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. , 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. 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 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 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. 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. - 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 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 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 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. 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, 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. - 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 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= - 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. 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 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.

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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. - 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, 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 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. ;