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What is the average speed per miles per hour for a trip? Multiple RTD Chart, NEVER just calculate the average of 2 trips ! Pick a number for unknown variable d) Work backwards to find the T for each trip R * T = D A to B 40 t1 = SMART NUMBER 200 B to A 50 t2 = SMART NUMBER 200 1. total time t1 + t2 = 9 hours 2. The average speed = total distance / total time
Back solving: How much WAS in account Y? P has money in 2 bank accounts. Account x earns 3% interest and account y earns 15% interest. The total interest earned was 53. If the initial amount was 400, how much was in account Y? Backsolving: 1. Draw a table and get equations: x + y = 400 ; 0.08X + 0.15 Y = 53 2. Plug in ANSWER CHOICES and CHECK with the 2. Equation !
Using Charts to organize values: A circus earned 150000 in ticket revenue by selling 1800 VIP and Standard tickets. They sold 25% more standard tickets than VIP tickets. If the revenue from standard tickets is 1/3 of the Total Revenue, what is the price of a VIP ticket? DRAW A CHART price per ticket * quantitiy = revenue Standard 1.25 x 50000 VIP x 100000 Total 1800 150000
Venn Diagrams Work from inside out 1. Substract overlapping data points Total number = add up all the numbers
DATA SUFFICIENCY: Weighted averages Tickets to a play cost 10$ for a child and 25$ for an adult. If 100 tickets were sold, were more adult than child tickets sold? i) Average revenue was 18 => DRAW A LINE AND SEE THAT 18>average ii) The Total revenue exceeded 1750 => SEE THAT this means 17.5<x
PROPERTIES of Evenly spaced sets 1. Arithmetic mean = the Median 2. Arithmetic mean = Median = (First + Last) / 2 3. Sum of elements = Arithmetic Mean * n
Evenly spaced sets Consecutive integers; Increment = 1 (1,2,3,4,5) a. Consecutive even integers = (8,10,12,14) Increment = 2 b. Consecutive primes = (11,13,17,19) EXCLUDING 15 as it is not prime Consecutive multiples; Increment = multiples of increments (3,6,9,12,15) Evenly spaces sets; Increments = constant (2,5,8,11,14) All sets of consecutive integers are sets of consecutive multiples and evenly spaces sets. All evenly spaces sets are fully defined if First + Last number Increment n are knows.
PROPERTIES of consecutive integers Total number of integers n = Last - First + 1 Median / Arithmetic Mean = (Last * First) / 2 Median * Number of integers n = Total ODD numbers of Elements => Always an integer (middle number) EVEN numbers of Elements => Never an integer = average of 2 consecutive numbers is .5
What is the sum of all the integers from 20 to 100 inclusive? n * Median 1. n = ( Last - First ) / Increment + 1 ( 100 - 20) + 1 = 81 2. Median = 100+20/2 = 60 3. Sum = 81 * 60
How many integers are there from 14 to 765 inclusive? 764 - 14 + 1 = 751
How many multiples of 4 are there? ( Last - First) / 4 + 1
DATA SUFFICIENCY: Catch-up problem A and B are driving around a circular track. If A is 200 m behind B and bot drivers drive at their respective constant rate, how long, in seconds will it take for A to catch up to B? No Data given, CATCH UP PROBLEMS REPHRASE : "What is the difference in the two drivers rates?" i) lenght of the track = INSUFFICIENT ii) A's "GAINING" and speed, 25% quicker => 200/25 = 8 D SR
OPERATIONS on Consecutive Integers AVERAGE / MEAN / MEDIAN ODD n (Number of items) => the average is ALWAYS an integer EVEN n (Number of items) => the average is NEVER an integer SUM of n items S = n * average ODD n (Number of items) => the sum is ALWAYS a multiple of n EVEN n (Number of items) => the sum is NEVER a multiple of n PRODUCT ODD and EVEN n (Number of items) => the product of n consecutive integers is ALWAYS divisible by n factorial n! DIVISIBILITY ONE EVEN INTEGER in a series of consecutive integers: product of the series is divisible by 2 TWO EVEN INTEGERS in a series of consecutive integers: product of the series is divisible by 4 Set up prime boxes and see how often you can find even numbers
If x is an even integer, is x ( x+1) ( x+2) divisible by 4? 1. Set up a prime box 2. If there is only on even integer in a series, then the product is divisible by 2 3. As x is even, x+2 will also be even. There will hence be 2 even integers in a serien, which makes it divisible by 2*2 = 4 ! Alternatively : The product of n consecutive integers is always divisible by 2!
Is the sum of the integers from 54 to 153 divisible by 100? Number of integers n = (153-54) +1 = 99 + 1 = 100 100 is even RULE: For any set of consecutive integers with an EVEN number of items, the sum is NEVER a multiple of n!
DATA SUFFICIENCY: If S is a set of consecutive integers, what is the sum of the elements in S? 1) The largest element is 55 2) There are 11 elements in S TRY ALL THE FORMULAS FIRST : Number of items; Average; Sum SUM = n * Average Average = (FIRST + LAST) / 2 BOTH seem not to be sufficient BUT TAKE THE FORMULA FOR n n = (Last - First) + 1 11 = 55 - First + 1 First = 45
5 identical pieces of wire form a longer wire with pieces overlapping by 4 cm at each join. if the wire is 1 m long, how long is each piece? Computation Problems: DRAW a SKETCH and find out that 4 (x-4) + x = 100 x = 116/5
DATA SUFFICIENCY: How many days after the purchase of a product does its standard warranty expire? No leap year SCHEDULING PROBLEM: 1) Mark purchased product X in 1/1997, the warranty did not expire until 03/1997 Longest possible warranty period: 1.1. - 31.3. = 89 days later Shortest possible warranty period: 31.1. - 1.3. = 29 days later 2) Santos purchased product X on 5/1997 and it expired in 5/1997 Longest possible warranty period: 1.5. - 31.5. = 30 days later Shortest possible warranty period: 1.5. - 1.5. = 1 day later THERE ARE STILL POSSIBILITIES: 29 or 30 days. NOT SUFFICIENT BUT if leap year, the two choices would have been sufficient!
GROUPING PROBLEMS: A company is breaking up its conference attendees into groups. Each group must have exactly one person from Division A, 2 from Division B and 3 from Division C. There are 20 people from Division A, 30 people from Division B and 40 from Division C. What is the smallest number that will not be assigned to a group? GROUPING PROBLEMS MAKE a CHART Ppl per group TOTAL max groupds take leftover Division A Division B Division C
Standard deviation How far from the mean do the data points fall Small SD = Data points are clustered closely Large SD = Data points spread out widely around the mean SD of 0 = Numbers are equal Changes of SD's ADDING NUMBERS = No Change, the proportion is the same MULTIPLY BY A NUMBER = CHANGE IN SD
The Median (non consecutive sets) The middle value of a set 1. Ordered set (increasing or decreasing order): Median depends on the 2 middle values ODD n = Median is the unique middle value EVEN n = Median is the average of the two middle values 2. Unordered sets with unknows: Does not matter, it is always the middle value 3. Entirely unknown sets: Label the items A;B;C;D;E and draw them in a chart
How many numbers are there of digits available for the third digit and 2 given digits to be the same Make a table 701-799 800-899 900-999 ones and tens 9 -1 (weil größer als 700) 9 9 tens and hundredts 9 9 9 ones and hundreds 9 9 9
Sam earned a 2000$ commission on a big sale, raising his average commission by 100$. If his new commission average is 900, how many sales did he make? SET UP AN RTD table A*n = S Average * n = S Old total 800 n 800n This Sale 2000 1 2000 New total 900 n+1 900* (n+1)
The weighted average The average weight of 2 values will fall closer to whichever value is weightes more heavily weighted average: = a / (a+b) + b / (a+b) "a 20's and b 30's 1. Look for the differential => 2 and 5 -> 2. Set up an equation => 2X + Y(-5) = 0 3. Search a number which wll lead to 0 => 5 and 2: the ratio will be 5:2; for every 5 parts of y you have 2 parts of x
A mixture of "lean" ground beed (10%) fat and "super-lean" ground beef (3%) fat has a total fat content of 8%. What is the ration of "lean" ground beef x to super-lean ground beef? 1. Lean ground beef 10% = x+2 2. Super-lean gound beef = y-5 3. Mixture: 2X + (-5)Y = 0 4. Pick values so that it equals 0: 5 and 2 5. The ratio will be 5:2. For every 5 paty of lean beed we have 2 party of ground beef.
A group of men and woman in a room are in a ratio of 2:3. If men have an average age of 50 and the groups average age is 56, figure out the average age of the women Men: x-6 Women = ? Ratio 2:3 Equation: 2*(-6) + 3*(Aw) = 0 Aw = 4 => Average Age is 60
INTEGER CONSTRAINTS: If K received 1/3 more votes than mike in a student election, which could be the number of total votes? K = 4/3 M = 3/3 SET UP A COMMON DENOMINATOR: 4+3 = 7 Choose the multiples of 7 => The total number of votes could be : 7,14,21,28,35 etc.
A store sells erasers for 0.25 $ each and pencils for 0.11$ each. Hoch many erasers did J buy? i) J bought 5 erasers => 1.15$ => NOT SUFFICIENT ii) J spent $1.70 on erasers and pencils => TRICKY ! This is SUFFICIENT just because it does not look like it is sufficient, it does not mean that it is not sufficient TEST OUT: 3.23 = 69 4.23 = 92 5.23 = 115 = 5 erasers and 5 pencils
Harvey runs a 30 mile course at a constant rate of 4 miles. If Clyde runs the same track at a constant rate and completes the course in 90 fewer minutes, how fast did Clyde run? MULTIPLE RATES: SET UP A CHART R * T = MILES Harvey 4 * t = 30 Clyde R * t - 1.5 = 30 2.) Solve for R!
A research analyst was noting a few values down for his research. While noting the data, he mistakenly noted 42 instead of 24. Because of this mistake, the average increased by 2. Find the number of values that was present in the analyst's research. No mistake: x x x x x x 24 = A1 = A1 * n - Mistake : x x x x x x 42 = A1+2 = A1+2 * n 18 = 2n n values in each equations Average given Aim: find n and cancel unknown terms out FIND THE DIFFERENCE IN SUMS
A certain club holds 10 members, including Harry. One of the 10 members is to be chosen to be president, one of the remaining 9 member is to be chosen to be secretary any one is to be chosen to be treasurer. What is the probability that Harry will be either secretary or treasurer? 1. Probability that Harry will be chosen Secretary = 9/10 * 1/9 = 1/10 2. Probability that Harry will be chosen Treasurer = 9/10 * 8/9 * 1/8 = 1/10 3. Together: 0,1*0,1 = 1/5
BODIES MOVING TOWARDS EACH OTHER ! Safe time by creating a this RT=D equation for the rate at which the distance changes! Bodies move towards each other => decrease the distance + R * t = D Person A 3 t = d Person B 4 t = 14-d Total Distance Chane 3 + 4 = 7 t = 14 TIME SAFER!
BODIES MOVING AWAY FROM EACH OTHER ! Safe time by creating a this RT=D equation for the RATE at which the distance changes! Bodies move away from each other => increase the distance between themselves at Rate + R * t = D Person A 3 t = d Person B 4 t = 14-d Total Distance Chane 3 + 4 = 7 t = 14 TIME SAFER!
BODIES MOVE IN THE SAME DIRECTION ! Safe time by creating a this RT=D equation for the rate at which the distance changes! Bodies move towards each other => decrease the distance between themselves at a neg Rate - R * t = D Person A 3 t = d Person B 2 t = 14-d Total Distance Change 3 - 2 = 2 * t = 14 TIME SAFER!
Direct proportionality y = k*x with k as the proportionality constant x ist the input, y is the output
Inverse proportionability y = k/x reciprocal relationship k is proportionalityconstant x is input, y is outpus
Exponential growth function y (t) = y0 * kt k = factor the equation increases by " in a manner that the ratio of its values in any 2 following years is constant"
A quantity increases in a manner such that the rate of its values in any 2 following years is constant. If the quantity doubles every 6 years, by what factor does it increase in 2 years? EXPONENTIAL GROWHT "constant rate" y = y0 * kt quantity doubles : 2y = y0 * k6 l :y 2 = k6 = FACTOR EACH YEAR Factor in 2 years k = 3√2
Conjungation to rationalize denominators for a+√b = the conjuncate is a-√b for a-√b = the conjungate is a+√b SIMPLIFY 4/(3-√2) => 2*2 / (3-√2) * (3+√2)/(3+√2) = 4*(3+√2) / 9+2 = 12+4√2 / 9-2 = 12+4√2 / 7 (3-√2) wird also zu 9-2
FUNCTIONS: The Magic BOX Numerical Substitution f(5) Variable Substitution f(w+6) Compound functions f(g(3)) => WORK FROM THE "INSIDE OUT" => g(3) gives an input for f(x) Unknown constants f(x) = ax2-x Common function types Proportionability Linear groth Exponential growth
PS: The Population Problem : The population of a certain bacterium triples every 10 minutes. If the population of a colony 20 minutes ago was 100, in how many minutes will it reach 24000? PICK SMART NUMBERS AND TRY OUT! time elapsed population -20 minutes 100 -10 minutes 300 now 900 10 minutes 2700 20 minutes 8100 30 minutes 24000
Rebecca, who is 34 years old and her daughter, who is 8 years old celebrate their birthdays. How many years will pass before Rebecca's age is twice her daughters age? let x be the desired number of years. In x years, Rebecca will be 34+x years old In x years, her daughter will be 8+x years old It follows that 24+x=2*(8+x) = 34+x = 16+2x -> x=18
PS IMPORT TAX PROBLEM: Leo imported an item and he had to pay 7% import tax on the portion of the total value of the item in excess of 1000. If the amount of the import tay that Leo paid was 87.5, what was the total amount of the item? 7% import tax on the portion of the total value of the item in excess of 1000 = allen über 1000 mit 7% versteuern Total Value = Excess Value + 1000 and Tax = 0.07 * Excess Value 0.07 * (Value - 1000) = 87.5 x-1000 = 1250 x = 2250
A collection of 16 coins, each with a face value of either 10 cents of 25 cents has a total face value of 2.35$. How many of the coins have a face value of 25 cents? Let x represent the number of coins each with a face value of 25 cents. Face value of 25 cents = x Face value of 10 cents = 16-x 25x + (16-x)*10 = 235 x=5 Therefore, 5 of the coins have a face value of 25 cents.
The numbers of cars sold at a certain dealership on six of the seven business days were 4,7,2,8,3 and 6, respectively. If the numbers of cars sold on the 7. business day was either a) 2, b) 4, or c) 5, for which of the three values does the average equal the median number of cars sold for the 7 business days? 1. List in a numerical order for l, ll and lll NOTE n = 7 SUM DIFFERS a) 2,2,3,4,6,7,8 -> Median = 4, average = 32/7 b) 2,3,4,4,6,7,8 -> Median = 4, average = 34/7 c) 2,3,4,5, 6,7,8 -> Median = 5, average = 35/7 = 5
If a set of n objects is to be ordered from the 1st to the nth object If a set of n objects is to be ordered from the 1st to the nth object, then there are n choices for the 1st object n-1 choices for the second object and n-2 choices for the third object. => THEREFORE, factorial 2! = 2*1 = 2 NOTE: 0! = 1! = 1
If an object is to be chosen from a set of m objects and a second object is chosen from a different set of n objects, If an object is to be chosen from a set of m objects and a second object is chosen from a different set of n objects, then there are m*n ways of choosing both objects simultaneously => THEREFORE, 3*5 e.g. meal choice: one dessert and one entrée e.g. 8 consecutive coin flips = 2*2*2*2*2*2*2 = 2^8 possible outcomes
30 minutes after Car A startet traveling from A to B (distance of 62 miles), Car B started traveling the same road. The cars met each other on the road 15 minutes after Car B started its trip. If Car A traveled at a rate 8 miles per hour greater than B, how many miles had Car B driven when they met? Draw an RTD Chart, NEVER use 15 minutes and convert it into 0.25 hours R * T = D Car A r+8 0.75 = Car B r 0.25 = 1. total distance = 62 km SOLVE: (r+8)*0.75 + r*0.25 = 62 r = 56 Therefore, in 15 minutes, Car B traveled a distance of 0.25*56 = 14 miles
IN a recent election, Mr R received 8000 votes cast by independent voters (voters not registered with a political party). She also revoked 10% of votes cast by voters (registered with a political party). If N is the total number of votes cast in the election and 40% of the votes cast were cast by independent voters, which of the following represents the number of votes Mr R received? N represents the total number of votes cast and 40% of the votes cast were cast by independant voters => RECEIVED: 8000 of these then 60% of the votes cast or 0.6N were by voters registered with a political party. => RECEIVED 10% of these Total = 0.06 N + 8000 votes in all
Orangeade was made by mixing a certain amount of orange juice with an equal amount of water. On the next day, orangeade was made by mixing the same amount of orange juice with twice the amount of water. On both days, all the orangeade that was made was sold. If the revenue from selling the orangeade was the same for both days and if the orangeade was sold at 0.60 $ per glass on the first day, what was the price per glass on the second day? The ratio of orangeade made and sold on the first day to amoung of orangeade made and sold on the second day is 2:3, because the orangeade on the first day was 1 part orange juice and 3 parts water. Since the revenues on each day were equal and 2 glasses were sold for every 3 glasses 2*($ 0.60) = 3p, where p represents the price per glass p = 2/3 * 0.60$ = 0.40$

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GMAT (Subject) / Quantitative - Word Problems (Lesson)