Math - Weakness Hitlist

The Ultimate Percentage Formula

(Original #) (1.00 +/- the % change) = Final Number

percent decrease formula

(actual decrease/original amount)x100 ex. if the price of a $100 DVD player is on sale for $80, the percent decrease is... (20/100)x100 =20%

Percent increase formula

(actual increase/original amount)x100 ex. if the price of a DVD rises from $80 to $100, the percent increase is... (20/80)x100 (1/4)x100 25%

How many 5 digit positive integers satisfy the following conditions: -each digit is a 1, 2, 3 , 4, or 5 -no digit is repeated -the first and last digits are both odd

-Since the first digit is odd, there must be 3 choices for the 1st digit: 1, 3, and 5 -There are 2 choices for the last digit: one of the two odd integers not chosen for the 1st digit. -There are 3 choices for the 2nd digit, 2 choices for the 3rd digit, and 1 choice for the last digit. 3x2x3x2x1=36

12.5%=

1/8 0.125

[picture: points A, B, and C all on one line] In the figure about, the length of AC is 20 and the length of AB is 3/5 the length of AC. What is the length of BC? (a) 6 (b) 8 (c) 12 (d) 14 (e) 16

20 x (3/5) = 12 therefore, the length of AB is 12. AC-AB=BC so 20-12=8 (b) is the correct answer. Read the question, bozo.

The average of the test scores of a class of p students is 70, and the average of the test scores of a class of n students is 92. When the scores of both classes are combined, the average score is 86. What is the value of p/n?

3/8 Explanation: The correct answer is 3/8 or 0.375 . The total of the p test scores is 70p and the total of the n test scores is 92n. The overall average score for both classes combined, 86, is the total of all the test scores, 70p+92n, divided by the total number of scores, p+n. Therefore, [(70p+92n)/(p+n)]=86 which simplifies to 70p+92n=86p+86n and further to 6n=16p. Therefore, p/n = 6/16 = 3/8.

.375=

37.5%

What is the equation to set up for "the probability that a red glass bead with be chosen is 3 times the probability that a blue glass bead will be chosen."

3b = r (where b is blue glass and r is red glass)

Five times a number is the same as a number added to five. What is the number?

5x=x+5 subtract x from both sides 4x=5 x=4/5

90 is 45% of what number?

90=45% of x .45x=90 x=200

Ellie is dropping marbles into a box one at a time in the order: r, w , w, b, b, b. How many marbles will be put in the box right after the 100th blue one is put in?

After 33 repetitions of the pattern there will be 6*33=198 marbles in the box, of which 99 will be blue. When these are followed by 4 more marbles, there will be 100 blue marbles, and a total of 202 marbles (198+4)

When you can't make up your own numbers...

Ask yourself for each question: can I make a variable? If so, to replace what?

In 1970 the populations of town A and town B were the same. From 1970 to 1980, however, the population of town A increased by 60% while the population of town B decreased by 60%. In 1980, the population of town B was what percent of the population of town A?

Assume that the populations of BOTH towns were 100 is 1970. Then in 1980 Town A's population would be 100+60=160 and Town B's population would be 100-60=40. 40/160=1/4=25%

When you need to know what two or more things equal together... (what is the value of X + Y? What is the value of A - B? What is the average of Z, X, and Y? How much money did Ben and Chris raise together?)

Before you ever solve the value of ONE thing, see if you'll have an easier time finding the value of BOTH, first! ALWAYS HAVE THE FINISH LINE IN MIND!

if a and b are positive integers and their product is 3 times their sum, what is the value of 1/a+1/b?

By adding the fractions we get 1/a+1/b=(a+b)/ab But given that ab is three times (a+b), (a+b)/ab=1/3

The price of ground coffee beans is d dollars for 8 ounces and each ounce makes c cups of brewed coffee. In terms of c and d, what is the dollar cost of the ground coffee beans required to make 1 cup of brewed coffee? (a) d/8c (b) cd/8 (c) 8c/d (d) 8d/c (e) 8cd

Choice (A) is correct. The price of ground coffee beans is d dollars for 8 ounces, which means that 1 ounce costs d/8 dollars. Since 1 ounce makes c cups of brewed coffee, the dollar cost of the coffee beans required to make 1 cup of brewed coffee is (d/8) x (1/c) = d/8c

If j and k are integers, j is greater than k , and (j^2) + (k^2) = 25, what is the least possible value of j ? (A) -5 (B) -4 (C) -3 (D) 0 (E) 3

Choice (C) is correct. All possible values of and satisfying the equation (j^2) + (k^2) = 25, where j and k are both integers and j>k are (-3, -4) (0, 5) (3, -4) (4, -3) (4, 3) and (5 ,0). The least possible value of j in this set is -3. [make a table with all possible values, circle ones where j > k]

The total daily profit p in dollars from producing and selling x units of a certain product is given by the function p(x)=17x-(10x+b) where b is a constant. If 300 units were produced and sold yesterday for $1,900 in profit, what is the value for b?

Choice (E) is correct. Putting in the given values for x and p(x) yields the equation 1,900=17(300)-(10(700)+b) This simplifies to 5100-3000-b which further simplifies to 1900=2100-b It follows that b=200

If |a-b|=5 and |a-c|=3, which of the following could be the value of |b-c|? I. 2 II. 4 III. 8 (A) I. only (B) II. only (C) III. only (D) I. and II. (E) I. and III.

Choice (E) is correct. The expression |a-b|=5 means that the distance between a and b on a number line is 5. Similarly, |a-c|=3 means that the distance between a and c on a number line is 3. If a> b and a>c, then the distance between b and c is 2. Also, if a<b and a<c then the distance between b and c is 2. [draw a number line with these possibilities] If c<a<b, then the distance between b and c is equal to 8. Also, if is less than a<c, then the distance between b and c is equal to 8. Therefore, as illustrated below, in both of these cases. [draw a number line with these possibilities] There are no other possible arrangements of a, b, and c on the number line. If a<b<c or if c<b<a, then the distance between a and c would be greater than the distance between a and b. However, this contradicts the information given in the question and therefore cannot be true. Therefore, the only possible solutions are |b-c|=2 and |b-c|=8.

In a group of 300 people, 60 percent are women. If 10 percent of the people in the group speak Chinese, how many of the women in the group speak Chinese? (A) 6 (B) 15 (C) 18 (D) 30 (E) It cannot be determined from the information given.

Choice (E) is correct. The item does not indicate how the percentage of people in the group who speak Chinese is distributed. Since this is not known, there can be no determination of the number of women in the group that speak Chinese.

When you find an answer...

DROP your pencil, make sure you've ANSWERED the QUESTION!

Five points, A, B, C, D, and E, lie on a line, not necessarily in that order. AB has a length of 24. Point C is the midpoint of AB, and point D is the midpoint of AC. If the distance between D and E is 5, what is one possible distance between A and E?

Draw a picture. The only possible answers are 1 or 11 Since segment AB has length 24 and C is the midpoint of AB, segment AC has length 12. Since D is the midpoint of AC, segment D has length 6 and points A and D must be 6 units apart. Point E is 5 units from D, but E could be between A and D or between D and C. If E is between A and D then the total distance between A and E will be (6-5=) 1 unit. If E is between D and C then the total distance between A and E will be (6+5=) 11 units. Either answer is correct.

A grocery customer spent a total of $9.60 for ground beef and coffee. The coffee cost 2 times as much per pound as the ground beef, and the customer bought 3 times as many pounds as ground beef as pounds of coffee. How much, in dollars, did the customer spend on coffee?

Everything can be defined in terms of the price of ground beef by weight, so the price of a unit of ground beef is x. Then the price of a unit of coffee is 2x. The customer buys 3 times as much ground beef as he does coffee. So say he buys 3 units of beef (cost: 3x) and one unit of coffee (cost: 2x). So he spent 3x + 2x = 5x on groceries, and that came to $9.60. A little quick division tells us that x = $1.92. Since 2x was the amount he spent on coffee, we can conclude he spent $3.84 on coffee.

If k, n, and y are positive numbers satisfying x^-4/3 = k^-2 and y^4/3 = n^2 what is (xy)^-2/3 in terms of n and k?

For this problem, you want to get to the finish line - which is finding x and y to the -2/3 power. Since x^-4/3=k^-2 , it follows that x^-2/3=k^-1. Since y^4/3=n^2, it follows that y^2/3=n^1 and y^-2/3=n^-1. Therefore, (xy)^-2/3=(nk)^-1 which is equivalent to 1/(nk)

A jar contains 20 marbles: 4 red, 6 white, and 10 blue. If you remove 1 marble at a time, randomly, what is the minimum number you must remove to be certain that you have at least 2 marbles of each color?

In a problem like this one, the easiest thing to do is see what could go wrong in your attempt to get 2 marbles of each color. If you were really unlucky you might remove 10 blue ones in a row, followed by all 6 white ones. At this point you would have 16 marbles and no reds. The next 2 marbles, however, must both be red. The answer is 18.

For any cube, if the volume is V cubic inches and the surface area is A square inches, then V is directly proportional to which of the following? (a) A (b) A^2 (c) A^3 (d) A^(2/3) (e) A^(3/2)

In order to have direct proportionality, both variables need to increase or decrease at the same rate. For the purposes of this question, it's best to use the standard math textbook definition of direct proportionality: y = kx, where k is a constant. You want to be able to say that V = [constant][something]. Think of both V and A in terms of s, the length of one edge of the cube. V = s^3, and A = 6s^2. Buckle up: V = s^3 A = 6s^2 s = V^(1/3) s = (A/6)^(1/2) V^(1/3) = (A/6)^(1/2) <— both equal s, so they equal each other V = (A/6)^(3/2) <— raise both sides to the 3rd power V = [1/(6^(3/2))] x A^(3/2) V = kA^(3/2) <— 1/(6^(3/2)) is a constant There's your direct proportionality. The answer you seek is (E).

For all numbers x and y, let the operation [] be defined by x[]y=xy-y. If a and b are positive integers, which of the following can be equal to zero? I. a[]b II. (a+b)[]b III. a[](a+b)

In these types of problems, you're trying to prove that it CAN equal zero. The correct answer is I. and III. 1. a[]b=ab-b this can equal zero if a=1 and b= any number 2. (a+b)[]b=(a+b)b-b factored--> b(a+b-1) this can equal zero if b=0 or a+b-1=0, but a and b have to be positive integers, so this one is incorrect. 3. a[](a+b)=a(a+b)-(a+b) =(a-1)(a+b) this can equal zero if a=1 and b is any number

Brian gave 20% of his baseball cards to Scott and 15% to Adam. If he still had 520 cards, how many did he have originally?

In total, Brian gave away 35% of his cards (15+20). 100%-35%=65%. Now rephrase the question: 520 is 65% of what number? 520=.65x x=800

In the afternoon, Judy read 100 pages at the rate of 60 pages per hour; in the evening, when she was tired, she read another 100 pages at the rate of 40 pages per hour. In pages per hour, what was her average rate of reading for the day?

Judy's average rate of reading is determined by dividing the total number of pages she read (200) by the total amount of time she spent reading. (rate = d/t) In the afternoon she read for ( time = distance/rate) (100/60)=5/3 hours, and in the evening for (100/40)=5/2 hours for a total time of 5/3+5/2=25/6 hours Her average rate was 200/(25/6)= 48 pages per hour

In a lottery, 4% of the tickets printed can be redeemed for prizes, and 4% of these tickets have values in excess of $100. If the state prints 40,000 tickets, how many of them can be redeemed for more than $100?

Let x be the number of tickets worth more than $100. Then x=4% of 4% of 40,000 or .4x.4x40,000=64

Mark drove to a meeting at 60 miles per hour. Returning over the same route, he encountered heavy traffic and was able to drive at only 40 miles per hour. If the return trip took 1 hour longer, how many miles did he drive each way?

Let x represent the number of hours Mark took to go, and make a table: Going: r=60 t=x d=60x Returning: r=40 t=x+1 d= 40(x+1) Since he went the same distance going and returning, 60x=40(x+1) x=2 hours BE SURE TO ANSWER THE QUESTION! 60(2)=120 miles

Two printing presses working together can complete a job in 2.5 hours. Working alone, press A can do the job in 10 hours. How many hours will press B take to do the job by itself?

Let x represent the number of hours press B would take working alone. Make a table. Press A can complete 1/10 of a job in 1 hour and 2.5/10 of a job in 2.5 hours. Press B can complete 1/x of a job in 1 hour and 2.5/x of a job in 2.5 hours. Together, they can complete 1/2.5 of a job in 1 hour and 1 job in 2.5 hours. Make an equation: (2.5/10)+(2.5/x)=1 and solve to get 3 1/3 hours.

In 1980, Judy was 3 times as old as Adam, but in 1984 she was only twice as old as he was. How old was Adam in 1990?

Make a table: 1980: J=3x A=x 1984: J=3x+4 A=x Judy's age was twice Adam's age in 1984, so 3x+4=2(x+4) x=4 +10 (to bring him to 1990) = 14

John buys a shirt for 50% off. The price of his new shirt is $70. How much was the shirt before the discount?

Original Number = Unknown Percentage Change: Minus 50% Final Number = 70 (Original Price) (1.00 - 0.50) = (70) Original Price = 70/(0.50) Original price = $140.

When you see language like this: "What is a number which satisfies these conditions?" "Which of the following..." "What is the largest/smallest possible value of..." "What could be a value of..." "Each of the following [has this or that quality] EXCEPT..."

Pick from the potentially correct answers right in front of you! Remember to check all the answers and eliminate.

Absolute value and inequality problems...

Remember that inequalities function almost exactly the same as regular equations with ONE key twist: the sign is reversed for the negative possibility! In other words, to convert this into my "or" propositions: |X-3| > 6 X-3 > 6 or X-3 <-6

If p, r, and s are three different prime numbers greater than 2, and n = p x r x s, how many positive factors, including 1 and n, does n have?

Since p, r, and s are primes, and since n = p x r x s the only possible positive factors of n are: 1, p, r, s, p x r, p x s, r x s, and n = p x r x s. Since the primes numbers p, r, and s are different, all 8 factors listed above are different. Therefore, there are 8 positive factors of n.

Let [x] be defined as [x] = x^2-x for all values of x. If [a]=[a-2] what is the value of a?

Since the function is defined as x^2-x we have a^2-a=(a-2)^2-(a-2) This equation is equivalent to a^2-a=a^2-4a+4-a+2. This can be simplified to 4a=6 or a=3/2

A school ordered $600 worth of lightbulbs. Some of the lightbulbs cost $1 each and the others cost $2 each. If twice as many $1 bulbs as $2 bulbs were ordered, how many lightbulbs were ordered all together?

Since there are twice as many $1 bulbs as $2 bulbs, 2/3 of the total number of bulbs cost $1 each, and 1/3 of the total bulbs cost $2 each. If n represents the total number of bulbs, then $1(2/3n)+$2(1/3n)=600, which simplifies to 4/3n=600. Therefore, the total number of bulbs is 450.

If the rent on an apartment goes up 10% every year, next year's rent will be how many times last years rent?

Since this is a percent problem, assume the rent last year was $100. Since 10% of 100 is 10, this year the rent went up $10 to $110. 10% of 110 is 11, so *next year* the rent will go up $11 to $121. 121 is *1.21*x100

Two sides of a triangle are 3 and 8. If all the side lengths of the triangle are integers, what COULD NOT be the side length of the 3rd side?

So assuming that the largest side is 8, that means that the third side would have to be at least 6 long. So it can be 6. It can also be 7, since 3+7 is bigger than 8. It could be 8, since 3+8 is bigger than 8 Now we're getting into "bigger" territory. 9 works, but now 9 is the biggest side, so it's 8+3 >9. 10 works, since 8+3 is bigger than ten. But 11 does not work, since 8+3 is the same size. Therefore, the smallest a side length can be is 6, and the biggest it can be is 10. Therefore, any number that is smaller than 6 or larger than 10 is the answer to this question! I don't have to think about this any further.

If the ratio of red jellybeans to black jellybeans is between 4:11 and 8:11, and if you have 100 black jellybeans, what's a possible number of red jellybeans?

So the ratio of R:B is between 4x:11x and 8x:11x. If we know there are 100 blacks, that means that 11x = 100. So first, we figure out what X is: 11x = 100 x = 100/11 x= 9.09 Now we multiply 4 by that: We get 36.36. When we multiply 8 by that, we get 72.72 That means that the number of red jellybeans is somewhere, anywhere, between 36.36 and 72.72 - so I'll go with.....45.

Absolute value problems...

THE ABSOLUTE VALUE BRACKETS DOES NOT CHANGE - THE STUFF ON THE OTHER SIDE OF THE EQUAL SIGN DOES!!! |X-8|=4 X-8 = 4 or X-8 = -4

When 25 students took a quiz, the grades they received ranged from 2 to 10. If exactly 22 of them passed, by earning a grade 7 or higher, what is the highest possible average the class could have earned on the quiz?

The class average will be highest when all grades are as high as possible. Assume that all 22 students who passed earned 10's. Of the 3 who failed, one received a grade of 2 (to maintain the range) and the other three you can assume had 6's. (22x10)+2+(2x6)=234 234/25=9.36

The measures of the angles of a triangle are x degrees, y degrees, and z degrees. If the ratio x of y to is 2 to 3 and the ratio of y to z is 3 to 5, what is the value of x?

The correct answer is . Since the ratio of x to y is 2 to 3, it follows that x/y = 2/3. Thus 3x=2y and so y=(3/2)x. Since the ratio of y to z is 3 to 5, it follows that y/z = 3/5. Thus 5y=3z, and so z=(5/3)y. Since y=(3/2)x, the equation z=(5/3)y can be rewritten as z=(5/3)((3/2)x)= (5/2)x. The sum of the measures of the angles of any triangle is 180 degrees. Hence x+y+z=180. This can be rewritten in terms of as x+(3/2)x+(5/2)x=180 which simplifies to 5x = 180. Therefore, the value of x is 36.

f(n) = 1/sqrt(n-1000) The function f is defined above. For what integer value of is function f(n) also an integer?

The correct answer is 1001. Since the question asks for what integer n will function f(n)=1/sqrt(n-1000) be an integer, the value of the denominator sqrt(n-1000) must be equal to 1. Therefore, n-1000 = 1^2 = 1 and so n=1001.

The height of a plant at the end of June was 4 inches, its height was 6 inches at the end of July, and its height was 9 inches by the end of August. If the plant continues to grow by the same percent each month, how tall, in inches, will the plant be at the end of September?

The correct answer is 13.5 or 27/22. Since the plant grows by the same percent each month, the growth rate is found by dividing the height of the plant at the end of one month by the height of the plant at the end of the preceding month. Since 6/4 = 9/6 = 1.5, the plant grows by 50 percent each month. Hence at the end of September, the height of the plant will be 50 percent greater than its height at the end of August. Since the plant's height at the end of August is 9 inches, the plant will be 9 x 1.5 = 13.5 inches tall at the end of September.

The three-dimensional figure has two parallel bases and 18 edges. Line segments are to be drawn connecting vertex V with each of the other 11 vertices in the figure. How many of these segments will NOT lie on an edge of the figure?

The correct answer is 8. There are only 3 edges that connect vertex V to other vertices. Therefore, of the 11 segments to be drawn, 8 do NOT lie on an edge of the figure.

If A is the set of integers between -50 and 50, and a number is in set B if it is the cube of a number in set A, how many elements of set B are in set A?

The elements of B that are in set A are the perfect cubes between -50 and 50. There are 7 of them: -27, -8,-1, 0, 1, 8, 27

h(t) = c - (d-4t)^2 At time t = 0, a ball was thrown upward from an initial height of 6 feet. Until the ball hit the ground, its height, in feet, after t seconds was given by the function h above, in which c and d are positive constants. If the ball reached its maximum height of 106 feet at a time t=2.5, what was the height, in feet, of the ball at time t=1?

The height of the ball, in feet, at time t seconds is h(t)=c-(d-4t)^2. Since the ball was at height 6 feet when t was equal to 0, we have 6=c-(d-4(0))^2 which simplifies to c=6+d^2. In addition, we know that the ball reached its maximum height of 106 feet at time 2.5 so 106=c-(d-4(2.5))^2. This last equation simplifies to 106=(6+d^2)-(d-10)^2 if you plug in c. Thus, d=10. This gives us c=106 if you plug it back in. Therefore, the formula for the height of the ball, in feet, at time t seconds is h(t)=106-(10-4t)^2. When t=1 we have h(1)=106-(10-4(1))^2. Thus, the ball was at a height of 70 feet at time t=1

For how many ordered pairs of integers (x,y) is 2x+3y<6? (a) one (b) two (c) three (d) five (e) seven

The inequality 2x+3y<6 is equivalent to 2x<6-3y. Since y is a positive integer, its value must be at least 1. If y=1 then the inequality becomes 2x<3. The only positive integer x that satisfies 2x<3 is x=1. Therefore, (x,y)=(1,1) is the only pair of positive integers that satisfies the inequality.

The Acme Plumbing Company will send a team of 3 plumbers to work on a certain job. The company has 4 experienced plumbers and 4 trainees. If a team consists of 1 experienced plumber and 2 trainees, how many different teams are possible?

The number of teams consisting of 1experienced plumber and 2 trainees will be the number of ways of choosing the experienced plumber multiplied by the number of ways of choosing a pair of trainees. Since there are 4 experienced plumbers, there are 4 ways to choose the experienced plumber. Label the 4trainees as A, B, C, and D. There are 6 ways to form a group of 2 of them: AB, AC, AD, BC, BD, and CD. Teams consisting of 1 experienced plumber and 2 trainees is 4x6=24

Any time you're asked to find the area of a shape you don't know...

Think of the shapes you do know and then use the power of subtraction to solve!

If X is larger than the square of Y, and if X is the product of at least two different prime numbers, which of the following CANNOT be X and Y, respectively? A) X = 14 Y = 3 B) X= 15 Y = 2 C) X= 16 Y = 2 D) X = 21 Y = 4 E) X = 33 Y = 5

We want to prove that they DO work so that we can kill them. A) is 14 the product of two different primes? 2 and 7. Check! Is it bigger than the square of 3 (9)? Check! Kill A. B) 15 is a product of 3 and 5. Check. 15 is larger than 4. Check. Kill it and move on. C) 16 is a product of...4 and 4, 8 and 2, 1 and 16....this doesn't seem to be working. So let me leave it alone for a second. D) 21 is a product of 7 and 3. Check. It's bigger than 16. Kill it. E) 33 is the product of 3 and 11. It's bigger than 25. Kill it. C is the right answer because we can't kill it. We can kill A, B, D, and E instantly and with great ease.

x+y=10 y+z=15 x+z=17 What is the average of x, y, and z?

When you have two or more equations, add them: 2x+2y+2x=42 x+y+z=21 to find the average, DIVIDE BY 3! (x+y+z)/3=21/3=7

If you see "Cannot be"

You want to find the answers that are WORKING to eliminate them-- steal from the answer choices and prove that they DO work so that we can kill them

for any positive number a%...

a% of 100 is a

For any positive numbers a and b...

a% of b = b% of a

plus, more than, sum, increased by, added to, exceeds, received, got, older than, farther than, greater than

addition +

Mr. Howard was planning on depositing a certain amount of money each month into a college fund for his children. He then decided not to make any contributions during June and July. To make the same annual contribution that he originally planned, by what percent should he increase his monthly deposits?

assume that Mr. Howard was going to contribute $100 each month, for an annual total of $1200. Having decided not to contribute for 2 months, he would have to contribute the $1200 in 10 monthly deposits of $120 each. This is an increase of $20, and a percent increase of actual/original=20/100=20%

As you use information in a problem...

cross it out from the prompt

time is calculated by...

distance/rate

rate is calculated by...

distance/time

in terms of sequences and series, when k numbers form a repeating sequence, how do you find the nth number?

divide n by k and take the remainder t . The rth term is the same as the nth term.

divided by, quotient, per, for

division /

If A is 25 km east of B, which is 12 km south of C, which is 9 km west of D, how far, in km, is A from D?

draw a diagram. you should get that triangle AED has the measurements of DE=12 and AE=16, so using the pythagorean theorem you can see that DA=20.

is, was, will be, had, has, will have, is equal to, is the same as

equals =

If you see a question with "could be" or if it's asking for possible values...

establish a range!

Which way does the graph move from f(x) to f(x-2)?

f(x-2) moves the graph f(x) 2 units to the RIGHT--> remember, h is always opposite

If a>b, the percent increase going from a to b is ALWAYS...

greater than that of b to a

how many rectangles with perimeter 12 can be inscribed in the circle? (Given that there are already 2 inside)

infinite, because the points don't have to be restricted to the ones already there. You could rotate the 4 points around the center wherever you want.

[picture: triangle ABD, point C on same line as AD creating triangle ABC] [y is angle on point C, x is angle DBC] In the figure above, if y=40, AB is perpendicular to BC, and AB=BD, what is the value of x? (a) 10 (b) 20 (c) 25 (d) 30 (e) 35

is angle ABC is 90 degrees, angle A+y have to equal 90. 90-40=50, so angle A is 50 degrees. Since triangle ABD is isosceles, (AB and DB are the same length) angle A and angle D must be equal. This means that angle D is also 50 degrees. 180-50-50=80, so angle ABD is 80 degrees. We are given that AB is perpendicular to BC, so angle ABC is 90 degrees. Angle ABC-ABD=angle x, therefore 90-80=10 degrees. (a) is the correct answer.

formula for percent

is/of = percent/100 ex. what IS 45% OF 200? x/200 = 45/100

The sum of two numbers is smaller than their difference. Which of the following MUST be true? A) Neither number is negative B) Both of the numbers are negative C) At least one of the numbers is negative D) One of the numbers is zero E) The product of the numbers is negative

let's try to think of numbers that DON'T work, realizing that the problem could be restated like this: X + Y < X - Y A) If neither number is negative, we'd have two positives. Let's try: 5+9 < 5-9 False. So let's move on. B) A lot of people here might be tempted to try two negative numbers here. But instead, I'm going to try to prove the OPPOSITE!!! So instead of using two negatives, I'm going to try one positive and one negative and say X= 3 and Y = -3: 3+ - 3 < 3 - (-3) This is TRUE, meaning that both numbers do not HAVE TO BE negative for this to work. So we can kill answer B. C) In example B, I showed that this can be true with a negative and a positive, and that with two positives it doesn't work (from example A). I also know that it doesn't HAVE TO BE two negatives, so I'll leave C alone for the time-being. D) I already proved that this worked WITHOUT a zero in example B. So this is wrong. E) Hmmm. My working example in B does have a negative product. So let me try to disprove this by creating a working example that has a positive product: X = -3 and y = -3 Is -3 + -3 less than -3 - -3? Why, yes it is! And is (-3)(-3) positive? Why, yes it is! So E is wrong! As it turns out, the numbers here could either both be negative or just one of them could be negative but only one of them MUST be negative. And so we're good to go!

If there's a value in a problem that's NOT defined...(variables like x, y, m, n, z, etc.)

make up that value! Try to come up with "un-simple" or "non-reactive" numbers. Bad reactive numbers are 0, 1, 2, 3, and 4. Picking weirder numbers like 10 and 35 rather than 2 and 1 make sure you don't get false positives.

times, of, product, multiplied by

multiplication x

how to take fraction out of an equation

multiply EVERYTHING by the denominator FIRST.

to increase a number by k%...

multiply it by (1+k%)

to decrease a number by k%...

multiply it by (1-k%)

If multiplying k by 7 gives the same result as squaring k, which of the following must be true? (a) 7 + k = k^2 (b) k^2 + k = 7 (c) k^2 + 7k = 1 (d) 7k = 1 (e) 7k = k^2

multiplying k by 7 = 7k squaring k = k^2 7k = k^2 read the question, bozo.

rule for sequences

never answer a sequence question without writing out at least the first 5 terms

9(1/3)^n = 3^m, what is m+n?

plug in 1 for n, you get 9(1/3) = 3^m, or 3 = 3^m, so m=1. 1+1=2.

how to represent k percent of a commission for 2 cars?

plug in 10 for k

distance is calculated by...

rate x time

how to count how many integers there are between two integers, IF neither endpoint is included

subtract then subtract 1 more

minus, fewer, less than, difference, decreased by, subtracted from, younger than, gave, lost

subtraction -

exterior angle rule

the sum of two angles in the triangle (not using the angle by the exterior angle) adds up to the exterior angle

If you see "Must be"

try as hard as possible to find ONE instance of the answer choices NOT WORKING-- steal from the answers and try to think of numbers that DON'T work for each condition

In any problem involving percents...

use the number 100

From 2003 to 2004, the number of applicants to a college increased 15% to 5060. How many applicants were there in 2003?

x(1.15)=5060 x=4400

if you have a graph of y=f(x), what would y=-f(x) look like? what about f(-x)?

y=-f(x) flips across the x axis

If you have two-variable equations...

you can add or subtract two or more equations together so long as you add or subtract all the values to the left of the = sign with all the values to the right of the = sign. ex. If X + 20 = 60 and 2Y - 20 = 10 Then X + 2Y = 70

is 1 a prime number?

NO.

The sum of three different positive integers is 12. Let g be the greates possible product of these integers and l be the least possible product of the integers. What is g-l?

Systematically list all the ways of expressing 12 as the sum of 3 different positive integers and calculate each product. (9,2,1) --> l = 18 (5,4,3) --> g = 60 60-18=42

How many four-digit numbers have only even digits?

The easiest way to answer this question is to use the counting principle. The first digit can be chosen in any of 4 ways (2,4,6,8) and the second, third, and fourth digits can be chosen in any of 5 ways (0,2,4,6,8). Therefore, the total number of four-digit numbers with only even digits is 4*5*5*5=500.

what is a circle graph?

a pi chart

simplify a^3 = 3a

a^2=3

largest/smallest value ?'s strategy

always start with the largest or smallest value in the answer choices

what is the rule for arithmetic sequences?

an = a1 + (n-1) d // where where a1 is the first term of the sequence, d is the common difference, n is the number of the term to find.

what is the rule for geometric sequences?

an = a1*r^(n-1) where a1 is the first term of the sequence, r is the common ratio, n is the number of the term to find.

how to set up a "proportion" problem algebraically

centimeters representing miles: 1.5C = 2M then plug in 35 miles to see how many cm

how to calculate the percent increase in a quantity

(actual increase)/(original) * 100%

10 is what percent of A?

10 = (x/100)A therefore 1000 = xA therefore 1000%/A --> substitute easy to use number. 10 is 100% of 10. Which choice is equal to 100% when A = 10?

from 90 to 360 is a ____% increase

300% increase because 90/360 = 1/4, and 4 is 3 more than 1

There are 27 students in Mr. White's homeroom. What is the probability that at least 3 of them have their birthdays on the same month? (a) 0 (b) 3/27 (c) 3/12 (d) 1/2 (e) 1

If there were no month in which at least 3 students had a birthday, then each month would have the birthdays of at most 2 students. But that's not possible; even if there were 2 birthdays in January, 2 in February,..., and 2 in December, only 24 students would be accounted for. It is guaranteed that, with more than 24 students, at least 1 month will have 3 or more birthdays. The probability is 1.

Tameka cut a circular pizza into wedge-shaped pieces. The tip of each piece is at the center of the pizza and the angle at the tip is always greater than 20 but less than 30. What is one possible value for the number of pieces into which the pizza is cut?

The five correct answers are 13, 14, 15, 16, or 17. If the pieces each had a tip angle measure of 20 there would be 360/20= *18* pieces. If each piece had a tip angle measure of 30 there would be 360/30=*12* pieces. Since the tip angle measure must be between 20 and 30 the number of pieces must be between 12 and 18.

how to represent MPH

miles/hours

how do I determine how many tables can seat 4 or 5 people with 19 tables and 84 people?

set up a systems of equations

how to count how many integers there are between two integers, IF exactly one of the endpoints is included

subtract

A car going 40 miles per hour set out on an 80 mile trip at 9am. Exactly 10 minutes later, a second car left from the same place and followed the same route. How fast, in miles per hour, was the second car going if it caught up with the first car at 10:30 am?

At 10:30 AM the first car had been going 40 miles per hour for 1.5 hours, or 40x1.5=60 miles. The second car covered the same 60 miles in 1 hour 20 minutes, or 4/3 hours. Therefore, its rate what 60/(4/3)=45 miles per hour.

In a venn diagram with 3 circles X Y and Z, which overlaps include the elements inside Y and Z?

Both the intersection of Y and Z and the intersection of X, Y, and Z

In the figure above (4 squares to make 1 big square, A at bottom left and X at top right) how many paths are there from A to X if the only ways are up and right?

Either label all the vertices and systematically list the possibilities, or systematically trace the diagram.

A, B, C, and D lie on the same straight line, and AC=2CD=3BD. What is the value of the ratio BC/AD?

Draw a diagram. Since this is a ratio problem, immediately plug in a number. To avoid fractions, use 6. Let AC=6; then CD=3, with D on either side of C. BD=2, but B could be on either side of D so we have no way of knowing length BD. The value of the ratio BC/CD cannot be determined by the information given.

Of the 410 students at Kennedy High School, 240 study Spanish and 180 study French. If 25 students study neither language, how many students study both?

Draw a venn diagram. Let x represent the number of students who study both languages, and write x in the part of the diagram where the two circles overlap. Then, the number of students who only study Spanish is 240-x, and the number who only study French is 180-x. The number who study at least one language is 410-25=385, so 385=(240-x)+x+(180-x) 385=420-x x=35

If the 5 cards shown above are placed in a row so that [O] is never placed at either end, how many different arrangements are possible?

One way to solve this problem is to calculate the number of different possible arrangements of the 5 cards in a row and then subtract the number of possible arrangements of the 5 cards in a row where [O] is at the right end or at the left end. There are 5! =120 ways of arranging the 5 cards in a row. If the shaded card were placed at the left end, the remaining cards could be arranged in 4!=24 ways. Similarly, there are 24 ways in which 4 remaining cards could be arranged if the shaded card were placed at the right end. Therefore, there are 120-24-24 = 72 different possible arrangements in which the shaded card is not at either end.

Probability calculation

P(E) = number of favorable outcomes/total possible outcomes. In all cases, 0<= P(E) <= 1

If x+2y=a and x-2y=b which of the following is an expression for xy? (a) ab (b) (a+b)/2 (c) (a-b)/2 (d) (a^2+b^2)/4 (e) (a^2-b^2)/8

Plug in numbers for x and y to solve. Let x=2 and y=1. Then xy=2, a=4, and b=0. Now, plug in 4 for A and 0 for B, and see which of the answer choices works. Only E does.

In a three circle venn diagram where circle A is the set of positive integers less than 20; B is the set of positive integers that contain the digit 7; and C is the set of primes, how many numbers are at the intersection of A and C?

Positive integers less than 20 and prime include 2, 3, 5, 7, 11, 13, 17, and 19; but we can't include 7 and 17 because they would be in the intersection of all 3 circles. There are 6.

rule for geometric series?

Sn = [a1(1-r^n)]/1-r where Sn is the sum of n terms (nth partial sum), a1 is the first term, r is the common ration.

what is the rule for arithmetic series?

Sn = [n(a1 + an)]/2 where Sn is the sum of n terms (nth partial sum), a1 is the first term, an is the nth term.

how to find altitude of a square pyramid in terms of the base?

Use 45-45-90 triangle rule to find diagonal, which is root 2 m. Half of root 2 m is (root 2 m)/2, then use pythagorean theorem using that value, side length, and hypotenuse

Is 0 an even number?

YES.

is 0 an integer? is 0 a positive integer?

0 is an integer, but NOT a positive integer

if a speed of 1 meter per second is equal to a speed of k kilometers per hour, what is the value of k?

1 m / 1 sec = 3600 m / 3600 sec = 3600 m / 1 hr = 3.6 km / hr so k = 3.6

What is the first digit of (a three digit number (which is the sum of two 2digit numbers))?

1, same goes for relationship between 3 and 4 digit numbers.

A positive integer is said to be "tri-factorable" if it is the product of three consecutive integers. How many positive integers are tri-factorable?

1,000 = 10*10*10, so you know the number of tri-factorable integers is less than 10. use calculator: 1*2*3 = 6 2*3*4 = 24 3*4*5 = 60 etc. until you get to 9*10*11 = 990.

if (x-2)^2 = 25 and x<0, what is the value of x? (a) -23 (b) -7 (c) -5 (d) -3 (e)-2

Steal the answers for this one. If you plug in -7 you get (-9)^2, but if you plug in -3 you get (-5)^2 which is 25!

solve for hypotenuse - pythag or 30-60-90?

pythag!

how to find the 3rd side of a triangle

the two smaller sides of a triangle must be LARGER than the largest side of the triangle

The figure above shows an arrangement of 10 squares, each with a side length of k inches. The perimeter of the figure is p inches. The area of the figure is a square inches. If p = a, what is the value of k?

there are 10 squares of side length k the area of 1 square is k^2 the area of 10 squares is 10*k^2 since a = p, and a = the area of the figure, the area of the figure is also a, the area of the figure is also p, the perimeter of the figure so 10*k^2 = a = p p = perimeter, which is the sum of all the side's lengths if you count the number of uniform sides (the squares' sides) you get 16 sides of k length each, or 16k so perimeter = 16k = p 10*k^2 = 16k solve for k: k^2=1.6k k*k=1.6*k k=1.6

how to determine what 4-digit integer is based on 3 rules?

think through it logically, find biggest and smallest and plug in to see what works

From a jar containing 50 pieces of candy, of which 25 are red and 25 are green, Ari has taken 3 red and 4 green pieces. He takes an additional 13 pieces from the jar. What is the least number of these additional pieces that must be red in order for Ari to have more red candies than green candies among all the pieces he has taken? (free response)

25 red + 25 green = 50 -3 red and -4 green = 43 candies left. 22 reds + 21 greens leftover = 43 -13 = 30 left in the jar. Altogether, he has taken 20 pieces of candy. 3 reds, 4 greens --> trying to get up to 13. ask yourself: what adds up to 13? 7 + 6 = 13 1) 3 reds + 7 = 10 2) 4 greens + 6 = 10 So 7 & 6 is can't work because red has to be more than green. 8 + 5 = 13 1) 8 + 3 reds = 11 2) 4 greens + 5 = 9 11 reds and 9 greens. Answer is 8.

A printer that can print 1 page in every 5 seconds shuts down for 3 minutes to cool off after every hour of operation. How many minutes will the printer take to print 3600 pages? (a) 300 (b) 312 (c) 315 (d) 18000 (e) 18897

After all the math, you find that 3600 pages takes 5 hours. The mistake I made here was adding 5 rest breaks in between rather than 4 in between each hour. The correct answer is B, not C.

Two circular tables have diameters of 36 inches and 24 inches, respectively. The area of the larger table is what percent more than the area of the smaller table?

First you READ THE QUESTION then you calculate the area of both tables, or 324pi/144pi. Simplify to get the ratio of the two, or 9pi/4pi. Convert the ratio to a percent: 9/4 = 2.25 or 225% The area of the larger table is 225% OF the area of the smaller one, or 125% MORE than the area of the small one.

(x-8)(x-k) = x^2-5kx+m In the equation above, k and m are constants. If the equation is true for all values of x, what is the value of m? (a) 8 (b) 16 (c) 24 (d) 32 (e) 40

First, you need to multiply out the left side (x-8)(x-k) = x^2 - 5kx + m x^2 - 8x - kx + 8k = x^2 - 5kx + m -5kx is the middle number in the equation, so it follows that: -5kx = -8x - kx factor out x -5k = -8 - k 8 = 4k k = 2 m is the last number in the equation, so it follows that: m = 8k m = 8(2) m = 16 So, k = 2 and m = 16.

how to represent wins and losses in variable form (ex. Adam has won 6/30 games, if he's going to lose 1/3 of additional games, how many more must he play so he has wins>losses?)

He loses x games and wins 2x games, so he wins a total of 6+2x and loses 24 plus x, so you would set up the inequality as 6+2x>24+x

Let P and Q be points 2 inches apart, and let A be the area, in square inches, of a circle that passes through P and Q. Which of the following is the set of all possible values of A? (a) 0<A (b) 0<A<pi (c) A=pi (d) A>pi (e) A>=pi

If PQ is a diameter of the circle, then the radius is 1 and A, the area, is pi. This is the smallest possible value of A but A can actually be any number larger than pi if the radius is made arbitrarily large. Draw a picture!

Jordan has taken 5 math tests so far this semester. If he gets a 70 on his next test, that grade will lower his test average by 4 points. What is his average now?

If a represents Jordan's average after five tests, then he has earned a total of 5a points. A grade of 70 on the sixth test will lower his grade average 4 points to a-4. Therefore, a-4 = [5a+70]/6 6(a-4) = 5a+70 6a-24=5a+70 6a=5a+94 a=94 remember the question doesn't say he GOT a 70, so you don't have to subtract the answer by 4 points.

If c carpenters can build a garage in d days, how many days will it take e carpenters, working at the same rate, to build 2 garages? (variable form)

If c carpenters build a garage in d days then 1 carpenter will take c times as long or cd days, and 2 cd days to build 2 garages. Finally, the work is divided up among e carpenters, they will take 2cd/e days. ...you can also plug in numbers for this one

The counting principle

If two jobs need to be completed and there are m ways to do the first job and n ways to do the second job, then there are m*n was to do one job followed by the other. Ex.: there are 10 paintings to be hung in 4 rooms. The first job is to choose 1 of the 10 paintings. The second is to choose 1 of 9 paintings for the next room. The third is to choose 1 of 8, and the last to choose 1 of 7. These 4 jobs can be completed in 10*9*8*7 = 5040 ways.

At the audition, n people tried out. If k people went before Judy, who went before Liz, and m people went after Liz, how many people tried out between Judy and Liz? (represented with variables)

It may help to draw a line and label it. Since k people went before Judy, she was number k+1 to try out; and since m people went after Liz she was number n-m to try out. Then, the number of people to try out between Judy and Liz was (n-m)-(k+1)-1 =n-m-k-2

Blair has 4 paintings in the basement. She is going to bring up 2 of them and hang 1 in her den and 1 in her bedroom. In how many ways can she choose which paintings go in each room? (a) 4 (b) 6 (c) 12 (d) 16 (e) 24

Label the paintings 1, 2, 3, and 4, write B for bedroom and D for den, and make a list. B-D: 1-2, 1-3, 1-4, 2-1, 2-3, 2-4, 3-1, 3-2, 3-4, 4-1, 4-2, 4-3 There are 12 ways to choose (c).

The members of the French club conducted a fund-raising drive. The average (arithmetic mean) amount of money raised per club member was $85. Then Jean joined the club and raised $50. This lowered the average to $80. How many members were there before Jean joined?

Let n represent the number of members before Jean joined. These members raised a total of 85n dollars. After Jean was in the club, the total raised was 85n+50, the average was 80, so: (85n+50)/(n+1)=80 ...math... n=6

A number of people boarded the bus at the terminal. At the first stop, half of the passengers got off and 1 got on. If the bus then had 15 passengers, how many were there left at the terminal?

Let x=the number of passengers originally on the bus, and keep track of the comings and goings. At the first stop half the people got off and 1 got on: 1/2x+1. At the second stop 1/3 of the passengers got off, leaving 2/3 on the bus and 1 person got on: 2/3(1/2x+1)+1 Do math... x=40

If X is a multiple of 5, a multiple of 11, even, positive, and smaller than 1,000, what is one possible value of X?

Next, we know that it must be a multiple of 5, 11, and 2. What happens if I just multiply 5×11×2? I get 110. Did I follow all the rules? Yes. Does my number therefore fit all the criteria? Yes. So long as you establish a "range of possibilities" and make sure to follow all of them, you'll end up with an answer that is certain.

Any 2 points determine a line. If there are 6 points in a plane, no 3 of which lie on the same line, how many lines are determined by pairs of these 6 points? (a) 15 (b) 18 (c) 20 (d) 30 (e) 36

READ THE QUESTION- NO 3 OF WHICH!! The best way to do this is draw 6 points and connect them. The first point has 5 connections to the other points, the next has 4 (already connected to the first one), the next has 3, and so on. 5+4+3+2+1=15.

There are 12 men on a basketball team, and in a game 5 of them play at any one time. If the game is 1 hour long, and if each man plays exactly the same amount of time, how many minutes does each man play?

Since the game takes 1 hour, or 60 minutes, and there are always 5 men playing, there is a total of 5x60=300 man-minutes of playing time. If that amount of time is evenly divided among the 12 players, each one plays 300/12=25min

If, in figure above, BC is the longest side of ABC (triangle, angle BC is 80, x is the exterior angle of C) and x is an integer, what is the smallest possible value of x?

Since the measure of an exterior angle is equal to the sum of the two opposite interior angles, x=80+y. BUT, since BC>AC, y>80. So, x must be greater than 160. Since x is an integer, it must be at least 161.

if x varies inversely with y and varies directly with z, and if y and z are both 12 when x=3 what is the value of y+z when x=4?

Since x varies inversely with y, there is a constant k such that xy=k. Then k=3*12=36, and, when x=4, 4y=36, y=9. Also, since x varies directly with z there is a constant m such that x/z=m. Then m=3/12=1/4 and when x=4, 4/z=1/4, z=16. 9+16=25.

A man has to drive from Jonesville to Marcustown in under 70 hours, but more than 30 hours. If Marcustown is 2100 miles away from Jonesville, what's one possible speed that the man could drive, in miles per hour?

So first, we remember the MPH formula. Miles per hour is just: Miles/Hours = Miles Per Hour We already know our miles: 2100. So we plug that into the equation: 2100/Hours = Miles Per Hour Next, we have our maximum and minimum hours that the man is allowed to drive for. Let's plug them both in: 2100/30= 70. 2100/70 = 30. That means that the man must go faster than 30MPH, but slower than 70MPH. So if I have to pick an answer, I'll go with.....45.

If x is an integer greater than 1 and if y = x + 1/x which of the following must be true? I. y cannot equal x II. y is an integer III. xy>x^2

Statement I. is true because if y=x+1/x y cannot equal x. If you plug in numbers, statement III. must also be true. Statement II. is NOT true b/c if you plug in numbers y almost always comes out as a fraction.

On a square game-board that is divided into n rows of n squares each, k of these squares lie along the boundary of the game-board. Which of the following is a possible value for k? (a) 10 (b) 25 (c) 34 (d) 42 (e) 52

Steal the answers for this one. If you try out 2x2, 3x3, and 4x4 game-boards, you'll find that they all increase by 4, so k has to be divisible by 4 with no remainder. 52 is the only answer choice that follows that.

The square of x is equal to 4 times the square of y. If x is 1 more than twice y, what is the value of x? (a) -4 (b) -1/2 (c) -1/4 (d) 1/4 (e) 1/2

Substitute 2y + 1 for x in the first equation so that you only have one variable. (2y+1)^2 = 4y^2 (2y+1)(2y+1) = 4y^2 etc. until you get y=-1/4 plug that into x=2y+1 and you get x=1/2

If x^2-y^2=77 and x+y=11, what is the value of x?

The equation x+y=11 implies that y=11-x. This can be substituted for y in the equation x^2-y^2=77. x^2 - (11-x)^2 = 77 x^2 - (121-22x+x^2) = 77 22x-121=77 22x=198 x=198/22=*9*

2, -4, 8, ... The first term of the sequence above is 2, and every term after the first term is -2 times the preceding term. How many of the first 50 terms of this sequence are less than 100? (a) 22 (b) 25 (c) 28 (d) 30 (e) 37

The first 6 terms in this sequence are all less than 100 (2, -4, 8, -16, 32, -64). We still have (50-6) 44 terms to go. We know that half of the rest of the terms will be negative & automatically less than 100 (-256, -1024, etc.), but the rest of the positive terms are over 100 (128, 512, ...). So, (44/2) 22 of the remaining terms are under 100, for a total of (22+6) 28 terms.

For how many integers, x, is the function sqrt(x^2-9) undefined? (a) none (b) 4 (c) 5 (d) 7 (e) infinitely many

The function sqrt(x^2-9) is defined for all real numbers except those that cause x^2-9 to be negative, so -3<x<3. Four integers satisfy this inequality: -2, -1, 0, 1, 2

In the xy-coordinate plane, lines l and q are perpendicular. If line l contains the points (0,0) and (2,1), and line q contains the points (2,1) and (0,t) what is the value of t? (a) -3 (b) -2 (c) 2 (d) 3 (e) 5

The slope of line l is 1/2, so the slope of line q would be -2 (because it's perpendicular - opposite and reciprocal) So, eqn of line q is y = -2x+c Plug in the point (2,1) to see what the y-intercept is 5.

There are 75 more women than men enrolled in Linden College. If there are *n* men enrolled, then, in terms of *n*, what percent of those enrolled are men? (a) n/(n+75)% (b) n/(2n+75)% (c) n/100(2n+75)% (d) (100n)/(n+75)% (e) (100n)/(2n+75)%

The total number of students is (n+w), where w represents the number of women. We know that the number of the women is the number of men plus 75, so *the total number of students is n+(n+75) or (2n+75).* The percent of men enrolled is given by *the number of men (n) divided by the total number of students, all multiplied by 100*. Thus, in terms of n the percent of men is (100n)/(2n+75)

The estate of a wealthy man was distributed as follows: 10% to his wife, 5% divided equally among his three children, 5% among his 5 grandchildren, and a balance to a charitable trust. If the trust received $1,000,000, how much did each grandchild inherit?

The trust received 80% of the estate. If E represents the value of the estate, then .8E=1,000,000 and E=1,250,000. Each grandchild received 1% (one fifth of 5%) of the estate, or $12,500.

A list of numbers consists of p positive and n negative numbers. If a number is picked at random from this list, the probability that the number is positive is 3/5. What is the value of n/p?

There are p+n numbers on the list, so the probability of a positive number being picked at random is (p/(p+n)). It is given that the probability is equal to 3/5, so cross multiplying (p/(p+n)) = 3/5 we get 5p=3p+3n, which simplifies to 2p=3n. Therefore, n/p=2/3

How many integers are there between 100 and 1000, all of whose digits are odd?

Use the counting principle. Think of writing a 3 digit number as 3 jobs to be done. The first is to select one of the 5 odd digits and use it as a digit in the hundreds place. The second job is to select one of the five odd digits to use in the tens place. The last is to select one of the five odd digits to be the digit in the units place. Each of these jobs can be done in 5 ways, so the total number of ways is 5*5*5 = 125.

In the xy-coordinate plane, the distance between B(10,18) and A(x,3) is 17. What is one possible value of x?

Use the distance formula! 17=sqrt[(10-x)^2+(18-3)^2] 289=(10-x)^2+255 64=(10-x)^2 Therefore, 10-x=8 or 10-x=-8, or x=2,16 respectively.

In 1980, Elaine was 8 times as old as Adam, and Judy was 3 times as old as Adam. Elaine is 20 years older than Judy. How old was Adam in 1988?

You set up this equation as 8A=E 3A=J J+20=E Substitute and solve and you get Adam as being 4 in 1980. Add 8 to get to 1988 and you get 12.

In the infinite sequence in which the six digits [1,4,2,8,5,7] keep repeating in that order, what is the 500th term of the sequence?

You would divide 500 by 6 to get 83 remainder 2. Therefore, the first (83*6=) 498 terms are just the numbers repeated 83 times, then the pattern repeats again: the 499th is 1, and the 500th is 4.

[n] represents the sum of the integers 1 to [n]. If [1000]=a and [10]=b, what is the value of [1010]? (in terms of a and b)

[1010] = (1+2+...+1000) + (1001+1002+...+1010). The sum of the first parenthesis is (1000+1) + (1000+2) +...+ (1000+10) which can be written as (1000+1000+...+1000) +(1+2+...+10) = 10,000 +b. The total is a+b+10,000

The cost of a telephone call using long-distance carrier A is $1.00 for any time up to and including 20 minutes of $0.07 per minute thereafter. The cost using long-distance carrier B is $0.06 per minute for any amount of time For a call that lasts t minutes, the cost using carrier A is the same as the cost using carrier B. If t is a positive integer greater than 20, what is the value of t?

carrier A: 1.00+0.07(t-20) *this way, if t=21 it is 1 minute over 20, so you would add 1 plus .07 for the extra minute* carrier B: 0.06t 1+0.07(t-20)=0.06t 1+0.07t-1.4=0.06t 0.01t=0.4 t=40

how to find 2 points with same y coordinate on f(x) quadratic mathematically

find known point and plug in numbers until get that known point, or use graphing calculator

The eggs in a certain basket are either white or brown. If the ratio of the number of white eggs to the number of brown eggs in 2/3, each of the following could be the number of eggs in the basket EXCEPT: (a) 10 (b) 12 (c) 15 (d) 30 (e) 60

for every 2 white eggs there are exactly 3 brown eggs, all the eggs can be split up evenly into groups of 5 (2+3=5) each with 2 white and 3 brown eggs. This means that the total number of eggs in the basket must be a multiple of 5 because we do not consider fractions of eggs. The only one of the choices that is not a multiple of 5 is 12.

t^2-k^2 < 6 t+k>4 If t and k are positive integers in the inequalities above, what is the value of t? (a) 1 (b) 2 (c) 3 (d) 4 (e) 5

for this one, plug in values for t and k and see what works. 2 and 1 don't work because added together they are less than 4, but 3 and 2 work because 9-4=5 and 3+2=5, which are less than 6 and greater than 4. Also, READ THE QUESTION- GREATER THAN 4

the value of 10 pounds of gold is d dollars, and a pound of gold has the same value as p pounds of silver. What is the value, in dollars, of one pound of silver?

proportion: dollars/pounds. d dollars/10 pounds of gold = d dollars/10p pounds of silver = x dollars/1 pound of silver. Then, d/10p = x/1 to x=d/10p.

how to represent a clock

remember that any time past the hour o' clock, the hour hand is a fraction more towards the next hour. Account for that fraction in calculations.

how to find largest number in an average set?

replace largest number with a variable, find total sum, then make other numbers as small as possible

how to count how many integers there are between two integers, IF both endpoints are included

subtract then add 1

In a figure, a small circle is inscribed in a square which is inscribed in a larger circle. What is the ratio of the area of the large circle to the area of the small circle?

the r and R be the radii of the two circles. From the firgue, you can see that OAB is a 45-45-90 right triangle, so R=r(sqrt2). Therefore, the area of the large circle/area of the small circle = piR^2/pir^2 = pi(r(sqrt2)^2/pir^2 = 2pir/pir^2 = 2 The ratio os 2:1

what is the largest integer, x, such that x < 10,000 and (sqrt.x)/5 is an even integer?

x < 10,000 --> sqrt.x < sqrt. 10,000 --> sqrt. x < 100 --> (sqrt.x)/5 < 100/5 = 20. Since (sqrt.x)/5 has to be an even integer, the greatest possible value of (sqrt.x)/5 is 18 --> (sqrt.x) = 90, 90^2 is 8100.

In the equation 1<= x which side of 1 is x on in a number line?

x is GREATER than 1, so it would be on the positive (right) side of 1 on a number line.