Word problems
nCn =
1
In a 2X2 local ratio problem...
1. Identify whether to use Lcm or 100. Make LCM(V) or 100(V). 2. Identify the clues in the question. NOTE: boxes can be either something (V) or just something. Does not have to have a variable attached.
Work Word problems, rate together formula
1/a + 1/B = 1/t
5C4 is the same thing as
5C1 NCR = NCN-R
If they give a number line and say each point is an equal distance apart
Find that distance, D and then count the number of Ds to the point you want
What is a complementary event? Probability
Probability of an event happening is equal to 1 - probability if event not happening = 1 - number of relevant cases/total number of cases
Gross Profit
Revenue - expenses ( or selling price - cost)
For speed rate questions involving 2 people remember that
The same amount of time passes for both people
If a question looks like a classic 2 by 2 question but says that there's no elements that do neither
Then do the Venn diagram approach and define the middle as X
X is 200% greater than Y
There is a "of Y" omitted after 200%. Greater than —> addition X=(200*1/100)Y+ Y = 3Y
Work formula
Work = rate x time
nCn-1 =
nC1 = n nCn-n+1
average speed formula
total distance/total time
X is 200% as great as Y
"As _____ as" axiom X = 200*(1/100*Y) = 2Y
Picture 8X10 surrounded by picture frame of uniform width with area 144
(8+2X)(10+2X) = 144+ 80
Which of the following cannot be the factor of x 4 + 10x3 + 35x2 + 50x + 24?
(X+A)(X+B)(X+C)(X+D) ABCD = 24 Therefore the answer will have an number that doesn't divide into 24
A box has 10 balls, either red or blue. If two balls are to be drawn simultaneously from the box, what is the number of red balls? (1) The probability that the two balls drawn will be red is 1/15. (2) The probability that one of the balls drawn will be red and the other will be blue is 7/15.
1. RC2/10C2 = 1/15. R/(R-2)!*2/10C2 2. RC1*BC1/10C2 = 7/15; R*B/10C2
Oil in a tank leaks at the rate of m gallons per s minutes. If oil costs $8 per gallon, what is the cost, in dollars, of the amount of the oil that will leak out in t minutes?
1. Set up proportion. Gallons: Minutes : Cost 2. Plug in what you know M : S 1: : 8 3. Multiply bottom one by M. M: 8M S: 8M = T: something S(something) = 8MT Something = 8MT/S
If a car is drove at a constant rate of 41.2 kilometers per liter of gasoline, approximately how many miles per gallon of gasoline did the car drive at the constant rate? (1 mile = 1.6 kilometers and 1 gallon = 3.8 liters, both rounded to the nearest tenth.)
1km = 1/1.6 miles; 1 liter = 1/3.8 gallons 41.2 km:1 liter = X miles: 1 gallon 41.2/1.6:1/3.8 = X miles: 1 gallon 41.2/1.6 = X/3.8 X = 41.2* 3.8 / 1.6 = 184*2/3 approx 100
John traveled westward past a certain bus stop at a constant speed of 40 miles per hour at 12:00. Then, 10 minutes later, Tom passed the same bus stop westward at a constant speed of 50 miles per hour. If both traveled at their constant rates, at what time after John passes the bus stop will Tom catch up with John?
1st step. Find out the distance John traveled in 10 minutes. 20/3 miles in 10 minutes 2nd step. Recognize that the time elapsed for John and Tom is the same. 3rd Step. Set Tom's distance = John's distance. 20/3+40T = 50T. T = 40 minutes 4th step. Recognize that this needs to be added to the 10 minutes that had already elapsed since Tom hasn't started. This 40 minutes is the amount of time elapsed that Tom and John travelled together. Answer: 50 minutes after 12 PM 12:50PM
The number of su sets with m elements is?
2^m
There are 4 machines with the same work rates. If it took M hours for 3 machines together to work and did. M-3 for 4 machines together, what is m?
3 machines * m hours * rate = 4 machines * m-3 hours * rate The rates cancel out. 3M= 4M -12. M=12
Four identical machines can fill up a water tank W in 7 days at a constant rate. How many machines will it take to fill up the water tank W in 4 days?
4 machines X rate X 7 days = N machines X rate X 4 days Rates machines and days cancel. Left with N = 7
If a code number is a sequence of different numbers selected from the 9 numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9, what is the ratio of the number of 5-digit code numbers to the number of 6-digit code numbers?
5. When it comes to code numbers, apply permutation.In order to make 5-digit code number out of nine numbers, it is efficient to substitute directly. Because the possible number of cases is 5- digit, for 5-digit code numbers ( )( )( )( )( ), 9 numbers can be placed in the first place, then 8 numbers can come in the second. After that, 7 numbers and followed by 6 and 5 numbers, which makes (9)(8)(7)(6)(5). In the same logic, for 6-digit code numbers, (9)(8)(7)(6)(5)(4) is derived. Therefore, the number of 5-digit code numbers : the number of 6-digit code numbers=(9)(8)(7)(6)(5) : (9)(8)(7)(6)(5)(4) = 1 : 4.
A certain machine produces 500 units of toys per hour. When the machine will produce the toys working non stop at this constant rate, how many units of the toy will be produced in 5 days?
500 units: 1 hour 5 days = 5* 24 hours = 120 hours 500 units : 1 hours = something : 120 hours Something(1 hour) = 500*120 = 60000
A motor vehicle traveling at a certain constant speed takes 10 seconds shorter to travel 1 mile than it would take to travel 1 mile at 60 miles per hour. What is the vehicle's speed in miles per hour?
60 miles per 1 hour = 60 miles per 60 minutes = 1 mile per 1 minute = 1 mile per 60 seconds. The questions is asking for 10 seconds shorter. So 1 mile per 50 seconds = x miles per 1 hour = x miles per 3600 seconds. 1 miles * 3600 seconds = 50 seconds * x miles. All the seconds and miles cancel 50X = 3600 x = 72 miles per hour
simple interest
= principal * interest rate * time
Compound interest semi annually
A(1+ r/100*1/2)^2n
Compound Interest Formula
A(1+r/100*1/n)^nt where N is amount compounded per year
In a question that tells you in a race 5 contestants run, A must finish before B and B must finish before C?
A>B>C Set these all equal to A AAADE 5!/3!= 20
by means
Addition
If a question says that a list of numbers each 2 numbers added up equals 8
All the numbers are 4
In 2by2 Venn diagram questions
Always start with the intersections. Let X equal the middle
John traveled the first 100 miles of a 200 mile trip at a constant average speed of 50 miles per hour. At what average speed must he travel the remaining 100 miles to finish the entire 200 mile trip at the constant average speed of 60 miles per hour?
Average speed = total distance over total time. Trip split into two parts. 1st 100 miles is at 50 miles per hour. t1 = 2 hours 2nd 100 miles is at x miles per hour. t2 = 100/x Apply formula 200/t1+t2 = 200/(2+100/x) = 60
What is the difference between increases by and increases to?
By: indicates addition or subtraction To: become finally
When you're given a number line with 4 points and it says each point is an equal distance from eachother?
Call the distance D. Then equate each term as a multiple of D away from the first point. Now you have 2 variables
In arranging the letters, P, R, E, S, and S, how many cases are there where at least one letter in between S and S?
Complementary event. # total - #complementary Total = 5!/2! = 5*4*3 = 60. Complementary is all the cases in which S and S move together. This means S and S form 1 letter in theory, lets call it X. PREX. this is 4! = 4*3*2*1 = 24 60-24 = 36 Note that if the question asks for the case where the letters PR travel together, then you have to consider both slots
When 100 CM is 1 meter, 200 square centimeters is equal to?
Consider united as variables. Given M= 100 * CM 200 CM^2 = X M^2 Then 200 CM^2 = X (100 CM)^2 200= X* 100^2. X = 200/10000 = 2/100
Mixture problems involving buying and selling products that are mixed together
Determine the total cost and divide by total amount
Distance formula
Distance = Rate x Time
Exceeds means
Equal to
In a problem that gives you a fraction of multiple factorials
Find the lowest number, expand out the factorials to that number and cancel out
Mixture problems involving salt water concentration
Find total salt concentration and divide by total amount
4 men and 4 women sit around a table. How many seating patterns with alternating men and women?
First find the total permutations of just men: (4-1)! Then find the fixed arrangements of the women: 4 remaining sports, 4 women—> 4*3*2*1= 4! Multiple (4-1!)(4!)
The code numbers consist of 4-digit and 2 digits are different and 2 digits are the same and none of them is zero, for example, 1122, 1212. How many such code numbers are possible?
First, the number of possible ways of to list 1, 1, 2, 2 is 4!/2!2! = 6. Then, from the total 4 digits, each 2 digits are the same, from the 9 numbers 1,2,3,4,5,6,7,8,9 , the number of ways to select two numbers is 9 C 2 ==(𝟗)(𝟖)/𝟐! 36 , and since each of the code numbers can be in 6 ways as shown above, you get ( 6)(36)=216. The
Machine A can produce toys at a constant rate of 3 units per hour and machine B can produce toys at a constant rate of 4 units per hour. If at least one of machine A or machine B produces toys, what are the smallest possible hours when machine A and machine B work together at their constant rates so that two machines A and B can produce 77 units of toys in 14 hours?
For these types of questions, find how many the more efficient machine can produce by itself in the alloted hours. Take the difference and divide by less efficient machines rate. 4 units * 14 = 56. 77-56 = 21 21/3 = 7. There 7 hours together is the least amount.
10 students including Salvatore and Herold are in a dormitory. If 3 students are to be select ed randomly from the students to make an executive branch , what is the probability that the executive branch must include Salvatore and Herold ?
From the total of 10 students, 3 students are randomly selected, so Total= 10 C 3 = 120 and since Salvatore and Herold must be included in the executive branch, if assuming that Salvatore and Herold are already selected, from the rest 8 students, only 1 person needs to be selected, so you get Want= 8 C 1 = 𝟖. Thus, in order to make the executive branch, 3 out of 10 students are selected, and the probability that Salvatore and Herold are both selected becomes 8/120 = 1/15
X is 100 percent greater than Y, Y is what percent less than X?
Greater than —> addition. And "of Y" is omitted after 100 percent. X = 100*(1/100)*Y + Y = 2Y Less than —> subtraction AND an order change. "Of X" is missing after percent. Y = X - var*1/100*X Can then solve by plugging in x=2Y
The probability of two tossed coin s to be one head and the other to be tail is 4/9. Rewrite this to solve for T
HT + TH = 4/9 H = T -1
discount
If something gets N% discount, it means we have to multiply original price by (100-N)%
Each of the 15 boxes in a certain warehouse is either 5 pounds or 10 pounds, and the average (arithmetic mean) weight of the boxes in the warehouse is 8 pounds. If the average weight of the boxes in the warehouse is also 8 pounds when 5 boxes are removed, how many 10-pound boxes must be removed?
If the number of boxes that weigh 5 pounds is x, and the number of boxes that weigh 10 pounds is y, you get x+y=15 and (5x+10y)/15=8, then 5x+10y=8(15)=120. If you divide both sides by 5, you get x+2y=24, subtract x+y = 15 get y=9, x=6. In other words, 5 pounds (x=6) ---------> 6-a 10 pounds (y=9) ---------> 9-b Then you get a+b=5, and 5(6-a)+10(9-b)= 8(6-a+9 b)=8(15-a-b)=8(15-5)=80, then 30-5a+90-10b=80, 120-5a-10b=80, 120- 80=5a+10b, and then 40=5a+10b. If you divide that by 5, you get 8=a+2b. However, since it is a+b=5, from 8=a+2b=a+b+b=5+b, you get b=3 and a=2. However, since the question asks how many boxes that weigh 10 pounds should be removed, from b=3,
A new rectangular solid is former from 36 smaller cubes with 1 as side length. Translare
LWH = 36
John is collecting coins. If the amount he collected last year was 80% of the amount he collected this year, what is the percent increase of the coins that he collected from last year to this year?
Let This Year = 100 Last Year = 80 Last Year (Before) --> This Year (After) TY - LY/LY = 100-80/80 = 20/80 = 25%
A watermelon is comprised of 50% water. On a certain hot summer day, the water loses 40% of its water content. What percent of the watermelon on water after it's been dehydrated by the sun.
Let watermelon = 100W Water content = 50W Loses 40% water= 50W - 20W = 30W The water content decreases for both! So it's 30/80 NOT 30/100
Machine M take 6 hours to do x amount of work. If M does 2/3 of the x amount of work and the rest of the work is done by machine N that works 1/15 the speed of M, what is the total amount of time taken to do x amount of work?
Machine M: X/6 hours --> (2/3X)/(2/3*6 hours) --> (2/3X)/4 hours Rest of work = 1/3 X. Machine N rate = 1/15*X/6 = X/90 hours Machine N: (1/3)X/(1/3*90 hours) = (1/3X)/30 hours. 30 hours + 4 hours = 34 total hours
For path finding questions, given a grid
Make each column a variable A and each row a variable b. The. Apply coffee
At a certain kindergarten, children drank 141 glasses of milk for 7 consectuive days. If they drank the most glasses of milk on Monday , what is the least number glasses of milk the children drank on Monday?
Monday: = least (N) other 6 days = Max (n-1) so 6(n-1) + n = 141. N=21
In a certain conference, if all n attendees shake hands 120 times, what is N?
NC2 is the number of handshakes = 120 N!/(N-2)!2! = N(N-1)(N-2)!/(N-2)!*2 (N)(N-1)/2 = 120 N(N-1) = 240
repeated permutation
N^r If it's a repeated permutation problem, it's the amount of choices available ^ number of people Have to read the question for context. If every time you choose something the next person cannot choose it the normal N choose R formula applies. Otherwise this formula applies
A store had 10 loaves of bread, out of which 7 were baguettes. If the store sold 6 loaves of bread, what is the probability that the store sold exactly 4 baguettes out of these 6 loaves? (Assume that every loaf has an equal chance of selling)
Need chances that 4 out of 7 bagguetes multiplied by 2 out of 3 non bagguetes over 6 out of 10 7C4*3C2/10C6
In company K, the printer's speed is z percent faster than the old printer's speed. Then how many percent slower is the old printer's speed compared to the new one's?
New printer: N Old printer: O N = O + O(1+z/100) = O (1+z/100) "Compare to N" means this is the "Before" in the percentage change equation. Percentage change = old - n/n
A store currently charges the same price per pound of salad. If the current price per pound were to be increased by $0.2, 0.5 pound smaller of the salad could be bought for $9. What is the current price of salad per pound?
Old unit price = p 9/p - 9/p+0.2 = 1/2
If a work rate problem asks what the least or greatest possible hours 2 machines can work together to complete a task in a set amount of hours
Plug in the answer choices with the together rate and see if either of the machines can complete the rest of the work in the allotted time
If a committee of 4 people is to be selected from 10 people including Kelly and Salvatore so that the committee must include Kelly and Salvatore, how many possible committees can be formed?
Since 4 people are selected out of 10 people, and Kelly and Salvator must be included in the committee, if you assume that Kelly and Salvator is already selected, from the rest of the 8 people, only 2 people need to be selected. Hence 8C2 =(8)(7)/2! =(4)(7)=28.
There are 10 pairs of socks with each pair in different colors. If we select 2 from the 20 socks, what's the probability of selecting the same color socks?
Since there are 20 socks in total, you get Total= 20 C 2 = (𝟐𝟎)(𝟏𝟗)/𝟐! = 190 , and since you need to select same colored socks, from the total 10 pairs, you must select 1 pair, and from the pair, you must select both socks, so you get Want=( 10 C 1 ) (2 C 2 )=(10)(1)=10 . The probability of selecting 2 same colored socks from 20 socks becomes 𝟏𝟎/𝟏𝟗𝟎 = 𝟏/𝟏𝟗.
If Samson is filling a bathtub with COLD water, it will take him 6 minutes and 40 seconds, and if he fills it with HOT water, it will take him 8 minutes. If draining the tub takes 13 minutes and 20 seconds, how many minutes will it take to fill up the bath tub with both HOT and COLD water running while the plug is out, so the water is constantly draining?
Solve using reciprocals, and the draining is subtracted
A certain machine produces 50 units of item I per minute. If the machine works at a constant rate, what will be the daily production quantity of the machine?
Start out with 50 units: 1 minute Multiple by 60 min —> 50*60 units: 1 hour Multiply by 24 hours—> 50*60*24 units: 1 day 72,000 units:1 day
Tom is riding a bike at a constant rate of 60KM per hour. What is the speed rate of the bike, in meters per second? (1Km = 1,000 meters)
Start out with 60KM: 1 hour Rewrite as 60KM:60 minutes Divide by 60-> 1Km:1 minute Rewrite as 1,000 m: 60 seconds Divide by 60 —> 1,000/60 meters: 1 second
The positive numbers w, x, y and z are such that x is 20 percent greater than y, y is 20 percent greater than z and w is 20 percent less than x. w is what percent greater than z?
Start with Z, let it be 100. work backwords from there
In a certain school, 115 students played at least one sport including football and basketball. The number of students who played football is twice the number who played basketball. If 50 students played both football and basketball, how many students played only football?
Start with the intersection, venn diagram B + F -50 = 115 B+F = 165 F = 2B 3B = 165. B = 55 F = 110 THEN SUBTRACT OUT BOTH (50) Answer is 60
What does "than" represent In percentage increase or decrease questions?
The before. Also indicates the best place to start (ie make 100 and go from there)
What is a complementary event? Number of cases
The number of cases is equal to the total number of cases - the non relevant cases
A bus and a truck is moving while facing each other on a road 253 miles long. When the velocity of the bus is 70miles/1hr and that of the truck is 91miles/1hr, how far did the bus travel when the bus and the truck meet?
The time elapsed is the same for the bus and the truck. = t so the bus travelled 70t and the truck travelled 91t when they met. 161t = 253 (total distance). Solve for t and then plug into 70t
When you see the word "select"
Think Combination "N choose R" formula
When you see "at least" one male must be selected
Think complementary event
In a certain company, a manager gives 3 employees one day break from Mon to Fri. If 3 employees cannot get one day break at the same time, how many possible cases are there?
This is a repeated permutation problem. The employees choice does not affect the other employees Each employee may be rostered off on 5^3 different days. This gives 125 total days. Have to subtract out the days for which the employees have the same day off. 5 days (all monday, Tuesday, Wednesday, Thursday, Friday) Total= 120
John sent at least one email per day
This means that 1 is the minimum. That is one equation for DS questions.
Machines A, B, and C complete a task at their respectively constant rates in 2 hours, 4 hours, 5 hours. If machines A, B, C will complete the task when working together at their respective constant rates, what fraction of a total amount of working by machines A, B, C together is the amount of working by machine A alone?
Time is inversely proportional to work rate. Work rate of A:B:C = 1/2:1/4:1/5 LCM 20 = 10:5:4 A = 10K B=5K C=4K. The question is asking for A/A+B+C = 10K/19K = 10/19
If a certain coin is flipped, the probability of the flipped coin to land on head is 1/2. If the coin is flipped 5 times, what is the probability that it will land heads up on 3 flips and not on the other 2 flips?
Total cases = 2^5 = 32 3 Heads 2 Not heads is HHHTT. Next find how many cases these have. 5!/3!2! = 10 Probability = 10/32 = 5/16
5 parking spots are arranged in a row. In how many different ways can these parking spots be filled with 3 identical blue cars, one red car, and one white car?
Total spots = 5! Fixed number = 3 So it's 5!\3! = 20 coFFEE: 6!/2!2!
For questions that ask about the divisibility of numbers
Treat them the same as remainder questions and use direct substitution. Plug in numbers
When you see the word probability
Write down the formula before starting the question Total relevant/total possible
If a question asks for number of subsets that excuses an element
You remove it and apply 2^m
If a question asks for number of subsets that includes one element
You remove it and apply 2^m
In a work problem if you see the terms "together" and "alone"
You solve using reciprocals
