Is there an algebraic way to solve this problem?

[miteshsholay]

If the original price of an item in a retail store is marked up by m percent and the resulting price is then discounted by d percent, where m and d are integers between 0 and 100, is the item’s final price (after both changes) greater than its original price?

(1) m > d + 10

(2) m = 1.5d

[tim]

Before we help with this question, we need you to show some effort of your own. What did you try on this question? Where did you get stuck?

[miteshsholay]

I approached somewhat like this, but got stuck in between coz it was getting too lengthy.
let original price=100
therefore, marked price MP= 100 + m
discounted price DP= 100 + m + [100+m]d/100

Question : Is DP>original price (i.e. 100)?
or DP-100>0 ?
m + [100+m]d/100 > 0
m + d + (md/100) >0

lets take 2nd statement (coz it looks easier :-))
2)m = 1.5d
put this in above inequality
1.5d + d + (1.5d*d/100) >0
2.5 + (1.5d/100) >0
25 + (15d/100) >0
5 + (3d/100) > 0
this reduces to d > (-500/3)

but i dont know if this makes any senses.
here i got stuck for sometime and then guessed the answer.
The official explanation took some values, which i cudnt hav thought for this problem (like first they took extreme values, then some mid value 45 etc)

[tim]

look, if you're getting stuck because something is "getting too lengthy", you're pretty much out of luck. you have to be willing to put in the effort to get all the way through a problem, and nothing we can do will help you if you don't have that drive yourself..

you've got the right idea with the algebra, you just need to remember that a discount will lower the price (i.e. use a minus sign instead of a plus sign). if you get to the end and are being asked something about a variable you don't have any information about, that means the statement is insufficient..

[miteshsholay]

you just need to remember that a discount will lower the price (i.e. use a minus sign instead of a plus sign). ..

Sorry for the discount fumble.

let original price=100
therefore, marked price MP= 100 + m
discounted price DP= 100 + m - [100+m]d/100

Question : Is DP>original price (i.e. 100)?
or Is DP-100>0 ?
Is m - [100+m]d/100 > 0 ?
Is m - d - (md/100) >0 ?------------------------------(EQ)

lets take 2nd statement (coz it looks easier :-))
2)m = 1.5d
put this in above inequality
1.5d - d - (1.5d*d/100) >0
2.5 - (1.5d/100) >0
25 - (15d/100) >0
5 - (3d/100) > 0
this reduces to d < (500/3)
but this gives no new information, its already given that 0<d<100
so we just check two extreme values of d
lets say d=10, 60, (at 70 the corresponding m becomes >100)
that makes m=15,90
if we substitute d=10, m=15 in (EQ)
we get 15-10-1.5>0 ?
or 3.5 > 0 ? YES
if we substitute d=60, m=90 in (EQ)
we get 90-60-54 >0
or -24 > 0 ? NO
Hence statement 2 is INSUFFICIENT

Now lets take statement 1
m > d + 10
lets borrow the values we took earlier
d=60
m=90
Answer to the question (we already checked earlier) NO
lets take values which already satisfy statement 2
d=30
m=45
Answer to the question
substitute in (EQ)
45-30-13.5>0 ?
or 1.5>0 ? YES

So both are insufficient.
ok. :-)

But this again relies on taking values, which took me about 5+mins to think :-(
which brings me to my original concern.
is there any algebraic alternative?

look, if you're getting stuck because something is "getting too lengthy", you're pretty much out of luck. ..

I mistakenly wrote lengthy. I meant the time was running out so i had to guess and move on.

[tim]

thanks for the clarification about what you meant by "lengthy". please keep in mind that after your time is up and you have to guess on an answer, it is imperative that you go back *before* you look at the solution and see if you can solve the problem on your own given unlimited time, regardless of how long it takes..

as for your algebra, you have made a mistake going from the first line below to the second line:

1.5d - d - (1.5d*d/100) >0
2.5 - (1.5d/100) >0

see if you can find your mistake and re-do the calculations. your approach however is fine..

[mokura]

i am almost sure that most people can't solve this in less than 3 minutes, let alone 2 minutes! I think the guessing of "extremes" is a good way to approach this problem, but still very time consuming.

[jlucero]

i am almost sure that most people can't solve this in less than 3 minutes, let alone 2 minutes! I think the guessing of "extremes" is a good way to approach this problem, but still very time consuming.

I'd tend to agree with you, mokura, but I've worked with a lot of students who are far better than me at algebra. Ultimately, it's always about finding a way that works best for you. Personally, I like picking numbers and I like your thought of picking extremes.

1) m = 100 and d = 5 (smal discount) or 85 (huge discount- more than the original 100% markup)

2) m = 1.5 and d = 1 or m = 15 and d = 10. No matter what numbers you pick here, the ratio is going to pack quite a punch because you ultimately have to multiply (1-d/100) times the increased price. Ultimately, since you are going to multiply these variables together, knowing the ratio of these two values will allow you to answer the question.

[hkparikh09]

i am almost sure that most people can't solve this in less than 3 minutes, let alone 2 minutes! I think the guessing of "extremes" is a good way to approach this problem, but still very time consuming.

I'd tend to agree with you, mokura, but I've worked with a lot of students who are far better than me at algebra. Ultimately, it's always about finding a way that works best for you. Personally, I like picking numbers and I like your thought of picking extremes.

1) m = 100 and d = 5 (smal discount) or 85 (huge discount- more than the original 100% markup)

2) m = 1.5 and d = 1 or m = 15 and d = 10. No matter what numbers you pick here, the ratio is going to pack quite a punch because you ultimately have to multiply (1-d/100) times the increased price. Ultimately, since you are going to multiply these variables together, knowing the ratio of these two values will allow you to answer the question.

I did not see the official answer posted here, but I got E. Is this correct? I tried the problem in the way suggested above by Joe.

(1) With a small discount (e.g., 10%), the final price is higher. With a higher discount (e.g., 50%), the final price is lower. Insufficient.

(2) With a small discount (e.g. 10%), the final price is higher. With a higher discount (e.g. 50%), the final price is lower. Insufficient.

(1 & 2 Combined) The lowest combination of m & d that satisfies both statements is m=30, d=20, and in this case, the final price is higher. However, if m=75 and d=50, the final price is lower. Insufficient.

I will agree with most people on this thread in that going through all the scenarios took me way too much time than would be advisable on the exam. Can anyone suggest a faster way to think about this problem? Are there any underlying fundamentals that I should remember?

Thank you,
Hardik

[RonPurewal]

If the original price of an item in a retail store is marked up by m percent and the resulting price is then discounted by d percent, where m and d are integers between 0 and 100, is the item’s final price (after both changes) greater than its original price?

(1) m > d + 10

(2) m = 1.5d

actually, i'm going to be the voice of dissension here, and say that there's actually something to the original poster's "too lengthy" complaint.
in general, the solutions to GMAT problems shouldn't be full-pagers, or even half-pagers. if a solution is not clearly going somewhere by the time you're a few lines into it, then it may be best (from the standpoint of time management) to abort that approach and try something else, like testing cases.
no matter what, one thing is for sure: if you do not know exactly what you are trying to accomplish and exactly how you are trying to accomplish it, then you should QUIT, immediately.

so, if you are just "pushing variables around" with no real purpose, then heck yes you should quit, and fast. then you'll have time to try other things.

--

if you're lazy like me, but have a modest level of intuition about percentages, then you don't have to calculate everything involved in the "testing extremes" situation.
e.g.

statement 1:

* extreme of "small d, big m":
let's say m = 99, d = 1.
in this case it's obvious that the final price is greater than the original price; doing the math here would be a waste of time.

* extreme of "big d": (notice that m still has to be bigger than d + 10)
say d = 80, m = 90something (like 91, or 95, or whatever).
say your original price is 100. then after the markup, it's 191 or 195 or whatever.
if you discount that amount by 80 percent, then it's clearly less than 100. (even a 50% discount would bring the price below 100.)

so, the statement is insufficient.

statement 2:

* "small" extreme:
say d = 10, m = 15. (you could try even smaller numbers, like d = 2 and m = 3, but i like 10% because i'm lazy.)
start with 100. after the 15% markup, it's 115.
a 10% markdown is 11.5, giving a final value of 103.5. that's greater than 100.

* "big" extreme:
say d = 60, m = 90.
start with 100. after the 90% markup, it's 190.
if you mark this down by 60%, again, it should be clear that you wind up with less than 100. (as in the previous example, even a 50% markdown would be more than enough to bring the price below 100.)

insufficient.

together:

* small extreme:
say d = 30, m = 45. (as long as they are greater than 20 and 30, respectively, the values d and 1.5d will satisfy statement 1.)
start with 100.
after the 45% markup, that's 145.
30% of 145 is 3 x 14.5 = 43.5. (alternatively, you can just notice that this is less than 3 x 15 = 45, which would be the necessary discount to bring the price back to its original level.)
if you knock 43.5 off of 145, you are still greater than 100.

* big extreme:
we can just re-use the case d = 60, m = 90 from above. that goes below 100.

still not sufficient.

so, (e).

--

the above is really not that much work.

to the posters who are complaining that testing numbers "takes too long" -- two responses:

1/
if testing numbers really does take too long, then that's not a
problem with the number-testing method; that's a problem of lack of organization.
if you are organized in choosing the cases to test, and you have a CLEAR GOAL in mind the whole time, then, no, it won't take too long. it just won't.

2/
more likely, you may just be trying to make up excuses not to test numbers.
this happens a lot with my students from classes and tutoring: they are reallyreallyREALLY in love with algebra, and they want a monogamous relationship with algebra. to the point where it seems almost morally wrong to "cheat" by doing anything that isn't algebra, like testing cases.
so, these students try to manufacture every excuse in the world not to even try testing numbers.
"it'll take too long!"
"i won't know what numbers to test!"
"but, it won't always work!"
etc. etc.
folks, this is not the way to get better at the gmat. the way to get better at the gmat is to expand your skill set, not to be super-stubborn about sticking to the few things you're most comfortable with.

in other words, if testing numbers "takes too long" ...
... that DOESN'T mean that you shouldn't test numbers
... that DOES mean that you should practice testing numbers and get faster at it.

[beakdas]

I saw this question in a different light.

Let us take the initial price : p
Marked up by m % we get: (1+m/100)p
Discounted by d % we get: (1-d/100)(1+m/100)p

We need to check whether (1-d/100)(1+m/100)p>p?
i.e whether (1-d/100)(1+m/100)>1?

We can get a Yes and No for both Statement (1) and (2)

[jlucero]

I saw this question in a different light.

Let us take the initial price : p
Marked up by m % we get: (1+m/100)p
Discounted by d % we get: (1-d/100)(1+m/100)p

We need to check whether (1-d/100)(1+m/100)p>p?
i.e whether (1-d/100)(1+m/100)>1?

We can get a Yes and No for both Statement (1) and (2)

This is exactly what Ron's talking about in his above post about student's not wanting to cheat on their beloved algebraic method.

I've seen students who are better at straightforward algebra than me in class and I'm not going to tell them to change their methods as long as they are consistently accurate and efficient.

However, I'd be surprised if more than 1% of students out there could come up with the algebra above in less than 2 minutes. Picking numbers is also tricky, but I'm 100% in agreement with Ron when it comes to picking extreme numbers. That's how I did math on my real GMAT and I found it to be much faster than coming up with the equations above (which would definitely take me longer than 2 minutes to make).

[beakdas]

I saw this question in a different light.

Let us take the initial price : p
Marked up by m % we get: (1+m/100)p
Discounted by d % we get: (1-d/100)(1+m/100)p

We need to check whether (1-d/100)(1+m/100)p>p?
i.e whether (1-d/100)(1+m/100)>1?

We can get a Yes and No for both Statement (1) and (2)

This is exactly what Ron's talking about in his above post about student's not wanting to cheat on their beloved algebraic method.

I've seen students who are better at straightforward algebra than me in class and I'm not going to tell them to change their methods as long as they are consistently accurate and efficient.

However, I'd be surprised if more than 1% of students out there could come up with the algebra above in less than 2 minutes. Picking numbers is also tricky, but I'm 100% in agreement with Ron when it comes to picking extreme numbers. That's how I did math on my real GMAT and I found it to be much faster than coming up with the equations above (which would definitely take me longer than 2 minutes to make).

The thing is after coming up with the algebra you have to get used to picking numbers to prove the consistency or rather inconsistency of the statements which are given.

I did resort to taking extreme number cases such as when d=90 and m = 80 or d= 10 and m =50 to get a Y/N.

I do believe using the so called 'smart numbers' is efficient to cheat your way to the right answer but being a graduate of mathematics converting everything in the light of equations comes naturally to my acumen.
I doubt whether that is a bane or boom to my study methodology for the GMAT.

[RonPurewal]

I do believe using the so called 'smart numbers' is efficient to cheat your way to the right answer but being a graduate of mathematics converting everything in the light of equations comes naturally to my acumen.
I doubt whether that is a bane or boom to my study methodology for the GMAT.

This issue seems to have been resolved, but I kjust wanted to comment on the irony of the red thing.

I also have a degree in mathematics. If I were asked what, exactly, is the difference between "higher-level" and "lower-level" mathematics, I would say that the whole idea of reliably being able to translate things into equations is exactly what characterizes "lower-level" mathematics.
In other words, the further I progressed through mathematics, the fewer opportunities there were to transform things into any type of equation, and the more proofs there were that required tedious testing of large numbers of cases. (Many of these proofs are the ones that are omitted from first-year calculus textbooks and such, under the guise of "This proof is beyond the scope of this book, but trust us on this result..." Ha.)

Perhaps I'm misinterpreting the phrase "a graduate of mathematics", but, that's my experience.

[beakdas]

I do believe using the so called 'smart numbers' is efficient to cheat your way to the right answer but being a graduate of mathematics converting everything in the light of equations comes naturally to my acumen.
I doubt whether that is a bane or boom to my study methodology for the GMAT.

This issue seems to have been resolved, but I kjust wanted to comment on the irony of the red thing.

I also have a degree in mathematics. If I were asked what, exactly, is the difference between "higher-level" and "lower-level" mathematics, I would say that the whole idea of reliably being able to translate things into equations is exactly what characterizes "lower-level" mathematics.
In other words, the further I progressed through mathematics, the fewer opportunities there were to transform things into any type of equation, and the more proofs there were that required tedious testing of large numbers of cases. (Many of these proofs are the ones that are omitted from first-year calculus textbooks and such, under the guise of "This proof is beyond the scope of this book, but trust us on this result..." Ha.)

Perhaps I'm misinterpreting the phrase "a graduate of mathematics", but, that's my experience.

Ron, I am happy to get a reply to my post.I wanted to enquire whether using ideas of higher level mathematics the so called theory of numbers or trigonometry to get answers quickly is advisable on the gmat.The GMAT doesn't cover a lot of higher level topics in the Quant section.Is it advisable to use higher mathematics in dealing with questions which are purely in the set of gmat quant?