1. Method one (short): we have 32 people so our median will lie i between the middle number: 32/2=16, there is no middle number becuase it's even so, our answer lies between 16 and 17 2. Now you must find the total score for each frequency but let's start from the third one but remember to keep adding becuase these scores are cumulative: 2(1) + 3(3) + 4(5) + 5(2) + 6(3) = 14 we know if we go any further, we will get into the range in witch our 16 and 17 lye. The next score that shows that is score 7.

A classroom has 10 tables that will seat up to 4 students each. If 20 students are seated at tables, and NO tables are empty, what is the greatest possible number of tables that could be filled with students?

1. No table can be empty so fill each table with one student first then fill the rest in with 3 student s

For every cent increase in price of a pound of apples, the grocery store sells 25 fewer pounds per day. The grocery store normally sells 800 pounds of apples per day at $1.09 per pound. Which of the following expressions represents the number of pounds of apples sold per day if the cost is increased by 3x cents per pound of apples? F. (1.09 + 3x) (800 - 75x) G. 800 - 25* H. 800 - 75(1.09)x J. 800 + 75x K. 800 - 75х

1. Notice how price and apple decrease are inversely proportional and can be written as 25 * 3x since 3x is the price increase 2. You could recognize that 800 is the y intercept or that we must start of from 800. 3. The expression is 800-75x

In the standard (x,y) coordinate plane shown below, what is the distance on the y-axis, in units, from point A to point B?

1. Notice how the question says on the y axis becuase I sure didn't 2. Subtract x1 from x2

Sample probability: Of the 16 cars on a rental-car lot, 6 are minivan, 7 are Sedan, and 3 are Hatchback. Thalia will rent 3 of cars, chosen at Random, for business associates. What is the probability that Thalia will rent 1 of each of the 3 types of cars?

1. Recognize that she will be choosing all 3 car together so we must multiply all probability together. 2. Once she picks a minivan, the total amount of cars she is picking from will decrease. 3. 6/16 * 7/15 * 3/14 = 3/80 4. But why start with minivans first ? Why not sedans. Think back to permutations. I need every possibility. Multiple this probability by possibilities there are: (3/80) * 3! = 9/40

Hang man method problem: How many 4 digit numbers have 1 or 6 as their 100t digit and 2 as their's 1st digit?

1. Set up 4 dashed lines to fill in to make a number:__ __ __ __ 2. 2 is the ones digit so 2 goes last: __ __ __ 2 3. in the tens place, we can have any number 0-9 4. In the hundreds place, we can have any number that is 1 or 6 5. In the thousands place, we can any number 1-9 6. Multiple possibilities of numbers: 9*2*10*1 = 180

Hard question

1. Simply set all solutions equal to the expression 2. Find n using quadratic formula until one yields a whole number

If 0 <pr < 1, then which of the following CANNOT be true? F. p<0andr < 0 G. p<-1 andr < 0 H. p<-1andr < -1 J. p< 1vandr < 1 K. p <1 and r > 0

1. So which one of the answers can never be true 2. From the given we know p*r= a Posotive fraction between 0 and 1 Choice f satisfies becuase -0.5 * -0.5 is 0.25 3. choice g works becuase -1/4 * -2 is 0.5 4. H cannot work becuase -1.1 * -1.1 = 1.21 this is not a fraction nor will any number make it a fraction

In the figure below, what is the sum of p and q? DO YOUR FIGURING HERE. F. 75◦ G. 150◦ H. 180◦ J. 285◦ K. 360◦

1. The 2 intersecting lines are not parallel so any line theorems will not work (ie alternate interior angles...) 2. Make equation: 180-p=x and 180-q=y 3. Another equation: x+y+105=180 4. substitute: (180-p) + (180-q) + 105 = 180 5. Solve for p+Q which equals 285

Sine graph question

1. The amplitude is 3 so the height of the graph = 3 2. Period = 2pi/b and period for tan = pi/b 3. The graph ends at pi becuase pi/2 + pi/4 = pi 4. Pi = 2pi/b = b=2 5. When it has a multiplier at the end (3x) it will multiply the number of cycles but if it has a factor (x/3) it will cut the cycles in half, extending them every time the denominator gets bigger 6. To check that when x= pi/2 then y = -3 we can plug it in our calculator using radians mode

A company prints contest codes on its fun-size bag candy. Each 6-character code consists of the letter A followed by the letter H followed by 4 of the digits O through 9. The digits may repeat. Which of the following expressions gives the number of different 6-character codes that are possible?

1. The first two spots must be 1 2. The digit may repeat so it can be anything from 0-1 3. 1*1*10*10*10*10

Hard question

1. The fixed amount is the amount that is charged every single time ie the y-intercept 2. Count backwards from 8.00, subtracting 0.50 each time until you get to 0 3. When x=0, y=5.50

Hard question

1. The greatest common factor between 2 numbers, is the one biggest number that can be divided out of the 2 2. This means that a^2 * b^2 is a big number and ab^3 is also a big number and both can have 45 taken out of them 3. These aren't numbers, there terms but we can also say that the gcf of both terms is ab^2, since that can be taken out of both number s 4. if both are gcf's then ab^2=45 the only numbers that satisfies this are 5 and 3 with a=5 and b=3

Hard question

1. The least common multiple is the smallest number that has all these in common ie that can be easily divisible by all numbers / terms 2. The only one that suffices is 60ab

Every harder

1. The question gives us the points earned per toss so that's a freebie 2. What is the probability of getting heads for the first toss: 1/2. And what about with the nickels : 1/2 and the dime: 1/2 3. So 1/2(3) + 1/2(3) + 1/2(3) = 9/2

Triangles question: Given that a is 5 and b is 7, which of the following inequalities gives all and only the possible values of c ?

1. The third side has to be less than than 5+7=12 2.the side "b" cannot be greater than a+c 3. Of this is true, then c must be greater than 2 becuase 2+5=7 and 1+5=6 but the sum of the 2 sides must be greater than the third so 2<c<12

When the vector ai + 4j is added to the vector -3i + bj, the sum is 5i - 5j. What are the values of a and b?

1. The two vectors (a,4) and (-3,b) need to be added to obtain the resultant, (5,-5) 2. When adding vectoryou combine like terms: ai + -3i = 5i 3. Solve for a which = 8 2. 4j + bj = 5i now solve for b which = -9

What is the 330th digit after the decimal point in the repeating decimal 4.6238 ?

1. There are 4 digits after . So divide 330/4 = 82.5 2. All 4 digits can be written 82 times before only .5 3. .5 is 2/4 of the way through the 4 digits which is 2

Hard east problem

1. These questions can be written using the equation: [one time charge] + [reoccurring charge]*[y-1] 2. This yields: y= 12y +3

In 3 fair coin tosses, where the 2 outcomes, heads and tails, are equally likely, what is the probability of obtaining exactly 2 heads?

1. They are going to toss the coin the times. Each time it is tossed, although there are 2 outcomes possible, if you flip the coin 3 times you will end up with heads or tails twice or head all 3 times. 2. If we have 2 outcomes with 3 possibilities in each outcome the 2^3 will give total outcomes = 8 3. Out of all of those outcomes only 2 have heads twice 4. Probability = 3/8

For all numbers x and y, let the operation & be defined as x @ y = 2xy - 4x. If a and b are positive integers, which of the following can be equal to zero? I. a @ b ii. (a-b) @ b iii. b @ (a-b)

1. Think of a @ b as f(x)so you want to make a = x and y = b and plug in the answer choices as numbers 2. First we must solve for a or b: 2(a)(b) - 4a then factor: 2a(b-2) 3. we know that 2 can't be zero but for this expression to be possible, 2 must be one and b must be 2 4. Let's try answer choice 1: 2(1)(2) -4(1) = 0 yes 5. Let's try choice 2: by plugging in to our factored version: 2(a-b)(b-2) Lon matter what, if b=2 then 2-2 is zero so this expression will always be zero 6. Same principle with the last choice

Identity problem

1. This problem is tricky becuase you just know the following already: [sin^2 + cos^2 =1 ] 2. It gives that that cos(2a) = cos^2 - sin^2 3. Rearranging the identity it can be written as: cos^2 -(1-cos^2) 4. Combine like terms: cos^2 - 1 + cos^2 5. It is given that cos(a)=b so plug it In: b^2 + b^2 -1 = 2b^2 - 1

Bad question

1. This question is bad becuase the angle 105 looks like it represents the unknown angle and the outside angle but in reality it only represents the outside angle 2. 180-30+x = 180 3. X=45

Counting repetition, how many prime factors appear in any prime factorization of the integer (63)^5

1. Use a prime factor tree ro seperate 63 into its prime numbers 2. 63 splits into 3 prime numbers (3,3 and 7) 3. 3 prime numbers * 5 = 15 prime numbers

Hard line question

1. Using the slope equation you yield: (s+t-s)/(r+2-r) = t/2 2. The slope of the line = 4 3. t/2 = 4 so t=8

Hard question

1. We are already given the equation for the sum of an infinite series. 2. Plug in: 200= 1/(1-0.15) solving for x, our first term, yields 170 3. We need the second term so, 170*0.15 = 25.5

Recursive formula question

1. We are give out first term s(1) = 4 2. Becuase it is a recursive formula, it tells use that n+1, our next term, will be obtained by doing 2(sn)-3 3. That means: 2(4)-3=5 which means our second term equals five 4. Doing this, the fifth term is 19

58. Carol has an empty container and puts in 6 red chips. She now wants to put in enough white chips so that the probability of drawing a red chip at random from the container is 3/8 F. 3 G. 6 H. 8 J. 10 K. 16 How many white chips should she add.

1. We need to change the probability of red. 2. So far that probability is 6/6 3. Make equation 6/(6+x)=3/8 4. Solve: x=10

Hard question

1. What 3 numbers multiply to get to 512 but also have roots of 2: 4*8*16 2. These numbers can be written as 2^2 * 2^3 * 2^4 which are all consecutive exponents 3. Plug into final equation: 4+8+16 = 28

Sample probability: A cookie jar contains 10 cookies of 3 types. There are 5 chocolate-chip cookies, 3 oatmeal-raisin cookies, and 2 sugar cookies. You reach into the jar and choose a cookie at random and then, without replacing the first cookie, reach into the jar and choose another cookie at random. What is the probability that both of the cookies vou choose are the same type?

1. What is the total and what is the success group: amount of cookies possibilities of taking out 2 similarly flavored cookies "without putting one back" 2. Notice it doesn't say and or or but you can infer that if you pick 2 cookies at the same time thst it's an "and" problem 3. Becuase we are looking at multiple outcomes (3 types of cookies we could pick 2 types from) we will need to add all the possibilities together

Probability

1. When dealing with probability questions, look to see if it's identify the total group picked from and the success of picking it 2. If the question asks What is the probability that this or this will be picked or happen: add If the question asks What is the probability that this and this will be picked or happen: multiply 3. look out for probability questions that say they pick "without replacing". This means when you add or multiply the fractions, the second fraction will have 1 less from the total amount and 1 less from the amount choosing from

Amy's best marathon time decreased by 10% from 2005 to 2006 and by 20% from 2006 to 2007. By what percent did her best marathon time decrease from 2005 to 2007? A. 28% B. 30% C. 50% D. 72% E. 10%

1. X(1-0.10) = 0.9x 2. 0.9x(1-0.20) = 0.72x 3. 0.72x is only the multiplier but we need the total amount decrease from beginning to end 3. Use: (new-old)/old *100 4. (0.72-1)/1 = -0.28 * 100 = 28% decrease

Explicit vs recursive formula question

1. a(1) = 10 mean our first term, when n=1, is 10 2. Plug n=1 into our answer choices and if a(n) = 10, then that answer choice is right 3. We can use the recursive formula to find duque tail terms: 2nd term = 2(10) = 20 4. When we plug n=2 in for the last one, since n=2 means our second term, we should get 20. We do.

Person is paid a regular hourly wage of 13.50per hour for working up to and including 40 hours in I week. For each additional hour he works in a week, he is paid 2 times his regular hourly wage. He worked 48 hours this week. What is his pay for this week?

1. regular wage * hours worked not including overtime: 13.50*40= 540 2. Find overtime rate: 2 * 13.50 = 27 3. Find overtime pay: 27*(40-48) = 216 4. Add 216+540 = 756

Harder hangman method: A jar contains 2 green mints and 3 white mints. If Renée randomly takes 2 mints out of the jar to eat, what is the probability that both of these mints are green?

1.probability = desired/total possibilities 2. There are only two possibilities becuase you draw twice: first that you draw 2/5 then that you draw 1 out of the 4 left 1. 2/5 * 1/4 = 1/10

Probability when 2 outcomes are desired

A jar contains 2 green mints and 3 white mints. If Renée randomly takes 2 mints out of the jar to eat, what is the probability that both of these mints are green? 1. (2/5) is the probability of picking the first green 2. (1/4) is the probability of picking the second one 3. (2/5)(1/4) = 1/10

Expected value

Expected value is defined as if I play this one carnival game a million times, what would I expect to be my average score overtime X = probability of getting score (points earned) If there are multiple probabilities of getting points, then add all of the x values together

Hard ellipse question

First let's convert our equations into a simpler form: E1: y <= -1/2x+3 E2: x2+y2 > 4 2. We can immediately eliminate choices F, J and G 3. The line has a y intercept of 3 and the radius of the circle is 2 meaning the line has to be above the circle and the circle has to be small 4. Answer is k because the shading is downward

What is there is a sequence of terms with a ratio difference

How to identify a geometric sequence: a, ab^1, ab^2, ab^3... Use: a(n) = a(1) * r^(n-1) a(1) = the first term r = the common ratio a(n) is the nth term If there is a summation of the first n terms of the geometric sequence, use: S(n) = a(1)*[(1-r^n)/(1-r)] If there is a summation of a fraction sequence where -1<r<1 use: S(n) = a(1)*(1/(1-r))

Factorials

If you see these you will solve by doing the following respectively 1.hangman method 2. multiply your number of options formulas 3. Formulas for permutation and combination: Equation for permutation, distinct ones, use: (#!)/(# of times one thing repeats!) Equation for for regular permutations, use: (#!)/(#-chosen)! Equation for combinations use: (#!)/(#-chosen)!*(chosen)! Equation for. Probability = (combination or permutations)/(total combinations or permutations)

Inscribed angles on a circle

Inscribed angles are always half of the corresponding full angle

Not hard question

Just plug n chug

Trapezoid properties

The middle length of a trapezoid is the average of the 2 bases

The recursive formula

The recursive formula will always tell you how to get to the next term from the one prior Written as: a(n)(+/-) 1 = equation to obtain next number

In a game, 84 marbles numbered 00 through 83 are placed in a box. A player draws 1 marble at random from the box. Without replacing the first marble, the player draws a second marble at random. If both marbles drawn have the same tens digit (that is, both marbles are numbered between 00 and 09, or 10 and 19, or 20 and 29, etc.), the player is a winner. If the first marble Dave draws is numbered 23, what is the probability that Dave will be a winner on the next draw?

The rules of the game state that a player is a winner if two marbles drawn have the same tens digit. The player has already drawn the marble numbered 23, which has a 2 in the tens digit. In order to win, the player must draw another marble with a 2 in the tens digit. The possible winning marbles are 20, 21, 22, 24, 25, 26, 27,28, and 29. Therefore, the player has nine chances to draw a winning marble. Since he has already drawn one of the 84 marbles and did not put it back, he has nine chances out of 83 to draw a winning marble. The probability is 9/83.

If a green light flashes every 6 seconds and a red light flashes every 9 seconds, how many times will they flash if they start at the same time?

To find out how many times the green and red lights will flash at the same time, we can find the least common multiple (LCM) of 6 and 9. The LCM of 6 and 9 is 18, which means both lights will flash at the same time every 18 seconds. Therefore, they will flash at the same time every 18 seconds, and the number of times they will flash depends on the duration of time for which they are observed. If you have a specific time frame in mind, I can calculate the number of times they will flash within that time frame.

Casey has buckets of 3 different sizes. The total capac- ity of 12 of the buckets is g gallons, the total capacity of 8 buckets of another size is g gallons, and the total capacity of 4 buckets of the third size is also g gallons. In terms of g when g>0, what is the capacity, in gallons, of each of the smallest-sized buckets?

To solve this problem, first determine the size of each bucket. Because the total capacity of 12 buckets is g gallons, each bucket can hold ( 1/12 )g, or g/12 gallons. Because the total capacity of 8 buckets is g gallons, then each of those buckets can hold ( 1/8 )g, or g/8 gallons. If the total capacity of 4 buckets is g gallons, then each bucket is (1/4g), or g/4 gallons. Therefore, the capacity of the smallest buckets is 1/12 or g/12

50. If the value, to the nearest thousandth, of cos a is -0.385, which of the following could be true about a?

We are given options so let's go through them. You must be in radians to convert you answer choices to numbers 1. Remember our answers tell of that a is between 2 fractions 2. On the unit circle, cos(x) = -0.385 is given so: cos( an answer choice) < x < cos( an answer choice) 3. Let's plug in cos(pi/2) = 0 < x < cos(2pi/3)= - (1/2) 4. This is true becuase -0.385 is between 0 and negative -1/2

Logarithms

Y = a^X is the same as X = log(a) *Y Log(b) *1 = 0 Log(b) *b =1 log(b)*(xy) = log(b) *x + log(b) *y and vise versa Log(b)*(x/y) = log(b) *x - log(b) *y Log(b) *x^n = n*log*x To cancel log: raise both sides to the power of b

Linear arithmetic sequence

a sequence where the difference is simply plus or minus a number Use: a(n) = a(1) + d(n-1) The sum of them use: s(n) = n/2(a(1) + a(n))

Vectors

i = Posotive x axis (east) -i =negative x axis (west) j = Posotive y axis (north) -j = negative y axis (south) Vectors have a terminating end and a initial end: term. - init. = magnitude of vector Represented by a coordinat written in the form i+j so the coordinate 2i + 3j is (2,3) When adding vectors, you bring one tails of a vector to the tip of another When you add vectors, the resulting diagonal is called the resultant vector Qualitatively, you can solve for the resultant vector using pythag theorem

Hard probability question

since you cannot have a partial coin, the total number & coins in the bagmust be divisible by both 6 and 4 (1/6 are quarters and 1/4 are nickels). The only answer choice that is divisible by both 6 and 4 is 24.