Is the number of seconds required to travel d1 feet at r1

Is the number of seconds required to travel d1 feet at r1 feet per second greater than the number of seconds required to travel d2 feet at r2 feet per second?

We need to find whether \(\frac{d_1}{r_1}>\frac{d_2}{r_2}\).

(1) d1 is 30 greater than d2 --> \(d_1=d_2+30\). Nothing about the rates. Not sufficient.
(2) r1 is 30 greater than r2 --> \(r_1=r_2+30\). Nothing about the distances. Not sufficient.

(1)+(2) The question becomes whether \(\frac{d_2+30}{r_2+30}>\frac{d_2}{r_2}\). Now, if \(d_2=r_2\), then \(\frac{d_2+30}{r_2+30}=\frac{d_2}{r_2}\), thus in this case the answer would be NO but if \(d_2=1\) and \(r_2=2\), then in this case \(\frac{d_2+30}{r_2+30}=\frac{31}{32}>\frac{1}{2}=\frac{d_2}{r_2}\), thus in this case the answer would be YES. Not sufficient.

Answer: E.


Generally if \(a\), \(b\) and \(c\) are positive numbers and \(a>b\) then \(\frac{a}{b}>\frac{a+c}{b+c}\) (\(\frac{a+c}{b+c}\) is closer to 1 then \(\frac{a}{b}\), but as \(\frac{a}{b}>1\) then \(\frac{a+c}{b+c}\) is getting less to be closer to 1). For example: \(\frac{3}{2}>\frac{3+30}{2+30}\);

But if \(a\), \(b\) and \(c\) are positive numbers and \(a<b\) then \(\frac{a}{b}<\frac{a+c}{b+c}\) (again \(\frac{a+c}{b+c}\) is closer to 1 then \(\frac{a}{b}\), but as \(\frac{a}{b}<1\) then \(\frac{a+c}{b+c}\) is getting bigger to be closer to 1). For example: \(\frac{4}{5}<\frac{4+30}{5+30}\).

Hope it's clear.
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Hi Absar,

As with all DS: translating the given information is critical

So the real question is: D1/R1 > D2/R2?

(1) D1 = D2 + 30 You can substitute one of the variables. BUT R1 and R2 could be anything so you could answer yes by making R1 very tiny number and R2 a huge number or you could do the opposite and the answer would be no. Insufficient.

(2) R1 = R2+ 30 The same logic as a above for this statement

(1) + (2) If you put the statements together the question becomes: D2+30/R2+30 > D2/R2

You should test very big values and very small values to see what happens. If R2 is a very tiny value and D2 is a massive value then the left side will be smaller than the right side because R2 will be increased by a much greater factor than will D2.

D2 = 1000 --------> 1030/31 > 1000/1 NO
R2 = 1

And vice-versa, if DS is a tiny value and R2 is a massive value than the left side will be greater because D2 will have a greater percentage increase than R2.

D2 = 1 ---------> 31/1030 > 1/1000 YES
R2 = 1000

So, you are able to say Yes and No to the original question so the the answer is E

Let me know if you have any questions on this!

HG.

Hi,

St1 and st 2 alone are not sufficient so combining we get

IS

(d2+30)/(r2+30) > d2/r2

The asnwer to this Question will depend upon the ratio of d2/r2.

If d2/r2 is <1, then on adding the same + ve qty to Num & Den will increase the ratio value
if d2/r2 is > 1 ,then on adding the same qty will decrease the ratio

Hence ans E
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Is the number of seconds required to travel d1 feet at r1 feet per second greater than the number of seconds required to travel d2 feet at r2 feet per second?
(1) d1 is 30 greater than d2.
(2) r1 is 30 greater than r2.

Now, when I solved this question, I was surprised to see that the solution was (E), i.e, none of the statements provide sufficient information.

I recall, however, that when you add the same number to both the numerator and denominator of a fraction, the resulting fraction is always closer to 1. Using that logic, statements (1) and (2) TOGETHER provide sufficient information. Am I approaching this wrong here?

EDIT:

questions is t1 > t2?
or (d1/r1) > (d2 / r2)?
=> (d2 + 30)/(r2 + 30) > (d2/r2)?
=> [[(d2+30)-(r2+30)]/(r2 + 30)] > (d2-r2)/r2 -- subtract 1 on both sides
=> [(d2-r2)/(r2+30)]>(d2-r2)/r2
=> 1/(r2+30) > 1/(r2)
=> (r2+30) < r2
=> 30 < 0 ?
This questions IMO is not answerable but justified in OG . Any thoughts? Am I wrong?

This problem depends on the following concept.

If a,b are two positive numbers. and k is any positive number

a. if a/b is greater than 1 then a+k/b+k is less than a/b
b. if a/b is less than 1 then a+k/b+k is greater than a/b


Now consider the Above question

we have d1/r1 and d2/r2.

but d1 = d2 + 30
r1 = r2 + 30

so we have d2+30/r2 + 30 and d2/r2

but we don't know whether d2 is greater than r2. So together these statements are not sufficient

Is the number of seconds required to travel d1 feet at r1 feet per second greater than the number of seconds required to travel d2 feet at r2 feet per second?

Time = distance/rate

t1>t2?

(1) d1 is 30 greater than d2

We know nothing about the rate for r1 and r2.
INSUFFICIENT

(2) r1 is 30 greater than r2.

We know nothing about the distance of 1 an 2. Even if r1 is greater, the distance for 2 may be proportionally less so that even at a slower speed, d2 is covered in less time than d1.
INSUFFICIENT

1+2)

Hypothetical A:

1: Distance = 31 feet and rate = 31 feet/second
2: Distance = 1 feet and rate = 1 foot/second

In this case the time it takes for both is the same.

Hypothetical B:

1: Distance = 330 feet and rate = 60 feet/second
Time = Distance / Rate
Time = 330 / 60
Time = 11/3 seconds

2: Distance = 300 feet and rate = 30 feet/second
Time = Distance/Rate
Time = 300 / 30
Time = 10 seconds

It's possible that the time to cover d1 is the same as d2. It also may be greater.
INSUFFICIENT

(E)

Can someone verify the above cited approach for this particular DS question?
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hi samichange,
this kind of approach is not required, it will just complicate the matters .. take each statement separately and then combined to see if they satisfy the answer...
i think the approach was to show that the question is not justified as after simplifying it gives 30<0, which is not possible ...
however the person has gone wrong after the colored portion... u just cant cancel d2-r2 from each side..
you have to get the entire thing on one side and take d2-r2 as common term outside..
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Hi Bunuel,

I got the correct answer. However, i just pondered over this and found the below. Can you help me find out where am i wrong over here?

D2 + 30 / r2 + 30 > d2/r2 ( combining the 2 statements)

Now, r2(d2 +30) > d2(r2 +30)
r2d2 + 30r2 > d2r2 + 30d2
r2 > d2


Then 2 st are sufficient together right?

I would rather simplify this expression further to derive at the answer:
\(\frac{d2+30}{r2+30} > \frac{d2}{r2}\) --> \(r2*d2+30*r2 > d2*r2 + 30*d2\), r2d2 cancel out and we have 30*r2 > 30*d2, if r2=d2 the answer is no, and if r2 > d2 the answer is yes, thus Answer (E)

R1T1 =D1 AND R2T2 =D2 => THE QUESTION ASKS IF THE TIME FOR 1 IS FASTER THAN FOR 2. SO FORMULA ALONE YOU CAN SAY \(T1 = \frac{D1}{R1}\), \(T2 = \frac{D2}{R2}\)


STMT1 SAYS THAT D1> D2 AND IF I PUT THAT INTO THE FORMULA I AM NOT SURE WHAT THE Rs ARE FOR ME TO CONFIRM WHICH ONE IF GREATER. INSUFFICIENT.


STMT2 SAYS THAT R1>R2 AND SIMILAR TO STMT1 I DONT HAVE ENOUGH INFORMATION TO CONFIRM.


COMBINING TOGETHER WE KNOW THAT D1>D2 AND R1>R2, AND OUR FORMULA OF \(\frac{D1}{R1} > \frac{D2}{R2}\) BUT WHAT WE KNOW ABOUT THESE VALUES DOES NOT HELP BECAUSE D1>D2 HELPS THE LEFT SIDE OF OUR EQUATION TO BE BIGGER WHILE R1>R2 HELPS THE RIGHT SIDE OF THE EQUATION TO BE BIGGER. WITHOUT KNOWING MORE NUMBERS I CANNOT SUFFICIENTLY. ANSWER E.

time to travel d1 at r1 speed = d1/r1
time to travel d2 at r2 speed = d2/r2

is d1/r1 > d2/r2 ??

from (1)
d1 = d2+30
not sufficient
from (2)
r1 = r2+30
not sufficient

now from (1) + (2)
is (d2+ 30)/(r2+30) > d2/r2

first lets take d2 = 1 and r2 = 2
31/32 > 1/2 holds true
but if d2 = 2 and r2 = 1
32/31 < 2/1 does nt hold true.

so correct option is E .

Solution:

We need to determine whether the number of seconds required to travel d_1 feet at r_1 feet per second is greater than the number of seconds required to travel d_2 feet at r_2 feet per second. We can set this question up in the form of an inequality. Remember that:

time = distance/rate

Thus, we can now ask:

Is d_1/r_1 > d_2/r_2 ?

When we cross multiply we obtain:

Is (d_1)(r_2) > (d_2)(r_1) ?

Statement One Alone:

d_1 is 30 greater than d_2.

From statement one, we can create the following equation:

d_1 = 30 + d_2

Since d_1 = 30 + d_2, we can substitute 30 + d_2 in for d_1 in the inequality (d_1)(r_2) > (d_2)(r_1):

Is (30 + d_2)(r_2) > (d_2)(r_1) ?

We see that we still cannot answer the question. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

r_1 is 30 greater than r_2.

From statement two we can create the following equation:

r_1 = 30 + r_2

Since r_1 = 30 + r_2, we can substitute 30 + r_2 for r_1 in the inequality (d_1)(r_2) > (d_2)(r_1):

Is (d_1)(r_2) > (d_2)(30 + r_2) ?

We see that we still cannot answer the question. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two we have the following equations:

1) d_1 = 30 + d_2

2) r_1 = 30 + r_2

Since d_1 = 30 + d_2 and since r_1 = 30 + r_2, we can substitute 30 + d_2 for d_1 and 30 + r_2 for r_1 in the inequality (d_1)(r_2) > (d_2)(r_1):

Is (30 + d_2)(r_2) > (d_2)(30 + r_2) ?

Is (30)(r_2) + (d_2)(r_2) > (30)(d_2) + (d_2)(r_2) ?

Is (30)(r_2) > (30)(d_2) ?

Is r_2 > d_2 ?

Since we cannot determine whether r_2 is greater than d_2, statements one and two together are not sufficient to answer the question.

The answer is E.
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Clearly S1 and S2 are insufficient by themselves. The Key thing here is to make the right judgement on S1+S2. What this question is really testing is our understanding of - what happens when the Numerator and denominator of a fraction is increased by a fixed number?

To really hammer this concept and others related to it, do not forget to read this excellent post by Bunuel/MIKE McGARRY - https://gmatclub.com/forum/gmat-shortcu ... l#p1856037

I hope you find the above link useful
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Bunuel,

Since d2 and r2 both are positive integers, we can multiply the latest inequality : \(\frac{(30+d2)}{(30+r2)} > \frac{d2}{r2}\) --> \(30*r2 + d2*r2 > 30*d2 + d2*r2\). Thus we can eliminate \(d2*r2\) on the both side, we get : \(30r2>30d2\) --> \(r2>d2\). With this inequality, your first test case : r2=d2 should not apply and this bring me to the C answer.

What is wrong with my approach?
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The question becomes whether \(\frac{d_2+30}{r_2+30}>\frac{d_2}{r_2}\) or whether \(r_2>d_2\).

If \(d_2=r_2\), the answer would be NO
If \(d_2=1\) and \(r_2=2\), the answer would be YES.
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Why are we not considering negative rates in this problem? Bunuel

Nothing says d2 can't be 0.

Say d2=0 and r2=2, thus the time2 is 0
d1=30,r1=32, thus the time1 is about .9, YES

Now say d2=10 and r2=2, time2 is 5
d1=40,r1=32, time1 is 1.25, NO

Statements are still insufficient together.