GRE Quant

A bakery stocks 3 cookies for every 2 cupcakes and 6 pastries for every 5 cookies. What is the ratio of cupcakes to pastries?

First, you have to set up the given ratios, which is 3 cookies : 2 cupcakes and 5 cookies : 6 pastries. Then, you find a common multiple of cookies (i.e. 15) and convert the ratios to 15 cookies : 10 cupcakes and 15 cookies : 18 pastries. Since both ratios now have 15 cookies, you can infer that the ration of cupcakes to pastries is 10:18 or 5:9.

There are 16 members of a club. 4 will be selected to leadership positions. How many combinations of leaders are possible?

With permutations and combinations, you have to know if the order people are selected matters or not. If not, like in this case, you must take the number of people and positions available: 16*15*14*13 and divide by number of spots open 4*3*2*1 = 1820

Alex had an investment worth $1500. If the total investment increased 25% the first year and decreased 10% the next year, what is the final worth of the investment.

$1500⋅25% increase=$1875 $1875⋅10% decrease=$1687.50

There are 6 photos in a bag, numbered 1 to 6. The proportions of photos, P(i) for number i, are as follows: P(1) = 1/4, P(2) = 1/8, P(3) = 1/8, P(4) = 1/8, P(5) = 1/8, P(6) = 1/4. If a photo is drawn at random from the bag, what is the chance that the number on the photo is 3 or greater?

simply add up the proportions of the photos with numbers greater than or equal to 3. P(3 or 4 or 5 or 6) = P(3) + P(4) + P(5) + P(6) = 1/8 + 1/8 + 1/8 + 1/4 = 5/8

What is the interest rate on an account if an original balance of $12050 rises to $12670 after one year if it is only compounded yearly?

standard formula for simple interest (which is merely a derivative of standard applications of the formulas used for percentages in general). (1+r)⋅12050=12670 Divide both sides by 12050, but leave the fractions for the time being: 1+r=1267012050 Solve for r: r=(12670/12050)−1 This is: r=(12670−12050)/12050=0.05145228215768 Which is the same as: 5.145228215768%, or 5.15% when rounded.

Mary buys a car from a mean salesman who charges her 12% over the original price of a $15,000 car. Luke buys the same car from a much nicer salesman who gives him an 8% discount off of the original price. How much more does Mary spend on the car than Luke does?

12% of 15,000 is 0.12 * 15,000 = 1800. 8% of 15,000 is 0.08 * 15,000 = 1200; therefore in total, Mary spent 1800 + 1200 = $3000 more.

Column A: 25% Column B: 0.40

2/5% = 0.40% = 0.004. Therefore, Column B is greater.

In a bag, there are 10 red, 15 green, and 12 blue marbles. If you draw two marbles (without replacing), what is the approximate probability of drawing two different colors?

Calculate the chance of drawing either 2 reds, two greens, or two blues. Then, subtract this from 1 (100%) to calculate the possibility of drawing a pair of different colors. The combined probability of RR, GG, and BB is: (10 * 9) / (37 * 36) + (15 * 14) / (37 * 36) + (12 * 11) / (37 * 36) This simplifies to: (90 + 210 + 132) / 1332 = 432 / 1332 Subtract from 1: 1 - 432 / 1332 = (1332 - 432) / 1332 = approx. 0.6757 or 67.57%

Choose a number at random from 1 to 5. Column A The probability of choosing an even number Column B The probability of choosing an odd number

Column B is greater There are two even numbers and three odd numbers, so P (even) = 2/5 and P (odd) = 3/5.

A grade school pays Mr. Day a salary of $24,585 per school year. Each school year contains 165 days. Suppose Mr. Day is sick for a week (5 work days) and the school doesn't have to pay him for those days. Instead, they must pay a substitute teacher to teach his classes. They pay the substitute $90 per day. Totally, how much does the school save for the week Mr. Day is sick?

Divide Mr. Day's salary by 165 to determine how much the school pays him per day: Mr. Day makes $149 per day. They only have to pay substitute $90 per day, saving them $59 per day. To figure out how my they save totally, multiply by 5 to get how much they save for the week Mr. Day is sick: $295

A Super Sweet Candy Puff Roll has 1450 calories per roll. A man eats one roll in 10 minutes. During the work day, the man eats a roll at the start of the shift and then eats another a roll every two hours after finishing the last one. Since he is watching his health, he eats only until 3 PM but will not start eating another one at any time after 3 PM. If his shift begins at 8 AM and ends at 5 PM, how many calories per minute does he consume in Super Sweet Candy Puff Rolls® during the whole work day?

It is probably easiest just to write out the eating schedule: Roll 1: 8:00 AM - 8:10 AM Roll 2: 10:10 AM - 10:20 AM Roll 3: 12:20 PM - 12:30 PM Roll 4: 2:40 PM - 2:50 PM Therefore, he eats 4 rolls, or 1450 * 4 = 5800 calories. To get the rate, this must be divided across the whole day's minutes: 9 work hours * 60 = 540 minutes. The average calories per minute = 10.74.

Box A has 10 green balls and 8 black balls. Box B has 9 green balls and 5 black balls. What is the probability if one ball is drawn from each box that both balls are green?

Note that drawing balls from each box are independent events. Thus their probabilities can be combined with multiplication. Probability of drawing green from A: 10/18 = 5/9 Probability of drawing green from B: 9/14 So: 5/9 * 9/14 = 5/14

You are making a cake that requires, by volume, three times as much flour as sugar, twice as much sugar as milk, eight times more milk than baking powder and twice as much baking powder as salt. If you start with a teaspoon of salt, how many cups of flour do you need (there are 48 teaspoons in one cup)?

One teaspoon of salt requires 2 teaspoons of baking powder, which requires 16 teaspoons of milk and 32 teaspoons of sugar. 32 teaspoons of sugar requires 96 teaspoons of flour, which equals two cups of flour

In a sequence of numbers, the first two values are 1 and 2. Each successive integer is calculated by adding the previous two and mutliplying that result by 3. What is fifth value in this sequence?

Our sequence begins as 1, 2. Element 3: (Element 1 + Element 2) * 3 = (1 + 2) * 3 = 3 * 3 = 9 Element 4: (Element 2 + Element 3) * 3 = (2 + 9) * 3 = 11 * 3 = 33 Element 5: (Element 3 + Element 4) * 3 = (9 + 33) * 3 = 42 * 3 = 126

A singing group has 10 boys and 20 girls. Half the boys and half the girls have blue eyes. What is the probability that a student chosen at random from the group is a boy OR has blue eyes?

P(boy) = 10/30 = 1/3 P(blue eyes) = 1/2 P(boy and blue eyes) = 5/30 = 1/6, because we are told that half (or 5) of the 10 boys have blue eyes P(boy or blue eyes) = P(boy) + P(blue eyes) - P(boy and blue eyes) = 1/3 + 1/2 - 1/6 = 2/3

Harry borrowed $5000 from his parents at a rate of simple interest of 2% annually and paid it back in full in 30 months. What is the total amount of interest and principal his parents charged him?

Principal = $5000 (original amount) Interest = 30 months = 2 years and a half $5000⋅2%=$100⋅2 years=$200 half a year's interest is 1%=$50 Total interest = $250

In a bowl containing 10 marbles, 5 are blue and 5 are pink. If 2 marbles are picked randomly, what is the probability that the 2 marbles will not both be pink?

Solve for the probability of choosing 2 marbles that are pink and subtracting that from 1 to obtain the probability of selecting any variation of marbles that are not both pink. The probability of picking 2 marbles that are both pink would be the product of the probability of choosing the first pink marble multiplied by the probability of choosing a second pink marble from the remaining marbles in the mix. This would be 1/2 * 4/9 = 2/9. To obtain the probability that is asked, compute 1 - (2/9) = 7/9. The probability that the 2 randomly chosen marbles are not both pink is 7/9

A dessert is made using 2 parts cake and 3 parts icing. The cake contains 4 parts sugar, 5 parts milk, and 11 parts of other ingredients. The icing contains 3 parts sugar, 2 parts milk, and 15 parts of other ingredients. Which quantity is greater? Quantity A: Parts sugar in the dessert Quantity B: Parts milk in the dessert

Quantity A: To determine the parts of sugar in the dessert, we use the following process. Let's first figure out the amount of sugar in the cake. This is 4/20. Next, find the amount of sugar in the icing topping: 3/20. Then, we need to account for the amount of cake and icing in the dessert. Using the fact that there are 2 parts cake and 3 parts icing, we can say that 2/5 of the dessert is cake and 3/5 is icing. Combining this information with the amount of sugar in both the cake and the icing, we obtain: 2/5 * 4/20 + 3/5 * 3/20 = 17/100. So, there are 17 parts of sugar in the dessert. Quantity B: Use the same method to find the amount of milk: 2/5 * 5/20 + 3/5 * 2/20 = 16/100. So there are 16 parts milk in the dessert. Thus, Quantity A is larger.

The probability that events A and/or B will occur is 0.88. Quantity A: The probability that event A will occur. Quantity B: 0.44.

Relationship cannot be determined from information given The only probabilites that we know from this is that P(only A) + P(only B) + P (A and B) = 0.88, and that P(neither) = 0.12. We cannot calculate the probability of P(A) unless we know two of the probabilites that add up to 0.88

Fudge sells at $18.50 for 5 pounds. What is the cost for 2 pounds

Set up a proportion: 18.50/5 = x/2. Cross multiply and solve for x: 37 = 5x.... x = 7.40.

a is chosen randomly from the following set:{3, 11, 18, 22}b is chosen randomly from the following set:{ 4, 8, 16, 32, 64, 128}What is the probability that a + b = 27?

Since any of the first set can be summed with any of the second set, the addition sign in the equation works like a conjunction. As such, there are 4 * 6 = 24 possible combinations of a and b. Only one of these combinations, 11 + 16 = 27, works. Thus the probability is 1/24, or about 0.04

Michael owns 10 paintings. Michael would like to hang a single painting in each of five different rooms. How many different ways are there for Michael to hang 5 of his 10 paintings

This problem involves the combination of 10 items across 5 slots. The first slot (room) can have 10 different paintings, the second slot can have 9 (one is already in the first room), the third slot can have 8 and so on. The number of possible combinations is obtained by multiplying the number of possible combinations in each of the 5 slots together, which here is 10*9*8*7*6= 30,240

Consider the following sequence of integers: 5, 11, 23, 47 What is the 6th element in this sequence?

consider the change in each element. Notice that in each case, a given element is twice the preceding one plus one: 11 = 2 * 5 + 1 23 = 11 * 2 + 1 47 = 23 * 2 + 1 To find the 6th element, continue following this: The 5th: 47 * 2 + 1 = 95 The 6th: 95 * 2 + 1 = 191

If there are 20 students in a class and 2 people are randomly choosen to become class representatives, how many different ways can the representatives be chosen?

A combination is used when the order doesn't matter while a permutation is used when order matters. In this problem, the two class representatives are randomly chosen, therefore it doesn't matter what order the representative is chosen in, the end result is the same. The general formula for combinations is C(n,k)=n!/(n−k)!k!, where n is the number of things you have and k is the things you want to combine. Plugging in choosing 2 people from a group of 20, we find C(20,2)=20!/(20−2)!2!=190. Therefore there are a 190 different ways to choose the 2 class representatives.

A restaurant has a meal special that allows you to choose one of three salads, one of five sandwiches, and two of fifteen side dishes. How many possible combinations are there for the meal?

Although this is a permutation style problem, we have to be careful regarding the last portion (i.e. the side dishes). We know that our meal will have: (3 possible salads) * (5 possible sandwiches) * (x possible combinations of side dishes). We must ascertain how many side dishes can be selected. Now, it does not matter what order we put together the side dishes, so we have to use the combinations formula: c(n,r) = (n!) / ((n-r)! * (r!)) Plugging in, we get: c(15,2) = 15! / (13! * 2!) = 15 * 14 / 2 = 15 * 7 = 105 Using this in the equation above, we get: 3 * 5 * 105 = 1575

There are four aces in a standard deck of playing cards. What is the approximate probability of drawing two consecutive aces from a standard deck of 52 playing cards?

Answer: .005Explanation: The probability of two consecutive draws without replacement from a deck of cards is calculated as the number of possible successes over the number of possible outcomes, multiplied together for each case. Thus, for the first ace, there is a 4/52 probability and for the second there is a 3/51 probability. The probability of drawing both aces without replacement is thus 4/52*3/51, or approximately .005.

If the sum of a, b and c is 400, and a is 1/3 b and b is 1/4 c, what is the value of a?

Answer: 25 For this type of problem, build an equation that represents the relationships between the quantities and solve for the quantitiy you need. The problem states that b=1/4c and a = 1/3b. Thus, a = 1/3(1/4)c, or 1/12 of c. Now put everything in terms of c, thus 1/12c + 1/4c +c = 400. Now comes the tricky step--combine like terms and create the improper fraction (1/12c + 3/12c + 12/12c = 16/12c). Reduce the fraction to 4/3. So 400 is 4/3 of c. Thus c is 3/4 of 400, or 300. a is 1/12 of 300, or 25.

Beverly just filled up her gas tank, which has enough gas to last her, at her usual driving rate, about 45 days. However, Beverly becomes extra busy and begins driving 66.6% more than she usually does. How many days does the tank of gas last Beverly at her new rate?

Answer: 27 daysExplanation: Recall that the rate of gas consumption is INVERSELY related to the time it takes to consume the gas. Thus, if the rate of gas consumption increases to 5/3 of it's original rate (a 66.6% increase), then the time it will take to consume all of the gas decreases to the inverse, or 3/5 of the original. The answer is thus 27 (3/5 of 45).

What is the sum of the odd integers 1,3,5,...,97,99?

Begin by looking at the first and the last elements: 1 and 99. They add up to 100. Now, consider 3 and 97. Just as 1 + 99 = 100, 3 + 97 = 100. This holds true for the entire list. Therefore, it is crucial that we find the number of such pairings. 1, 3, 5, 7, and 9 are paired with 99, 97, 95, 93, and 91, respectively. Therefore, for each 10s digit, there are 5 pairings, or a total of 500. To get all the way through our numbers, you will have to repeat this process for the 10s, 20s, 30s, and 40s (all the way to 49 + 51 = 100). Therefore, there are 500 (per pairing) * 5 pairings = 2500.

Ben mows the lawn in 1 hour. Kent mows the lawn in 2 hours. How long will it take them to mow the lawn working together?

Ben mows 1 lawn in 1 hour, or 1/60 of the lawn in 1 minute. Ken mows 1 lawn in 2 hours, or 1/120 of the lawn in 1 minute. Then each minute they mow 1/60 + 1/120 = 3/120 = 1/40 of the lawn. That means the entire lawn takes 40 minutes to mow.

A five-year bond is opened with $5000 in it and an interest rate of 2.5%, compounded annually. This account is allowed to compound for five years. Which of the following most closely approximates the total amount in the account after that period of time?

Each year, you can calculate your interest by multiplying the principle ($5000) by 1.025. For one year, this would be: 1.025∗5000=5125 For two years, it would be: 5125∗1.025, which is the same as 1.025∗1.025∗5000 Therefore, you can solve for a five year period by doing: 1.025^5∗5000 Using your calculator, you can expand the 1.025^5 into a series of multiplications. This gives you 5657.041064453125, which is closest to $5657.

Aperture Industries is made up of 370 employees who work a total of 11,000 hours per week. If the number of weekly hours per employee has a normal distribution and standard deviation of 6 hours, approximately how many employees work more than 36 hours per week?

First find the average number of hours worked by every employee: 11000/370 = 29.7 ~ 30 hrs/week. Next, recognize that a single standard deviation encompasses 34% of the population on 1 end of the curve, or 68% of the population both above and below the mean. Since a standard deviation of 6 hours means that 68% of the population works between 24 to 36 hours per week, finding the amount that works over 36 hours is 100-68 = 32; 32/2 = 16% of the entire workforce on 1 end of the curve. .16(370) = 59.2 or 59 employees work over 36 hrs/week.

If Mary has $17 and gains a $2 weekly allowance, while Todd has $4 and gains a $3 weekly allowance, what is the least number of weeks that will pass before Todd has more money than Mary?

First we set up two equations, one for Mary and one for Todd. Mary's money growth is represented by y = 2x + 17 (she starts with $17 so this is our y-intercept and she gains $2 weekly so this is our slope). Todd's money growth is represented by y = 3x + 4 (he starts with $4 so this is our y-intercept and he gains $3 weekly so this is our slope). Set these two equal and solve for x. We find that after 13 weeks they have the same amount of money. But this is not what the question asked for. They want to know how many weeks it will take before Todd has MORE than Mary. Thus the answer must be 14 weeks.

Flour, eggs, sugar, and chocolate chips are mixed by weight in the ratio of 12 : 5 : 3 : 5, respectively. How many pounds of chocolate chips are there in 75 pounds of the mixture?

First, add up the four parts of the ratio. This equals 25 parts. These 25 parts make up the 75 pound mixture, which means the 75 pound mixture is composed of 3 times the 25 parts (25x = 75 so x = 3). This allows you to know that the number of pounds of chocolate chips is 3 times the ratio, i.e. 3 * 5 = 15. The mixture includes 15 pounds of chocolate chips. The answer is 15.

If the length of a rectangle is increased by 50% and its width is decreased by 20%, what is the ratio of the area of the new rectangle to the original rectangle?

First, pick a number for the original length and width. To make it easy, you can choose a length of 1 and width of 1, which would give it an area of 1. If we increased the length by 50% and decreased the width by 20%, then the dimensions of the new rectangle would be 1.5∗0.8, which would give it an area of 1.2. Thus, the ratio of the new rectangle to the original rectangle is 6:5

A train travels at 50 feet per second. If there are 5280 feet in a mile, how many miles will the train travel in an hour?

First, we must determine how many feet per hour the train travels.50 feet per second * 60 seconds in a minute * 60 minutes in an hour. 50 * 60 * 60 = 180,000 Then, it's just a matter of converting 180,000 feet to miles. Because there are 5280 feet in a mile, just divide. 180,000 / 5280 = 34.091

How many different committees of 3 people can be formed from a group of 7 people?

There are 7*6*5=210 different permutations of 3 people from a group of seven (when order matters). There are 3*2=6 possible ways to arrange 3 people. Thus when order doesn't matter, there can be 210/6 = 35 different committees formed.

A cake order cost $45.40 before tax. If the tax rate is 6.5%, what is the price of the cake after tax is applied?

For all percentage problems, you need to convert your percentage to a decimal before working on your equation. You can solve this problem one of two ways. The first way, which is less efficient, is to multiply the original amount by the tax rate and then add that to the original: 45.4⋅0.065=2.951 45.4+2.951=48.351 or: $48.35 The easier way to do this is to multiply your original amount by 1 plus the tax rate. (This does the addition for you in one step.) 45.4⋅1.065=48.351 or: $48.35 Notice that your problem is asking for the total new cost, not the amount of tax to be added!

Joe has a set of 10 books that he hasn't yet read. If he takes 3 of them on vacation, how many possible sets of books can he take?

He can choose from 10, then 9, then 8 books, but because order does not matter we need to divide by 3 factorial (10 * 9 * 8) ÷ (3 * 2 * 1) = 720/6 = 120

You have a rope of some length, but 2/3rds of it is cut off and thrown away. 1/4th of the remaining rope is cut off and thrown away. What proportion of the original rope remains?

If 2/3 is cut off and thrown away, that means 1/3 of the original length remains. Of this, 1/4 gets cut off and thrown away, meaning 3/4 of 1/3 still remains. Multiplying 3/4 * 1/3, we get 1/4.

Carol ate 3 pancakes in 5 minutes. If she continues to eat at the same rate, how many whole pancakes can she eat in 24 minutes?

If Carol ate 3 pancakes in 5 minutes, she can eat 3/5 of a pancake every minute. 3/5 pancakes * 24min = 14.4 pancakes. That means she ate 14 whole pancakes (and an additional 2/5 of another pancake).

Book sales increased by 50% from April 1st to 30th and then decreased by 50% from May 1st to May 31st. Determine the greater quanity

If book sales were 100/day then an increase by 50% means they would reach 150/day (by the end of April). A decrease at that point means a 50% decrease in 150/day which would lower book sales to 75/day. Thus sales were higher at the beginning of April than at the end of May.

Two fair dice are thrown. What is the probability that the outcome will either total 7 or include a 3?

If rolled twice, there are 6 * 6 = 36 possible outcomes. Each number is equally probable in a fair die. So only need to count the number of outcomes that fulfill the requirement of adding to 7 or including a 3. These include: 1 6 2 5 3 4 4 3 5 2 6 1 3 1 3 2 3 3 3 5 3 6 1 3 2 3 5 3 6 3 This is 15 possibilities. Thus the probability is 15/36 = 5/12.

It takes Mary 45 minutes to completely frost 100 cupcakes, and it takes Benjamin 80 minutes to completely frost 110 cupcakes. How many cupcakes can they completely frost, working together, in 1 hour?

In this rate word problem, we need to find the rates at which Mary and Bejamin frost their respective cupcakes, and then sum their respective rates per hour. In one hour Mary frosts 133 cupcakes. (Note: the question specifies COMPLETELY frosted cupcakes only, so the fractional results here will need to be rounded down to the nearest integer.) Benjamin frosts 82 cupcakes. 82 + 133=215

If a cash deposit account is opened with $7500 for a three year period at 3.5% interest compounded once annually, which of the following is closest to the positive difference of the interest accrued between the last two years and the first two years?

It is easiest to break this down into steps. For each year, you will multiply by 1.035 to calculate the new value. Therefore, let's make a chart: After year 1: 7500∗1.035=7762.5; Total interest: 262.5 After year 2: 7762.5∗1.035=8034.1875; Let us round this to 8034.19; Total interest: 271.69 After year 3: 8034.19∗1.035=8315.38665; Let us round this to 8315.39; Total interest: 281.2 Thus, the positive difference of the interest from the last period and the interest from the first period is: 281.2−271.69=9.51

For All Sweets Bakery, the daily sales ratio of bread to cakes is 5:2. If the bakery sells 12 more loaves of bread on Tuesday than its daily sale of 40 loaves, how many cakes were sold on Tuesday? (round to the nearest integer)

Since 40 loaves of bread are sold daily, and the ratio of bread to cakes is 5:2, then (40/5)*2 = 16 cakes are sold daily. Using the ratio in the same way, we can find the additional amount of cakes sold: (12/5)*2 = 4.8 is about 5 cakes with approximation. Thus, the total amount of cakes sold on Tuesday is 16+5=21 cakes.

At Jill's school fair, there is a game with 25 balloons hung on a dart board. 10 are blue, 8 are red, and 7 are green. Jill throws a dart and pops a blue balloon. What is the probability that the next balloon she hits will also be blue?

Since one blue balloon has already been popped, there are now 9 blue balloons left, and 24 balloons left overall. Therefore the probability that the next balloon Jill hits is also blue is 9/24 = 3/8.

Erin went to the movies with her friends. She spent 1/4 of her allowance on movie tickets and 3/5 of the remaining money on popcorn. If her allowance is $10, how much money remains?

Since she spent 1/4 on the ticket, 3/5 of the remaining 3/4 of the money was spent on popcorn: 3/5 x 3/4 = 9/20. This means 9/20 of the money was spent on popcorn so in total: 1/4 + 9/20 = 14/20 = 7/10 of her money was spent. This leaves 3/10 of her money behind: 3/10x 10 = 3.00

The ratio of x to y is 2 to 5, while the ratio of y to z is 2 to 3. What is the ratio of x to z?

Since the ratios are fixed, regardless of the actual values of x, y, or z, we can let 2x=5y and 2y=3z In order to convert to a form where we can relate x to z, we must set the coefficient of y of each ratio equal such that the ratio can be transferred. This is done most easily by finding a common multiple of 5 and 2 (the ratio of y to x and z, respectively) which is 10 Thus, we now have 4x=10y and 10y=15z. Setting the 10y values equal, we get 4x=15z, or a ratio of 4:15

At a widget factory, 60 workers produce 1,000 widgets per week using power from internal generators. If (f) cubic meters of fuel are required by (g) number of generators every day to power the factory, how long will (t) cubic meters of fuel last in days?

Suppose the generators consume 5 cubic meters of fuel per day and there are 10 generators. Then the number of days that 100 cubic meters of fuel will last is expressed as 100/5*10. Switching back to variables, that comes out to t/fg.

If there are 120 men and women on a committee, and the ratio of men to women is 5 : 1, how many more men are on the committee than women?

The best approach is to add the numerator and denominator of the ratio (5 + 1 = 6), and then divide the total by that sum (120/6 = 20). This gives you the value for 1 part when the total is divided into 6 parts and, luckily for us, there is only 1 part women for every 5 parts men. Now set up equal ratios: men/women = 5/1 = x/20. Solve for the number of men by cross multiplying (5 * 20 = 1x). So there are 100 men and 20 women, which makes 80 more men than women.

At 9 AM, the temperature is 65 degrees. At 2 PM, the temperature has risen to 100 degrees. What is the rate of temperature change in degrees per hour?

The change in temperature is 100 - 65 = 35 degrees. The change in time is 9 AM to 2 PM, or 5 hours. So the rate of change is 35 degrees / 5 hours = 7 degrees / hour.

What is the percent change in the area of a circle if its circumference increases from 20 to 45?

The circumference of a circle increases linearly with its radius. Therefore, an increase in circumference from 20 to 45 represents an increase of (45 - 20) / 20 = 25 / 20 = 125% increase. This means that if the radius was r, it is now r + 1.25r = 2.25r. Now, the original area, supposing r as our radius, would be πr^2 The new area would be π(2.25r)^2 = 5.0625πr^2 The percent change would be: (5.0625πr^2 - πr^2) / πr^2 = (4.0625πr^2) / πr^2 = 4.0625 or 406.25%

The cold-water faucet can fill a bucket in 30 minutes, and the hot-water faucet can fill a bucket in 60 minutes. How long will it take to fill a bucket when the two faucets are running together?

The cold-water faucet fills the bucket in 30 minutes, so in 1 minute it fills 1/30 of the bucket. The hot-water faucet fills the bucket in 60 minutes, so in 1 minute it fills 1/60 of the bucket. Then, when they're both running together they fill 1/30 + 1/60 of the bucket in 1 minute. 1/30 + 1/60 = 2/60 + 1/60 = 3/60 = 1/20, so they fill the whole bucket in 20 minutes.

A restaurant serves its steak entree cooked rare, medium, or well done. The customer has the choice of salad or soup, with one of two salads or one of 4 soups. The customer also chooses between one of three soft drinks as well as water or milk. How many unique variations are there to the entire steak dinner of steak + soup/salad + drink?

The customer has 3 choices on meat, 6 choices on side, and 5 choices on drink. This gives a total of 3*5*6 =90 choices for the meal.

What is the simple interest rate on an account that accrued $450 after a year if the original deposit was $7505? Round your answer to the nearest hundredth.

The easiest way to do this is to translate the question into "is / of language." The question is asking, "What percentage of 7505 is 450?" Remember, is means equals and of indicates multiplication. "What" means a variable like x. Therefore, we can rewrite our sentence as: x⋅7505=450 Solving for x, we get: x=450/7505=0.0599600266489 This is 5.99600266489%. Rounding to the nearest hundredth, you get 6.00 or 6.

10 people want to sit on a bench, but the bench only has 4 seats. How many arrangements are possible?

The first seat can be filled in 10 ways, the second in 9 ways, the third in 8 ways, and the fourth in 7 ways. So the number of arrangements = 10 * 9 * 8 * 7 = 5040.

The first term in a sequence of integers is 2 and the second term is 10. All subsequent terms are the arithmetic mean of all of the preceding terms. What is the 39th term?

The first term and second term average out to 6. So the third term is 6. Now add 6 to the preceding two terms and divide by 3 to get the average of the first three terms, which is the value of the 4th term. This, too, is 6 (18/3)—all terms after the 2nd are 6, including the 39th. Thus, the answer is 6.

A new painting company, Paintz, report ed a profit of six thousand dollars in 2011. In 2012, Paintz reported $10,000 more profit than 2011. What was the percentage increase in profit for Paintz between 2011 and 2012?

The formula for percentage increase is the difference in profit divided by the original profit multiplied by 100. Here the difference in profit between 2011 and 2012 is 10,000 and the original profit in 2012 is 6000, therefore the percentage increase is 10,000/6000 *100 = 166.67%.

A high school has 200 students. 120 are male, 50 are upper division students, and 40 are upper division male students. What is the probability of choosing a lower division female student, given the student is female?

There are 200 students in total, and 120 of them are male, so 80 must be female. We also know that there are 50 upper division students, and 40 of them are male, so 10 must be female. If 10 of 80 females are upper division, the other 70 females have to be lower division students, so the probability of choosing a lower division student, given the student is female, is 70/80 = 7/8.

Quantity A: The probability that all students have their birthdays in January. (There are 31 days in January). Quantity B: The probability that all students have their birthdays on a Saturday this year. (Assume there are 52 Saturdays in a year).

The probability of a single event happening is equal to the number of ways it can happen divided by the total number of outcomes. Because this problem asks about 31 students, we have to raise our probability to the 31st power because it's the probability of 31 different events happening. Now we can figure out the probability of one January birthday (31 days in January/365 days in a year) and the probability of all Saturday birthdays (52 Saturdays in a year/365 days in a year). So thus the probabilities that ALL of the students will have either type of birthday is: (31/365)^31 and (52/365)^31 Since the numbers are raised to the same power, we can simply look at the base to determine which is larger and which is smaller. Since the probability of having a birthday on Saturday (roughly 1/7) is greater than having a birthday in January (roughly 1/12), then Quantity B is greater.

The dealer gives you 4 cards from a regular 52-card deck, without replacement. What is the probability of receiving a spade, a heart, a diamond, and then a club, in that order?

The probability of choosing a spade is 13/52. Now we have 51 cards to choose from, so the probability of then choosing a heart is 13/51. Now we have 50 cards left, with 12 spades, 12 hearts, 13 diamonds, and 13 clubs, so the probability of choosing a diamond is 13/50. Lastly we choose a club out of the remaining 49 cards with probability 13/49. Therefore the answer is (13/52) * (13/51) * (13/50) * (13/49)

Alice has a puppy and a kitten. The puppy weighs 4 pounds and grows at a rate of 1 pound per month. The kitten weighs 2 pounds and grows at a rate of 2 pounds per month. Quantity A: Weight of the puppy after 8 months Quantity B: Weight of the kitten after 7 months

The puppy starts at 4 pounds and gains 1 pound per month for 8 months, so he weighs 4 + 8 = 12 pounds at the end of 8 months. The kitten starts at 2 pounds and gains 2 pounds per month for 7 months, so he weighs 2 + 14 = 16 pounds at the end of 7 months. Therefore Quantity B is greater.

If a sweater has a total cost of $54 after tax, what is the cost of the sweater before tax, if the sales tax is 4.5%?

The total cost of the sweater is the original cost plus a 4.5% tax, which can be expressed in the equation: TC=C+T Where TC is the total cost, C is the original cost, and T is the tax. Now, we will plug in the information we know. Since the original cost of the sweater is unknown, we will use the variable x. 54=x+.045x .045 is the decimal expression of 4.5%. Now, solve for x. Consolidate the x variable. 54=1.045x Divide both sides by 1.045. 51.67=x $51.67 is the cost of the sweater before tax.

A classroom has 9 boys and 9 girls. One student is chosen at random as the class leader, and a second student is chosen at random as a back-up leader. Quantity A: The probability of choosing a boy to be the leader and choosing a girl to be the back-up Quantity B: The probability of choosing boys for both roles

There are 18 students in the class, and 2 must be selected from the 9 girls and 9 boys. Key to this question is noting that the 2 students must be unique: ie once a student is selected to lead the class, he or she cannot be chosen to be the back-up. Since these are independent events, the probability of each event is found, and the events are multiplied by each other to find the total. Quantity A: P(boy leader) = 9/18 = 1/2 as there are 9 boys out of a possible 18 students. Once the boy has been chosen, there are 8 boys and 17 students total from which to choose the second student. P(girl back-up) = 9/17 because there are 9 girls and 17 students left. P(Quantity A) = (1/2)(9/17) = 9/34 Quantity B: P(boy leader) = 9/18 = 1/2 as there are 9 boys out of a possible 18 students. Once the boy has been chosen, there are 8 boys and 17 students total from which to choose the second student. P(boy back-up) = 8/17 because there are 8 boys and 17 students left. P(Quantity B) = (1/2)(8/17) = 8 / 34 = 4/17

What is the probability of drawing 2 hearts from a standard deck of cards without replacement?

There are 52 cards in a standard deck, 13 of which are hearts 13/52 X 12/51 = 1/4 X 12/51 = 12/ 204 = 3/51 = 1/17

Harry and Gerry each own equal shares in a gardening company. Their partner, Izzy, owns 40% of the company. What is the percentage increase of the sum of Harry's and Gerry's shares above Izzy's share in the company?

This basic percentage problem asks you to determine by what percentage the sum of Harry's and Gerry's share of the company is greater than ("exceeds") Izzy's share. Harry's and Gerry's share's are equal, and the sum is 60% of the company (100% - Izzy's 40% = 60%). 60% is 50% greater than 40% (0.40 * 1.5 = 0.60). Examinees may be tempted to go too far with the problem and mistake their task by finding how much greater Izzy's share is than either Harry's or Gerry's. Be careful to read instructions and translate keywords appropriately. You may think the answer is 20% (60% - 40%), but this is the difference in terms of total shares in the company, not in terms of Izzy's shares, which is what the question is asking.

What is the minimum amount of handshakes that can occur among fifteen people in a meeting, if each person only shakes each other person's hand once?

This is a combination problem of the form "15 choose 2" because the sets of handshakes do not matter in order. (That is, "A shakes B's hand" is the same as "B shakes A's hand.") Using the standard formula we get: 15!/((15 - 2)! * 2!) = 15!/(13! * 2!) = (15 * 14)/2 = 15 * 7 = 105

The President has to choose from six members of congress to serve on a committee of three possible members. How many different groups of three could he choose?

This is a combinations problem which means order does not matter. For his first choice the president can choose from 6, the second 5, and the third 4 so you may think the answer is 6 * 5 * 4, or 120; however this would be the answer if he were choosing an ordered set like vice president, secretary of state, and chief of staff. In this case order does not matter, so you must divide the 6 * 5 * 4 by 3 * 2 * 1, for the three seats he's choosing. The answer is 120/6, or 20.

There are 20 people eligible for town council, which has three elected members. Quantity A The number of possible combinations of council members, presuming no differentiation among office-holders. Quantity B The number of possible combinations of council members, given that the council has a president, vice president, and treasurer.

This is a matter of permutations and combinations. You could solve this using the appropriate formulas, but it is always the case that you can make more permutations than combinations for all groups of size greater than one because the order of selection matters; therefore, without doing the math, you know that B must be the answer

What is the number of possible 4-letter words that can be made from the 26 letters in the alphabet, where all 4 letters must be different? Assume non-sensical words count, i.e. "dnts" would count as a 4-letter word for our purposes.

This is a permutation of 26 letters taken 4 at a time. To compute this we multiply 26 * 25 * 24 * 23 = 358,800.

10,000 gallon shark tank is filled by two hoses. Hose A fills at a rate of 1,000 gallons per hour. When Hose A and Hose B are both on, they fill the shark tank in 4 hours. At what rate does Hose B fill the tank?

This problem gives the rate of Hose A instead of the time it takes to fill the shark tank. Let's convert the rest of the problem information into rates as well. The tank is 10,000 gallons, and it takes the two hoses together 4 hours to fill the tank. Therefore, the combined rate is 10,000/4 = 2,500 gallons per hour. Now we know that Hose A can deliver 1,000 gallons in an hour, and together Hose A and Hose B can deliver 2,500 gallons in an hour. So 1,000 gallons + Hose B gallons = 2,500 gallons. Then Hose B fills the tank at 2500 - 1000 = 1500 gallons/hour.

If a student borrows $200,000 at an interest rate of 6% compounded annually, when she graduates in 4 years how much money will she owe? Round to the nearest dollar.

This problem requires knowledge of the compound interest formula, A=P(1+r/n)^nt. Where A is the amount of money owed, P is the sum borrowed, r is the yearly interest rate, n is the amount of times the interest is compounded per year, and t is the number of years. We know that the student borrowed $200,000 compounded annually at a %6 interest rate, therefore by plugging in those numbers we find that after she graduates in 4 years she will owe $252,495

What percentage of profit is made on a product sold for $20 if its overall production cost was $17.50?

To find the profit percentage, you must first determine the amount of profit made on this transaction. If the sale price was $20 and the production cost $17.50, then the profit made was: 20 -17.5 = $2.50. The profit percentage is determined by dividing the amount of profit made by the original price, or 2.5 / 17.5 = (approx.) 0.14286 or 14.29%.

From a group of 8 students, 3 are attending a meeting. Quantity A: The number of different groups that could attend among the 8 students Quantity B: 336

To solve this problem, you would need to utilize the combination formula, which is C = n!/ (r!(n-r!)). C is the number of possibilities, n is the number of students, and r is the students attending the meeting. Thus, 8!/ (3!5!) = 56. 336 would be the result of computing the permutation, which would be incorrect in this case. relationship cannot be determined

In how many different orders can 8 players sit on the basketball bench?

Using the Fundamental Counting Principle, there would be 8 choices for the first player, 7 choices for the second player, 6 for the third, 5 for the fourth, and so on. Thus, 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 or 8! = 40, 320

Quantity A: The distance between the points with rectangular coordinates (6,0) and (10,0) Quantity B: The distance between the points with rectangular coordinates (1,1) and (-2,4)

We can see that the distance between the two points in Quantity A is 4 because they have the same y-coordinate and x-coordinates that are 4 apart (10 - 6). Quantity B is a little trickier to figure out and requires either the use of the formula below or creating a right triangle out of the two points. Using the formula √[(-2 - 1)2 + (4 - 1)2] is √[9 + 9] which equals √18. Although we don't know the square root of 18 automatically, we know that it will fall between √16 and √25 or 4 and 5. Since Quantity A is 4 and Quantity B has to be between 4 and 5, Quantity B is greater.

1 : 1 2 : 3 3 : 4 1 : 3 There are 28 students in a room. The ratio of boys to girls cannot be which of the above

When selecting ratios for two variables (boys and girls) the two sides of the ratio must add up to be a factor of the total student count. The factors of 28 include 14, 7, 4, and 2. (1 + 1 = 2), (2 + 3 = 5), (3 + 4 = 7), and (1 + 3 = 4). 5 is the only nonfactor and cannot be the ratio of boys to girls, thus making 2 : 3 the correct answer.

Two cars begin 500 miles apart and begin driving directly toward each other. One car proceeds at a rate of 50 miles per hour, while the other proceeds at a rate of 40 miles per hour. Rounding down, many minutes will it take for the two drivers to be 150 miles apart?

You know that the distance between these cars is defined by the following equation: Answer "250" = 500 - 90t. This is because the cars get 90 miles closer every t hours. You want to solve for t when the distance is 150: 150 = 500 - 90t; -350 = -90t; t = 35/9. Recall, however, that the question asked for the rounded-down number of minutes; therefore, multiply your answer by 60: 35 * 60/9 = 233 and 1/3 Rounding down, you get 233

A fence has a post that is 5 in wide. The fence is constructed with 5 foot pieces. What is a possible length of a straight fence if it starts and ends with a post with a post between each piece of fence

You must convert all the measurements to inches first so the fence pieces will be 60in (5ft∗12=60in). For one piece of fence, it would be two posts and one fence piece (5in+5in+60in=70in). For two pieces of fence, it would be three posts and two pieces of fence which would be the answer of 135in(5in+60in+5in+60in+5in=135in).

a is 15 percent of 20 7 is b percent of 140

a is 15 percent of 20: a=(15/100)(20)=300/100=3 7 is b percent of 140: 7=(b/100)(140)=b(140/100) b=(100/140)(7)=(100/20)=5 Quantity B is greater.

How many different license passwords can one make if said password must contain exactly 6 characters, two of which are distinct numbers, another of which must be an upper-case letter, and the remaining 3 can be any digit or letter (upper- or lower-case) such that there are no repetitions of any characters in the password?

consider the three "hard and fast conditions" - the digits and the one upper-case letter. For the first number, you will have 10 choices and for the second 9 (since you cannot repeat). For the captial letter, you have 26 choices. Thus far, your password has 10 * 9 * 26 possible combinations. Given remaining options, have 8 digits, 25 upper-case letters, and 26 lower-case letters (i.e. 59 possible choices). Since cannot repeat, will thus have for your remaining choices 59, 58, and 57 possibilities. Putting all of this together, you have: 10 * 9 * 26 *59 * 58 * 57 or 456426360 choices.

Max has 5 red marbles and 3 green marbles. He meets his friend Bob who has 4 white marbles and 4 green marbles. They combine them into a bag and pull individual marbles out one at a time with replacement. Quantity A: The probability of picking a green marble from only Max's collection of marbles Quantity B: The probability of picking a green marble from the combined bag of Max and Bob's marbles

evaluate Quantity A. Max has 5 red marbles and 3 green marbles, so the probability of picking a green marble is 3/8. Next let's evaluate Quantity B. The bag of combined marbles now has 5 red, 7 green, and 4 white, so the probability of picking a green marble is 7/16. To see which fraction is larger, we can make the two fractions have the same denominator. 3/8 is equivalent to 6/16, so now we can compare 6/16 to 7/16. Clearly Quantity B is bigger.

Three cars, A, B, and C, are in a race. A is twice as likely to win as B, and B is twice as likely to win as C. What is the probability that B or C wins?

find their individual probabilities of winning. Since B is twice as likely to win as C, P(B) = 2 * P(C). Since A is twice as likely to win as B, P(A) = 2 * P(B) = 2 * 2 * P(C) = 4 * P(C). We also know that the probabilities must sum to 1, so P(A) + P(B) + P(C) = 1, meaning P(C) + 2 * P(C) + 4 * P(C) = 1. Then P(C) = 1/7, so P(B) = 2 * P(C) = 2/7. Our answer is therefore P(B or C) = P(B) + P(C) = 2/7 + 1/7 = 3/7.

Kathy travels 40 miles in 1 hour and then 60 miles in 3 hours. What is her average speed in miles per hour?

looking at the two rates separately would give us 40 mi/hr and 20 mi/hr, making the trap answer here 30 mi/hr. The key is that Kathy goes 40 mi/hr for 1 hour and then 20 mi/hr for 3 hours, so the speed should be less than the middle number of 30. 18 mi/hr is lower than her lowest speed so that doesn't make sense, and 20 mi/hr is too low because the 1 hour at 40 mi/hr should bring the average up a little bit. Therefore the answer must be 25 mi/hr. We can also do the computations to find the answer. The correct formula is average miles per hour = total miles / total hours. Plugging in our values, average speed = (40 + 60) miles / (1 + 3) hours = 25 mi/hr.

Mike has a bag of marbles. 5 are green, 8 are red, and 3 are blue. He pulls one marble out of the bag and it is green. He pulls out another one and it is red. He does not return these marbles to the bag. What is the probability that the next marble he pulls out of the bag will be green?

need to find out how many marbles are in the bag in total. 5 + 8 + 3 = 16. He removes a green marble so now there are only 15 in total. When he removes the red one there are then 14 marbles in the bag. 14 is your denominator. The odds of picking a green one are 5 - 1 or 4 because there are only 4 green marbles left in the bag; therefore, the odds of picking another green marble is 4/14