equations

27 - 9

27 -9 = 28 -10= 18 the easiest way to add or subtract is increasing the two number or decreasing the two number in the same amount

1hr , 1.45hr write it in quarter form

4/4 ; 4/4 + 3/4= 7/4

8x^2 -6xy= 0

so 2x(4x - 3y) = 0 2X=0 so X=0 4x - 3y=0 so 4x=3y so x/y=3/4 or 8X^2=6XY divided by 2X so 4X=3Y

recursive sequence

- each term depends on the term immediately before it. In some sequences, such as the Fibonacci sequence mentioned earlier, these depend on the previous two terms. For the hardest recursive sequences on the test, a sub n will be defined in terms of both a sub n minus 1, which is the immediately previous term. And a sub n minus 2, which is the term before the previous term. So these are the two preceding terms.

undo

- he undid another button: unfasten, unbutton,the knot was difficult to undo. - they will undo a decision by the superior court: revoke, repeal, rescind, reverse, retract, countermand, cancel, annul, nullify, abrogate - she undid much of the good work done: ruin,

How many pages are there between 45 and 111, not including either of those pages?

110 - 46 + 1 = 65

|K| > E

1st equation: K>E so we just removed the sign of the absolute value 2nd equation: K<-E we removed the sign of the absolute value, change the direction and put the negative

Mike paints a fence in 9 hours Marty can paint the same fence in 5 hours Column A The time it take Mike and Marty, working at a constant rate, to paint the fence Column B 3 hours

Ask yourself, how much of the job does each person finish in one hour. With Mike, he finishes 1/9 of the job in one hour, because it takes him 9 hours to finish the entire job. With Marty he finishes 1/5 of the job in one hour. Add those two rates together, 1/9 + 1/5 = 14/45 and then Flip It! and you get 45/14. That is greater than 3, so the answer is (A).

Cleve is 4 times as old as Al. Bob is 3 years younger than Al. The sum of their ages is 81. column A= Al's age?

C= 4A B= A - 3 A + B + C = 81 A + A - 3 + 4A = 81 6A = 84 A= 84/6= 14 Notice B= A - 3 means B is 3 years younger than A not vice versa

Dimitri weighs x pounds more than Allen weighs. Together, Allen and Dimitri weigh a total of y pounds. Which of the following represents Allen's weight?

D = A + X A + D = Y A= ? A + A + X = Y so 2A = Y - X so A= (Y-X)/2

solution - solute - concentration

First of all it's a fact that many different materials in nature happen to dissolve in water. This forms a solution something dissolves, and it's a fact that chemists often dissolve these materials that can be salt, sugars, acid and bases in water. The dissolved substance is called the solute. The concentration indicates how strong the solution is, that is, how much solute is dissolved in the given quantity of water. On the test, concentration will always be expressed as a percent. So concentration equals the total amount of solute, divided by the total amount of solution and this is times 100%. Notice that concentration is not the ratio of solute to water but the ratio of solute to the total amount of solution, that is the amount of water plus solute. So it's not the ratio of the two materials involved, it's solute to the whole, that's what constitutes concentration.

what is the sum of all the multiples of 20 from 160 to 840 inclusive?

First, we have to know how many terms we have here. So, 160 is 8 times 20, so it is the eight multiple. 840 is 42 times 20, so that is the 42nd multiple of 20. And inclusive counting, we would do 42 minus 8 plus 1 that's 35. 160/20= the 8th multiple of 20 840/20= the 42nd multiple of 20 inclusive counting: 42-8+1= 35 number of pairs= 35/2= 17.5 sum of list= 17.5 * (160+840)= 17.5 * 1000= 17500

summary of work questions

For work-related word problems, use A equals RT. A is the amount, R is the work rate, T is the time. For proportion-related work problems, you need matching units on each side, and you need to understand the operations with proportions. For problems with multiple workers or machines, create rates for each one and then add the rates.

VICs: Picking numbers

It's very important to keep in mind that there is an art to picking numbers well. Picking numbers does not mean just plugging in the first numbers that come into your head. We really have to be very strategic to do picking numbers well

There are two broad categories of work problems.

One type involves using proportion, of course the work rate, like any rate, is a ratio, and can be used to set up a proportion.

When isotope QXW radioactively decays, it loses exactly half its mass in each three-day period. So that's our step interval, three days. Suppose scientists start with a 96 gram sample of a pure isotope of this on a certain day. What will be the remaining mass in 12 days?

So at the start, we have 96 grams. Three days later, there's half of this left. start ---- 96 g after three days --- 48 g after six days ----- 24 g after nine days ---- 12 g after 12 days ---- 6

Bob is training for a fitness competition. In order to increase his maximum number of pull-ups, he follows the following routine: he begins with 25 pull-ups, rests for thirty seconds, and then does 24 pull-ups and rests, dropping one pull-up each time (25, 24, 23, etc.) until his final set of 11 pull-ups. How many total pull-ups does Bob do?

This gives us a total of 270 pull-ups 25 - 11 + 1 = 15/2 the number of pairs first + last = 25 + 11 = 36 15/2 * 36 = 270 or the number of terms * first + last / 2

sums of long sequence

To add sequences of evenly spaced numbers, pair the numbers by first and last to produce a pair, pairs of a constant sums, then multiply by the number of pairs.

14, 23, 32, 41, 50, ... find the 41st term of this sequence

a1= 14, d= 23-14= 9 an= a1 + (n-1) * d a41- 14 + (40*9)= 14 + 360 = 374

rounding 1.66 to the nearest 0.1

if we're rounding to the nearest digit or decimal, we round 0 to 4 down and 5 to 9 up—simple as that. answer: 1.7 remember that 0.1 means the answer should be the tenth so 1.7 but in order to round, you gotta look at the nearest number

Let S be the set of all positive integers that, when divided by eight, have a remainder of five. What is the 76th number in this set?

remember that we said that the remainder five, that's gonna be the first term on the list. Because, of course, the lowest number in this whole set is gonna be five itself. If we divide five by eight, we get a quotient of zero and a remainder of five. And then eight will be the common difference, because we'll be adding eight each time to get the new numbers. So we have our starting term. We have our common difference. That's our general formula. We plug in n equals 76 for the 76th number. a1= 5, d= 8 an= a1 + (n-1) * d a76= 5 + (75 * 8)= 5 + 600= 605

350/2.016

350/(2016/1000)= 350000/2016

Pump X takes 28 hours to fill a pool. Pump Y takes 21 hours to fill the same pool. How long does it take them to fill the same pool if they are working simultaneously?

Again, what we need to do is figure out the rates and then combine the rates. So here the rates are gonna be in pool per hour. So we're talking about the same pool, that single pool in how many hours. The rate of X working alone is 1 over 28, 1 pool over 28 hours and Y working alone is 1, 21st, 1 over 1 pool in 21 hours. R= pool/hr so Rx= 1/28 and Ry= 1/21 We're gonna get the combined rate by adding those two individual rates, so we're gonna add them. In order to add them, we're gonna find a common denominator. Notice that 28 is 4 times 7, 21 is 3 times 7, 28 times 3 or 21 times 4 so the common denominator is 84 Rxy= Rx + Ry= 1/28 + 1/21= 1/12= 12hours they do one pool in 12 hours, so it takes them 12 hours together to fill the pool

how to solve growth and decay

These questions may refer to some specialized scientific ideas, such as radioactive decay. You do not need to know anything about the underlying science. All the mathematical information you will need will be given in the question itself. The basic format of this question is you will be given some starting value and told that the size gets multiplied or divided by a certain quantity in some fixed interval of time. Then you will be asked for the resultant amount after some slightly longer period of time. Take the calculations one change interval at a time. Whatever, whatever time interval they give in the question, use that time interval as a step. And just move by those steps. Step by step, figure out the amounts until you arrive at the final amount.

(3x-2)^2

pay attention to the - , this negative sign is not considered in making quadratic and notice also that when we have a quadratic equation on both sides, we should make them equal to zero

word problems: growth and decay

If you happen to see a growth or decay problem, don't panic. You will be given the multiplier you need and all you need to do is follow the calculations step by step, one change interval at a time.

arithmetic sequence

- One special kind of sequence is called an arithmetic sequence. An arithmetic sequence is one in which we add the same constant to get from each term to the next. So if you look at this sequence, you notice that to get from each term to the next, we're simply adding seven. 5, 12, 19, 26, 33, big idea. Any evenly spaced list is an arithmetic sequence. One special case is consecutive multiples of an integer P we would add P to get from each term to the next. Consecutive odd numbers consecutive even numbers or consecutive integers are also special cases of arithmetic sequences

sequence

A sequence is an ordered list of numbers. Sometimes it will be the job of the test-taker to discern the pattern. More often the question will present the pattern in some form and ask something else about the sequence. The value of a later term or the sum or difference of two terms something along those lines.

Sum of integers from 1 to 40 inclusive

to find the sum of integers from 1 to n inclusive = n/2 * (n+1)= guass= from 1 to 100 = 50 * 101= 5050 we should divide by 2 because we need the pairs, then we add the first and last so the answer is 20 * 41= 820

If a truck is traveling at a constant rate of 90 kilometers per hour, how many seconds will it take the truck to travel a distance of 600 meters? (1 kilometer = 1000 meters)

90 km/1 hr = 90000 m/3600s = 600m/X X= 24

Suppose we start with 5 liters of a 30% HCl solution. How much water must we add to create a 20% solution?

Again, 30% which is 0.3 times 5 we have 1.5 liters and of course, that's not gonna change if all we're adding is water. So we're gonna have a new amount, and it's gonna be 20% and it's gonna have the same amount of HCl in it. The same amount of solute in it and so here we need to solve for X and that would be, be the total amount of solution. in the new solution: amount of HCI = 0.2 * X = 1.5 L 1/5 * X = 1.5 L so X= 7.5 liters total added water = 7.5 - 5 = 2.5 L In that problem, we added water to the solution, to make a less concentrated solution. Notice we could also add pure solute to a solution to make a more concentrated solution. The test could ask about either of those scenarios, but the most common questions involve mixing two solutions of two different concentrations

Contract negotiations opened on the morning of March 20th cons, continued every day without break and ended late in the evening of May 10th, or how many calendar days were the contract negotiations in session?

In March, from the 20th to the 31st, that would be 31 - 20 + 1 = 12 days April = 30 days In may, 10 days 12 + 30 + 10 = 52 days So that's how many days the contract negotiations were in session.

A car moving at 72 kilometers per hour moves how many meters in one second?

So we're told that 1 kilometer equals a thousand meters. Well, we also have to change hours to seconds. Well, we know that one hour is 60 minutes, and each minute has 60 seconds per minute. So 60 times 60 is 3600. One hour equals 3600 seconds. 72 km/hr = 72000 m/ 3600 s = 20 m/s So 20 meters per second. So it moves 20 meters in one second.

Okay to detail a car means to clean it thoroughly, inside and out. When Amelia and Brad detail a car together, 1 car takes 3 hours to detail. When Amelia details the car alone 1 car takes 4 hours. How long does it take Brad working alone to detail one car?

The combined work rate of people or machines working together is the sum of the individual work rates. In other words, we add the rates, we can't add or subtract the times, we can't add or subtract the different amounts made in different times. Those are the numbers usually given in the problem, we have to construct the rates and add or subtract them. rates= cars/hours how many cars per hour?

FAQ: I did not get "undefined value." (in this problem, 15/0 is undefined value) so, what is undefined value?

When any fraction has 0 as the denominator (on the bottom) that creates a number with an undefined value. We cannot divide any number by 0. so the denominator should not be zero, if the variables on the numerator and the denominator are the same, the value of the variable on the denominator is a totally illegal value and you are responsible for knowing it, although it's rare on the exam, notice also that when the two sides of the equations are the same, the possible values for a variable for example x are absolutely everything except the values on the denominators.

what is the middle of |X-4|<= 7

X - 4 equal and less than 7 so X is equal and less than 11 X-4 equal and greater than -7 so X is equal and greater than -3 the numbers from -3 to 11 will be 3+11+1=15 and the 8th will be the middle 4

She asked him to add all the integers from 1 to 100, and to her astonishment, Gauss produced the answer in seconds. How did he do that?

he five-year old Gauss noticed the following. Suppose we have this sum, all the integers from 1 to 100. Well, take out the sum of the first and the last. Clearly we continue to pair all the numbers in the sequence this way, we would wind of with 50 pairs, each of which has a sum of 101. Well that makes it very easy. That means that the total sum is 50 times 101, or another words 5,050. sum= N(N+1)/2 5050= 100 * 101 /2 = 5050 Thus, the sum of the list will be given by that formula. So this is the sum of the list of any evenly spaced list of numbers. And notice we can also think of this as N, the number of items on the list, times the average of the first and last pair. Sum of list= N * (a1 + aN)/2

an= a1 + (n-1) * d. If a3= 17 and a19= 65, find a10.

let the initial term = b , and let the common denominator = d a3= b + 2d= 17 a19= b + 18d what we have here are two equations with two unknowns and in fact its very easy to solve. We're just gonna subtract the first equation from the second equation. Subtract. We get 16d equals 48. Divide by 16 we get d equals three. And then plugging into a3 we get, that the initial term is 11 So now we have the initial term. The common difference. We can set up the general formula. The 10th term will be initial term 11 plus common difference three times ten minus one, which is nine. a10= a1 + d * (n-1) a10= 11 + 3 * 9 a10= 11 + 27 a10= 38

To create paint with a certain shade of gray, one must combine 2.016 liters of black paint with every one liter of white paint. Approximately how many liters of white paint must be combined with 350 liters of black paint to create the certain shade of gray?

shade gray paint= black paint + white paint there is a ratio between the amount of black paint and white paint. so we use the fraction black/white = 2.016/1 = 350B/?W W= 350/2.016= 173.6 L If you don't want to use the calculator, you could say that 350/2= 175 + meaning the real value of must be smaller since the denominator is greater

The percent change or the percent increase or decrease

the % change= the difference of the old and new/old or the difference of the selling and buying/buying the point is you have to figure out which is the first and which is the second then the difference of them/the old or the first whatever

A certain taxi charges $0.85 for the first ½ mile and $0.25 for every ½ mile after that. The total cost of a trip was $8.85. Column A, The trip's distance in miles?

$0.25 per 0.5 miles this means taxi must charge $0.5 per 1 mile the question compares with 16 miles trip so let's the total cost for a 16 mile trip $0.85 + $0.5 (15.5 mi) = $8.60 but the question tells us that the total cost is $8.85 so the second column is smaller than column one Notice that in this kind of questions, use the column be as the default answer and then compare the result with the question

tip 1

when you calculate the equations, look at the both sides, sometimes you do not need to do anything if you want to cancel them out

Suits at a local clothing store are discounted 20 percent. Despite the discount, the store owner enjoys a net profit (discount price minus cost) equal to 60 percent of the cost of the suits. If the suits cost the clothing store owner $200 each, what was the prediscount retail price of each suit? (A) $300 (B) $400 (C) $500 (D) $600 (E) $700

(B) is the correct answer. If the suits cost the owner $200 each, then the net profit (discount price minus cost) on each suit is 60 percent times $200, or $120. Thus, the discount retail price of each suit must be $200 (cost) + $120 (net profit) = $320. Which of the (prediscount) answer choices discounted by 20 percent equals $320? (A) $300 times 20 percent discount equals $300-$60 = $240 (incorrect). (B) $400 times 20 percent discount equals $400-$80 = $320 (correct!). (C) $500 times 20 percent discount equals $500-$100 = $400 (incorrect) (D) $600 times 20 percent discount equals $600-$120 = $480 (incorrect). (E) $700 times 20 percent discount equals $700-$140 = $560 (incorrect). Notes: The Plug In approach demands common sense and confidence in the strategy. You will have time to use this strategy. You may choose to abandon your plugging in after securing (B) as the correct answer. Of course, the Textbook approach is always available: Let x = the prediscount price of a suit, then 0.8x = discount price. The net profit (in dollars) is expressed in two ways in the problem. Set these expressions equal: 0.8x - 200 = 0.6 x 200 Solving for x 0.2x = 400 X=

Operating at the same constant rate, 4 identical machines can produce a total of 220 candles per minute. At this rate, how many candles could 10 such machines produce in 5 minutes? (A) 512 (B) 2200 (C) 2750 (D) 1100 (E) 44000

(C) is the correct answer. A logical place to begin problem-solving analysis is at the end of the problem called the question stem. The stem asks for the production of 10 machines in 5 minutes. This is easily found if we know the production of 1 machine in 1 minute. Each machine produces 220/4, or 55 candles per minute. Ten machines would produce 55(10) = 550 candles per minute, and operating for 5 minutes would yield a total quantity of 5(550) = 2,750 Algebraic word problems demand an organized, systematic methodology. First, the question involves representation of an unknown quantity by a variable. Second, there's an equation to create and solve. The Textbook approach provides the best solution process for word problems.

Exactly 8 years ago, Jim's son was twice as old as Jim's daughter. If Jim's son is now 5 years older than his daughter, how old is Jim's daughter now? (A) 5 (B) 10 (C) 13 (D) 16 (E) 18

(C) is the correct answer. Again, start with the question stem. Let S stand for the son's current age and d for the daughter's current age. Then, the first sentence translates to: S - 8 = 2 (d - 8), and the second piece of information translates to: S= 5 + d . Plugging the second equation into the first yields one equation in terms of d: (5+d) - 8 = 2d - 16, so -3 + d = 2d - 16 so -3 + 16 = 2d - d and d= 13 and , so .

If (6 + 2/x)(X-4)=0 and x does not equal 4, x= ? C. -1/3

(C) is the correct answer. For the product of two factors to equal zero, one (or both) of the factors must equal zero. It's given that x does not equal 4, so let's examine the first factor. What value for x renders the first factor equal to zero? C= -1/3 6 - 6 * 6-4 = 0 The Plug In approach also works for hard-to-follow word problems when it's not easy to represent the unknown quantity and create the equation.

Two trains are headed along parallel tracks in the same direction at varying rates of 72 mph and 47 mph. In how many hours will the faster train be 100 miles in front of the slower train?

, let's imagine that one train is stationary, and the other moving. How? Let's say two cyclists (just to mix it up) are headed in the same direction. One is going 10 kilometer per hour, and the other is going 15 kilometers per hour. After one hour, the faster cyclist is going to be 5 kilometers ahead of the slower one. That's the same distance as if the slower cyclist decided not to bike at all, and the faster cyclist moved at 5 kph. The takeaway: when two entities (trains, cyclists, cars) are headed in the same direction, the difference between the two rates is the amount the faster entity is outpacing the slower one per hour (in the case that the rate is expressed in terms of per hour). Returning to the original problem, which asks for trains headed in the same direction, we want to take the difference of the two rates. This gives us 72 - 47 = 25. Therefore, for every hour the faster one is 25 miles ahead of the slower train. How long, then, will it take until the faster train is 100 miles ahead? 100/25 = 4 hours. Notice that the numbers in this question—72 and 47—were very daunting. But, if you imagine the slower train not moving at all, then the faster train is moving 25 mph. This number easily divides into 100.

VIC note

- And remember, even if all your algebraic work is correct, you may have to rearrange your correct answer to match what is listed among the answer choices. - algebraic expressions can be written in more than one form. Either by multiplying or dividing the numerator and denominator of a fraction, or expanding and simplifying an expression with an exponent.

mixture question

- Mixture problems. Some word problems concern mixing solutions of various concentrations Concentration is the ratio of solute to the total solution most often expressed as a percent, and here of course we have to be very fluent in going back and forth between percents and decimals. We can change concentration by adding water or adding pure solute to a solution, or by mixing two solutions of two different concentrations. If we mix two solutions in unknown amounts to get a total known, a total, a known total of a known concentration, we set up simultaneous equations. One equation is the total amount equation and the other equation is for the amount of solute.

here we have a question about H2S04, sulfuric acid, you don't need to know about that. Suppose we start with an, with unlimited supplies of a 20% solution and a 50% solution. We combine X liters of the first, with Y liters of the second, to produce 7 liters of a 40% solution, what does X equal?

- One equation will always be a total equation that is the total volume or the total mass or weight. The other equation will always be about the amount of solute. So those are the two equations you're always gonna set up and once you have these two equations, you use the techniques for simultaneous equations. - So clearly, X plus Y equals 7 that's one equation that we have, we have X liters of the first thing, Y liters of the second thing, so X plus Y equals 7 that's one equation. So what we're gonna do is focus on the solute, the X plus Y equals 7, that's the total equation. The solute, well, the resultant solution is 40% of 7 liters so that's 2.8 liters of solute. first, we calculate the total amount of the solute in the resultant solution: solute = 0.4 * 7 = 2.8 L Now, obviously, X + Y = 7 Also, solute from #1 = 0.2 * X and solute form #2 = 0.5 * Y Therefore, 0.2X + 0.5Y = 2.8 X= 7/3 L

cars P and Q are approaching each other on the same highway moving in opposite directions. So we add the speeds. Car P is moving at 49 miles per hour. Car Q is moving at 61 miles per hour. Very good, so we get both of the speeds. At 2 pm, they are approaching each other and are 120 miles apart. Eventually, they pass each other. At what clock time are they moving away from, are they moving away from each other and 44 miles apart in the other direction?

- So they are starting out approaching each other and they're 121 miles apart. Then they meet each other. Then they pass each other and at the end they're 44 miles apart.The gap that's shrinking will be shrinking at a speed of 49 plus 61, and then the second gap that's expanding will also be expanding at a speed of 49 plus 61. Cuz the whole time, their speeds are in opposite directions. So of course we're gonna add the speeds. So, we can actually just add the speeds and treat it as one big distance, 121 plus 44. The shrunken gap and the expanded gap both as one big descent because the speed is gonna be the same.So we're gonna add the speeds. And so the combined speed, the speed at which the gap is changing, is 110 mph. GAP shrinking at R= 49 + 61 = 110 mph We have the approaching gap, the receding gap. We're gonna add these together for a total distance of 165 mph. 121mi + 44mi = 165mi We're gonna divide that distance by that speed to get the time. T= D/R= 165mi/110mph= 15/10=1.5hr So 1.5 hours it makes this change. 1.5 hours after 2:00 pm, that would be 3:30. So when vehicles or travelers are moving in opposite directions, we add the velocities.

In a certain state, schools pay 2% tax on food and 8% tax on stationery. The school placed a combined order of $500 on food and stationery and paid $19 on tax on the order. How much of that money was spent on food?

- So, we could solve this, we could set up algebra and solve this. Instead of doing that, we're gonna explore a backsolving approach. So we're just gonna pick C, we're just gonna say pretend 300 is the answer, that means they spent 300 on food so they must've spent 200 on stationery. There's a 2% tax on food, so 2% of 300 gives us $6, there's an 8% tax on stationery so that gives us $16, the total tax there is $22 and that's too much. c= $300 on food so stationery = $200 tax on food 2/100 * 300= $6 tax on stationery 8/100 * 200= 16 Total tax= $22 Too Much They paid $19 in tax, so 22 is too much and so what this means is, to pay less in taxes the school must have spent more on food and less on stationery. Cuz food is taxed on a much lower rate, so if they spent more on food, they would be paying less in taxes. So right away we can eliminate A, B, and C, so this is great. Even if we were running out of time, we could guess from the remaining two and we would have a very high likelihood of, of guessing by chance the right answer because 3 answers have already been eliminated. food = $400 so stationery= $100 tax on food 2/100 * 400= 8 tax on stationery 8/100 * 100= 8 Total tax= 16 too little food = $350 so stationery= $150 tax on food= 2/100 * 350= $7 tax on stationery 8/100 * 150= $12 Total tax = $19 so that's the answer

Sets and Sequences, inclusive counting: A workshop's first day was April 8th, and it's last day was April 27th. How many days did the workshop run?

- This way of counting is known as inclusive counting. We always use inclusive counting when both endpoints, the starting value and the ending value, are included in what we are counting.

Shrinking and expanding gaps: Some motion-based word problems involve two travelers moving toward or away from each other.

- Two travelers are moving in opposite directions. Well if they're moving in opposite directions we always add the speeds. The speed at which they're approaching or the speed of the gap between them is always the sum of their speeds. If the two travelers are approaching each other then the sum of the speeds is the speed at which the gap is shrinking. If the two travelers are moving away from each other, the sum of the speeds is the speed at which the gap is expanding. - Now if they're moving in the same direction, we subtract the speeds. And we also do bigger minus smaller just so we get a positive number. If the faster traveler is in front, that means of course the faster traveler is pulling away from that slower traveler. Then the difference in speeds is the difference at which the gap is expanding. Over time the faster car is going to get further and farther from the slower car behind it. If the slower traveler is in front, then of course the faster car is going to be gaining on it, and eventually it's going to pass the slower car. The difference in speeds is the speed at which the gap between them is shrinking.

in summary Backsolving

- when all five answer choices are numbers, one alternative strategy is to solve by backsolving. - So the very straightforward approach is, start with answer C, try this as the answer to the prompt to see if it works in the scenario. - If C doesn't work, the information about too big or too small. Will allow us to eliminate other answers, and remember an alternate strategy would be to pick, for example, B and then you could, you might be able to determine that either B or A is the answer, depending on whether it was just right or too big, or too small and if it didn't work.Then you could pick D, and that second choice would allow you to narrow everything down.

How many integers are there in Set A, if Set A includes all the numbers 1 - 10?How many integers are there in Set A, if Set A includes all the numbers 5 - 15?

10 -1 + 1 = 10 15 - 5 + 1= 11

If I am at the top of page 125 of a book, then how many pages will I have to read if I want to read to the end of page 152?

152 - 125 + 1 = 28

how many multiples of 8 are there from 200 to 640, inclusive?

200/8= 25th multiples of eight 640/8= 80th multiples of eight and both are included number= 80 - 25 + 1= 55 + 1= 56 - We use inclusive counting whenever the situation demands that both endpoints, the lowest value and the highest value, are part of what we are counting. - We perform the ordinary subtraction of high minus low, then add one for the included lower endpoint.

If an 1/2 investment portfolio was invested in stocks, 1/5 in a mutual fund, 1/10 in bonds, and the remaining $25,000 in a savings bond, what was the total amount in the portfolio? (A) $100,000 (B) $125,000 (C) $150,000 (D) $200,000 (E) $250,000

Begin by focusing on the question stem. Let equal the total amount of money in the portfolio. Then, 1/2x + 1/5x + 1/10x + 25000 = x . Multiply each term by 10. 5x + 2x + x + 25000= 10x 25000 = 2x X= 12500

solution summary

Concentration is the ratio of solute to the total solution most often expressed as a percent, and here of course we have to be very fluent in going back and forth between percents and decimals. We can change concentration by adding water or adding pure solute to a solution, or by mixing two solutions of two different concentrations. If we mix two solutions in unknown amounts to get a total known, a total, a known total of a known concentration, we set up simultaneous equations. One equation is the total amount equation and the other equation is for the amount of solute.

What is the sum of the first 50 positive integers?

If I add up the first number and last number, I get 1 + 50 = 51. The next question I want to ask myself is, how many pairs of numbers are there in the first 50 integers? The logic is, if we pair numbers the way we did in the preceding paragraph, we always get 51. So, I'm asking myself, how many 51's are there. Dividing 50 by 2, we get the number of pairs: 25. Therefore, we have to multiply 25 x 51 to get the sum of the sequence, which is 1275. So, whenever we need to find a consecutive series, we simply add the first plus the last (e.g. 1 + 50), and then take the number of digits (e.g. 50) and divide by 2 (remember we are looking for the pairs). Next, I multiply this result (50/2) by the first and last (1 + 50), and I get 25 x 51 = 1275. Let's try that with another, easier problem: What is the sum of the numbers 1 - 10. Adding first plus last (1 + 10), I get eleven. Then, I take the number of digits, 10, divide by 2, and get 5. Next, I simply multiply 11 x 5 = 55.

onas takes 5 hours to paint a fence. Mark takes twice as long to paint the same fence. Working together, how long will it take them to complete the fence?

First off, we want to note that Mark takes twice as long as Jonas, so he takes 10 hours to paint the fence alone. With this information, we next need to find how much of a fence each can paints in one hour. By getting this hourly rate, we simply add up the amount of fence they paint in one hour. This numbers tells us how much of the fence they paint together in one hour. 1/5 the amount of fence Jonas paints in one hour. 1/10 the amount Mark paints in one hour. To find the work rate, we must first add the two independent hourly rates: 1/5 + 1/10 = 3/10, which is the amount of fence they paint together in one hour. At a rate of 3/10 of a fence together, how long is it going to take them to paint an entire fence? One approach is to set up a simple equation: 3/10 x = 1, where 1 stands for the entire job. Solving for x, or the combined work rate, we get 10/3, Answer C, or the reciprocal of 3/10. A good rule of thumb is that whatever the rate is in one hour, in this case 3/10 of a fence, just take the reciprocal of that fraction to find how long it would take them to paint an entire fence. An even quicker way is to set up a fraction.To recap: to find the work rate, first find the hourly rates for each individual. Then, add these two rates together, and then flip, or take the reciprocal of, that fraction.

age question: right now, Steve's age is half of Tom's age. In eight years, twice Tom's age will be five more than three times Steve's age. How old is Tom right now? So, what I'm gonna say about this is, the big mistake associated with this problem is thinking of the ages as a fixed number.

In summary age questions can be tricky when different mathematical relationships among the ages are specified at different times and so we have to very clear in defining the variables. It's not enough to say that F is Frank's age we have to specify when F is Frank's age now. We have to be very clear on that. Choose variables to represent the ages now and then use addition or subtraction to create expressions for ages at other times. S = 1/2T ; in 8 years 2T = 5 + 3S so 2(T + 8) = 5 + 3(S + 8) now we managed to set an equations, as we know T=2S and then we plugin ..

Multi-traveler questions

In summary, when a word problem involves multiple travelers, multiple trips, or a trip with multiple legs, remember that each traveler, each trip, and or each leg deserves its own D = RT equation. Sometimes, you will be able to solve for all the quantities in one equation, and use those numbers to solve for other equations. More often you will have to use the techniques for solving two or more equations with two or more unknowns. And these are the techniques of substitution and elimination, which we used in this video.

A transcontinental jet travels at a rate of x - 100 mph with a headwind and x + 100 mph with a tailwind between Wavetown and Urbanio, two cities 3,200 miles apart. If it takes the jet 2 hr 40 minutes longer to complete the trip with a headwind, then what is the jet's rate flying with a tailwind? (A) 500 (B) 540 (C) 600 (D) 720 (E) Cannot be determined by the information given.

Luckily, we can work with answer choice (C) 600. If the speed with a tailwind is 600 mph, then the speed with the headwind is 400. The distance between the two cities is 3200 miles. Using d = rt, where d stands for distance, r stands for rate, and t stands for time, we find that the time it takes to fly with a tailwind is 5 hr 20 min, and the time with a headwind is 8 hours. The difference in time is 2 hr 40 min. And there is our answer. Just like that. We know that he trip one way is 3200 miles. So we plug in the headwind speed of 400 and ask ourselves: how many hours does it take for a plane going 400 mph to travel 3200 miles. The answer is 3200/400 = 8hrs. Do the same for the tailwind (600 mph) and you get 3200/600 or 5hr 20 min. The question is asking for the difference between the two: 8hr - 5hr 20 min = 2hr 40 min.

equation for moving things just

Keep track of the units of numbers when you plug in. Don't ignore units. If you pay attention to units, that will really help you. Unit conversions can be written as fractions equal to one. We can multiply or divide by them. That's very convenient. Know the common unit conversions and change units, so that all units in the problem are consistent

If the sum of three consecutive integers is K , then which of the following is a possible value of K?

Let X = first number X + 1 = second number X + 3 = third number K = X + X + 1 + X + 2 K= 3X + 3 K = 3(X+1) so K must be divisible by 3 and the choice 3 is the number 201 is divisible by 3

A concession stand sells either hotdogs for $1.75 each or hamburgers for $4 each. If Charlie buys a total of 9 items from the concession stand for a total of 27 dollars, then how many hotdogs did he buy? (A) 3 (B) 4 (C) 6 (D) 7 (E) 8 On the GRE, you will most likely see a long, tedious word problem. Granted, this problem wasn't too bad. The question is, how did you go about trying to solve it? I want you to try something that may seem counterintuitive. I want you to turn off the algebra part of your brain. Instead, I want you to work with the answer choices by plugging them back into the problem.

Let's start with answer choice (C). The reason we start with this answer is because it is in the middle. If it turns out that the total price is less than 27 dollars, then we will need to plug in a lower number. The reason we would choose a lower number is because hotdogs are cheaper; therefore, the fewer hotdogs we have the more burgers we have, and this brings up the price. Plugging in 6 we get $1.75 x 6 = $10.50. That leaves us 3 hamburgers at $4 each = $12. Adding the two together gives us $22.50. That amount is too low, so we can eliminate answer choices (D) and (E), as well, because they will increase the number of hotdogs and lower the overall price. Plugging in (B) 4, we get $7 on hotdogs. That leaves us 5 hamburgers or $4 x 5 = $20. Add up $7 + $20 and we get $27, which is the correct answer.

a variable in the answer choice problem. At the store, Sam bought a shirt and a toaster. There was an 8% sales tax on each item, and with tax, Sam paid a total of Q. So that's the total amount he paid, the price of the items and the 8% tax. If the price of the toaster before tax was T, what, in terms of Q and T, is the price of the shirt?

Shirt= S Toaster= T S= ? well, we know (S + T) * 1.08 = Q so S(1.08)= Q - T(1.08) and the answer would be S= (Q/1.08) - T or you could divide both sides by 1.08 and then subtract T NOTICE that the prompt says an 8% sales tax on each price meaning an 8% increase that is 1.08 [multiplier for an 8% tax] this is a very important point. ----------------------------------------------- We can begin by eliminating answers that play on a common percent fallacy. Let S be the shirt price, T be the toaster price before taxes. The bill before taxes is S + T. With an 8% sales tax this is Q equals the multiplier for an 8% tax that's 1.08 times S plus T or 1.08(S+T). So that's the correct relationship. To solve for T, we have to undo that 8% increase. As we've learned in the module on percents, an 8% decrease does not undo an 8% increase. If we increase by 8% then decrease by 8%, we do not get back to where we started from. This is one of the principle fallacies of percents. We want to undo an 8% increase, but that does not involve an 8% decrease. So anything where 0.92 appears is wrong. We can immediately limit A, B, and C. And in fact, once we have that percent equation, this is very easy to do algebraically. All we have to do is just solve for S.

A machine, working at a constant rate, manufactures 36 staplers in 28 minutes. How many staplers does it make in one hour and 45 minutes?

So first of all I'm gonna change that time to a 105 minutes, so everything is in, in minutes now. So we're gonna set up a ratio, staplers over time, so 36 over 28 that's the initial ratio that we're given and this is gonna be S, the number of staplers over a 105 minutes. stapler/time= 36staplers/28min= S/105= Well, first of all it would be a mistake to cross multiply right here So do not start by cross-multiplying all those big numbers together. Instead, simplify the fraction on the left that's a very good place to start 36 and 28 are divisible by 4 cancel the 4 we get down to single digit numbers, 9 over 7. 9/7= S/105 Much better, but we'd still like to cancel more to make that 105 smaller before we cross multiply. Another huge mistake, we cannot cross-cancel between the 9 and the 105 that is an absolute nightmare, Instead we can cancel the sevens, a factor of 7 in the two denominators, so we're cancelling across the denominators so 9/1= S/15 so the answer is 135 staplers

A lichen advances 4 cm each year across a rock slab. So four centimeters per year, that's the rate. If this rate remains constant over time, how many years will it take to cross 30 meters?

So here we're looking for time, so we want to solve for time. Time equals distance over rate. But we have to take care of the inconsistent units because we have a speed. So our rate is 4 centimeters per year. Our distance is 30 meters. And the easiest thing to do is just right away change that distance to centimeters. T = D/R , R = 4cm/yr D= 30m = 30 * 100 = 3000cm T = D/R so 3000cm/4cm yr = 750 yr

Okay a car and a truck are moving in the same direction on the same highway. So same direction, we subtract the speeds. The truck is moving at 50 miles an hour. We know that speed. The car is traveling at a constant speed. At 3:00 pm the car is 30 miles behind the truck and at 4:30 the car overtakes and passes the truck. What is the speed of the car?

So in that time, that time from 3:00 pm to 4:30, which is one and a half hours, in that time, the car closed a 30 mile per hour gap. So we're gonna focus on the gap itself. And we can figure out the speed of the gap just by R equals D over T. Of course, the distance is 30. GAP R = D/T so 30mi/1.5hr = 20 mph That was the speed at which the gap was shrinking. And that would be the difference between the speed of the car and the speed of the truck. Well, of course, the car is moving faster cuz it catches up to the truck and passes it. Since the truck is moving at 50 miles an hour, the car must be moving 20 miles per hour faster than that. So 50 plus 20, it must be moving at 70 miles an hour. 20= car speed - 50 so car speed equals 70

Cars X and Y are traveling from A to B on the same route at constant speeds. Car X is initially behind car Y, but car X's speed is 1.25 times car Y's speed. Car X passes Car Y at 1:30. At 3:15 pm car X reaches the destination B and at that moment, car Y is still 35 miles away from B. What is the speed of car X?

So this is very interesting. So starting at 1:30 they were in the same place, and then by 3:15 car Y had opened up a gap of 35 miles. And in that time from 1:30 to 3:15, so that's not exactly two hours, it's less than two hours, it's one hour and 45 minutes or 1.45 hours. We want the speed of car X. Well, let's focus on that gap. We close a gap of 35 miles in, I'll write it as seven quarters of an hour. GAP R= T/D so 35/1.45=35*4/7= 20mph So the difference between the speeds is 20 miles per hour. X car = 1.25 car Y or X= 1.25Y we know that X- Y = 20 so 1.25Y - Y = 20 so Y= 80 so X = 20 + 80 = 100

variables in the answer choices.

So when you do the algebra, that is a guarantee that if we do the algebra correctly, you're gonna arrive at one unambiguous right answer. And so that's the big advantage of the algebra. That it gets you to the answer right away with no ambiguity. The problem with the algebra is if you don't know how to begin or you find the algebra confusing, that's a drawback of the algebra. Each round of picking numbers maybe relatively quick but it's hard to know how many rounds will be required to eliminate all four answer choices. So you might say that the algebra approach is a little more difficult, but it's efficient and gets you to an answer quickly. The picking number approach definitely is easier, makes things much easier, but the question is, how efficient will it be, how long will it take to narrow things down to one answer.

How much HCl and how much water must we use to create 5 liters of a 30% HCI solution?

So, the amount of HCl would be 30% as a decimal 0.3 times the whole, which is 5 liters and when we multiply out we get 1.5. So in other words we're gonna need 1.5 liters of pure HCl and that means that the rest of it which would be 3.5 has to be water, so 1.5 is HCl, 3.5 is water, we combine those, we get a 30% HCl solution. amount of HCI = 0.3 * 5 = 1.5 L amount of water = 5 - 1.5 = 3.5 L

Average Velocity; D = RT

V = total distance/ total time when finding the average velocity, do not fall into the trap of thinking it's a simple numerical average of two things. there are often two legs and the whole legs, the variables are velocity or rate; time, and distance V, D, and T respectively. rate is based on mph, time is based on hr and velocity or rate mph. when you know the Distance and Rate, clearly you can figure out the Time. Time total = Time 1 + Time 2 - when you have just one variable for example just R, for the other variable use D or T

A chemical supply company has 60 liters of 40% of some chemical, that chemical happens to be nitric acid. You don't need to worry about that. How many liters of pure undiluted acid must the chemists add so that the resultant solution is a 50% solution? A. 12 B. 15 C. 20 D. 24 E. 30

So, we're gonna add solute to increase the concentration and we want to increase it to 50%. Notice the amount of HNO3 is 60 * 4/10= 24 L So, C is a nice round number, so we'll just pick C as our answer. Before you even pick it, notice that the amount of solute in the beginning solution, 40% of 60 is 24 liters. There are 24 liters of the acid in the initial solution. So, now let's pick C. We add 20 more liters of acid, as that means that we have a total concentration of 44 liters, that is the amount of acid in the solution and of course, the total solution, the volume of the solution is gonna be 80 liters and so we wanna know do we have the right concentration. So C= 20 so 20 L added Total concentrate = 24 + 20 = 44 L Total solution = 60 + 20 = 80 L Well, we don't have to do the division. We can see that 44 is more than half of 80. So we don't even note, need to know the exactly number. All we need to know is the concentration would be over 50%, so this is too high. We've added too much acid. So right away, we can eliminate C, D, and E. All of those add too much acid. So now we have to pick either A or B, it doesn't matter. I'll just pick A. Suppose we add 12 liters of acid. Well now the total concentrate is 24 plus 12, 36 liters of acid. So that's how many liters of acid total would be in the solution, and at that point of volume, the total volume of the solution would be 72, and notice that 36 is exactly half of 72 so this would be a 50% solution, this answer actually works, so we picked the right answer here, the answer is A. ok we try A=12 total concentrate = 24 + 12= 36 L total solution=

So we have some kind of bacteria and it multiplies the size of its population by five and a half by five-halves every four hours. Four hours is the, is the time interval step that we're gonna take. So there are 24 billion at 9 a.m. And optimal conditions are maintained. How many are there at five p.m. Of the same day?

So, we're gonna step four hours at a time. So we're gonna start at 9 a.m. We're gonna have 24 billion four hours later, we multiply 24 by five-halves. We'll cancelling the half we get 12 times 5, that's 60. Then four hours later, we're at 5 p.m. 60 divided by a half is 30, so that will be 150 billion. So at 5 PM the same day there are 150 billion of this bacteria.

a sub 5 equals 28 a5= 28

We represent the sequence as a whole by an individual letter, usually a lower case letter, and the individual number in the sequence, the, the order in the list, by a numerical subscript. So, for example, if I say a sub 5 equals 28 that means, for some sequence, the fifth term, the fifth number on the list, equals 28. So 5 is the position on the list, I go down to the fifth number on the list, and the fifth number on the list is 28. - Notice that if a formula is given it makes it very easy to jump ahead to any term we want. So for example, the test could give us that formula and ask us for the 48th term on the list, and of course we could just jump right there just by plugging in n equals 48. ex: rn= n(n+2) r48= 48(48+2)= 48 * 50= 2400

an= a1 + (n - 1) * d

The nth term of an arithmetic sequence with an initial term of a sub one, and a common difference d is a sub n equals a sub one plus n minus one times d. - Notice that one context in which evenly spaced integers arise is the set of all positive integers that, when divided by one number, give a fixed remainder. For example, this sequence. The sequence that we've been looking at. Is the set of all integers that, when divided by seven, have a remainder of five. and 5 is a1 That's what all those numbers have in common, when we divide by seven, we have a remainder of five. Notice that the remainder is the first term, a sub one. And the divisor is the common difference,

Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21,

Think about the Fibonacci sequence again. This sequence has the two "seed" values. The first value is 1. The second value is 1. a1= 1, a2=1 And each term is the sum of the two previous terms. So algebraically, we would say a sub n equals a sub n minus 1 plus a sub n minus 2. That's a symbolic representation of the rule that creates the Fibonacci sequence an= an-1 + an-2 so In recursive sequences, each term a sub n [an]is defined in terms of one or two previous terms (an-1 and maybe an-2). It's defined in terms of a sub n minus 1 and maybe in terms of a sub n minus 2. numerical values for one or two terms will always be specified. With a recursive sequence, there is no way to jump immediately to the value of a term such as the sixth term. Instead, we have to find each and every term from the start up to the desired term.

Plug In (Substitution) Method

This strategy works best when a symbolic expression (like an equation) is provided and the answer choices are all numerical values.

plug-in approach for the above question

We're going to solve in a completely different way. Let's say that the original buy price per watch is $10 so B= $10. Let's make the mark-up 30%. So that's gonna be the actual answer to the question. A 30% mark-up. That makes the sell price $13, and the profit per watch is $3. If she sells 20 watches, she makes a profit of $60 or T= $60. So, there I just picked numbers. Now how did I come up with those exact numbers? We'll talk a little more about this in the picking numbers video coming up a couple videos from now, but right now let's just go with these numbers. This means that if we plug in B= 10, N= 20, and T= 60 into all the answer choices, the correct answer will yield a value of 30. So now we have to plug this in to all five answer choices. So those, that's what we're gonna plug in. 30 is what we're trying to get. Here are all the answer choices. We're gonna plug this in. So start with A. 100T/NB= (100 * 60)/(20 * 10)= 5 * 6= 30 so this is a valid possibility for the answer. This actually works at least when we plug this in. It doesn't guarantee that this is the right answer. This just happens to be one answer that works for this combination of numbers

Which of the following could be true of at least some of the terms on the sequence defined by that algebraic formula? bn= (2n-1)(2n+3) I. divisible by 2 II. divisible by 3 III. divisible by 5

Well, first of all, let's just notice if we plugin n equals 1, the simplest thing, the first number on the list, what we get is 5. So certainly it's possible for some of the numbers to be divisible by 5, so that is possible, 3 is possible. Now if we plugin n equals 2 then we get 21 which is divisible by 3 so it is possible that some of the numbers on the list are divisible by 3. Okay, so far so good. So II and III are possible. Now we get to I. Now notice, think about this, 2n. a1= 1 * 5 = 5 a2= 3 * 7 = 21 If n is an integer, then 2n has to be an even integer, 2n minus 1 has to be odd, and 2n plus 3 has to be odd. So b sub n is the product of odd times odd. So every number in this sequence is an odd number. So none of them are divisible by 2. So turns out it is not possible for any number on that list to be divisible by 2 because they're all odd numbers. note In the notation a sub n, n is the index, that is, the place on the list. An entire infinite sequence can be specified simply by giving an algebraic formula for a sub n in terms of n.

suppose we start with 8 liters of a 60% solution. We add 4 liters of a C percent solution. So C is the unknown and the result to 12 liters of a 50% solution. What is C?

Well, the first thing I'm gonna do is figure out how much solute did I begin with and how much solute did I end up with? Solute in 1st solution = .6 * 8 = 4.8 L solute in result = .5 * 12 = 6 L So in the first solution, I had 4.8 liters of solute. In the result in solution, the final solution, I add 6 liters of solute and that means that the amount of solute that was added, that amount of solute was 1.2 liters. therefore, solute added = 6 - 4.8 = 1.2 4 L * C% = 1.2 L so C= 0.3 = 30% Sometimes in solution problems, if amounts of two different solutions are initially unknown, we have to set up simultaneous equations. This is probably the most popular solution problem type that you'll see on the test.

Bob drives at an average rate of 50 mph from Berkeley to Los Angeles, a distance of 350 miles. How long does it take him to complete the trip?

When dealing with distance, rate and time, we always want to remember the nifty little formula, D = RT, in which D stands for the distance, R stands for the rate (or speed), and T stands for the time. Let's set up the equation, plugging in the values for D and R: 350 = 50T.

Two trains starting from cities 300 miles apart head in opposite directions at rates of 70 mph and 50 mph, respectively. How long does it take the trains to cross paths?

When you have any two entities (trains, bicyclists, cars, etc.) headed towards each other you must add their rates to find the total rates. The logic behind this is the two trains (as is the case here) are coming from opposite directions straight into each other. This yields 120 mph, a very fast rate. To find the final answer, we want to employ our nifty old formula: D = RT, where D stands for distance, R stands for rate, and T stands for time. We've already found R, which is their combined rate of 120 mph. They are 300 miles apart so that is D. Plugging those values in, we get 300 = 120T. Dividing 120 by both sides, we get T = 2.5 hrs.

FAQ: "Why is -1 an extraneous root? What is an extraneous root?"

When you plug x = -1 back into the original equation, we can tell it's not a true solution to the problem because √(8 * (-1)² + 17) = 3(-1) - 2, which becomes √25 = -5, so 5 = -5. This is false. (Remember that the roots are always positive when working with √). Whenever you have a square root function in an equation and you are solving for a variable, make sure to plug in the values you find back into the original equation to check for any extraneous roots. Why? It's because the tricky part of working with equations with square roots is that, unlike linear equations, plugging values back in will not always satisfy the original equation because when you square both sides to eliminate the root, you may end up with answers that did not exist originally.

work equations

Work Problems. Some word problems concern machines or workers and how fast they can get certain jobs done. A= RT A is the amount of work done, R is the work rate, the rate at which work is done, T is time. The amount A can be products manufactured or houses painted or pizzas made etc. The work rate could be the number of products per minute or houses per day or pizzas per hour etc.

Total T, Toral D, Velocity

You have to find D and T of each leg of the trip and add across the legs to find the total distance and the total time. The average velocity equals the total distance divided by the total time.

some basic patterns of sequence

a sub n equals n, or an= n that's just the sequence of all positive integers. So just the counting numbers, 1, 2, 3, 4, 5, 6, etc. a sub n equals 2n minus 1 [an= 2n-1]is the sequence of all positive odd numbers. So very interesting, and if we just add a sub n equal 2n,[an= 2n] that would be the sequence of all positive, positive even numbers. A sub n equals 7n [an= 7n] is the sequence of all positive multiples of 7, and similarly, if we had any factor times n it would be all the multiples of that particular factor. A sub n equals n squared [an= n^2] is the sequence of all positive perfect squares. A sub n equals 3 to the n [an= 3^n] is the sequence of all the powers of 3

We're not gonna solve this yet, but just to give you a feel of this question. Jennifer can by watches at a price of B dollars per watch, which she marks up by a certain percent before selling. If she makes a total profit of T by selling N watches, then in terms of B, T, and N, what is the percent of the markup from her buy price to her sell price? And notice all five answer choices involve the variables B, T, and N. A. 100T/(NB) B. TB/(100) ..

an algebraic approach: B is the buy price, and call S the sell price, which is not one of our variables. The profit per watch is (S - B), and for N watches that would be N (S - B) = T. That would be the total profit clearly. We can divide that to get the difference, (S - B) = T/ N. Well the % increase would be (S - B) / B * 100 = (T/N) * 100/B= 100T/NB notice that the difference between the selling and buying times number of the items would be the total profit And this is answer choice A. So that is an example of using a very straightforward algebraic approach. We introduced an extra variable, we fought through the whole thing in terms of variables, and we got to the answer right away. Now here's a plug-in approach to the same problem. Exact same problem.

Whenever Art Dealer sells a sculpture, he earns a 20 percent commission on the first $12,000 of the sale price plus 15 percent of the sale price in excess of $12,000. If Art earned a $3,900 commission on the sale of a certain sculpture, what was the sale price?

he earns a 20 percent commission on the first $12000 of the sale price means commission = 20% * 12000 or 0.2 * 12000 = $ 2400 plus 15 percent of the sale price in excess of $12,000. means 15% (X - 12000) FAQ: I don't understand the (x-12000). How does this show the "sales price in excess of $12,000? This question uses an English idiom that could appear on other GRE math problems. When we say "the price in excess of $12,000," it refers to the part of the price that is above $12,000. For example, suppose the total price of something is $14,000. Then $2,000 would be the amount that is in excess of $12,000. After we subtract $12,000 from $14,000, whatever is left is the part that's in excess of $12,000. That's precisely why (x - $12,000) is the expression for "the part of the price in excess of $12,000." x represents the total price, and when we remove the first 12,000 we have the excess.

VIC

when the problem gives you distance and speed, it means you have to calculate Time. for example, for the first leg of a trip, Fred travelled A miles at speed P. so your solution: 1st leg: Distance = A , Speed= P , so Time= A/p During the second leg, he traveled at a slower speed. There were only two legs in the trip. The entire trip took T hours, and the average speed for the entire trip was V. In terms of A, p, T, and V, what was the average speed of the second leg of the trip the whole trip: Time=T , speed= V, Distance= VT the 2nd leg: Time= T - A/p, Distance= VT - A V avg= Distance/Time= T - (A/p)/ VT - A The answer we found is not listed as one of the answers but it appears close to answer choice c. And in particular, the change I'm gonna make is I'm gonna divide the numerator and the denominator by T. So it's like I'm gonna multiply by a one over T over one over T

quadratic

when you see the quadratic, you should factor it into two binomials and use the sum and product technique

D= RT

you need to recall the formula for finding the time it takes to complete a certain trip. Simply take the fraction D/RT and remove the T (for time). This leaves you with D/R. In other words, Time = Distance/Rate. -