Is the number of seconds required to travel d1 feet

[BTGmoderatorDC]

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.

Which of the statements are the best option?

OA E

[ceilidh.erickson]

This is DS #350 in OG 2017. lheiannie07, you have consistently neglected to post your sources. Please start doing so - it's illegal not to.

Since distance = rate x time, the number of seconds required to travel d1 feet at r1 feet per second can be expressed as (d1)/(r1), and
the number of seconds required to travel d2 feet at r2 feet per second can be expressed as (d2)/(r2).

We can transcribe the question as: Is (d1)/(r1) > (d2)/(r2) ?

Cross-multiply to simplify further (we're allowed to do so here because we know that distance and rate will always be positive, so we don't have to worry about negatives & flipping the sign of the inequality):

Is (d1)(r2) > (d2)(r1) ?

To answer this question, we would need to prove that the left side of the equation will definitively be greater (or less) than the right side. If d1 and r2 are both greater than d2 and r1, then the left side will definitely be bigger (or vice versa).

(1) d1 is 30 greater than d2
Ignoring the 30 for a moment, this simply tells us that d1 > d2. Since we know nothing about the rates, this is insufficient:
(greater d1)(?? r2) > (lesser d2)(?? r1) ... We can't tell which side will be greater.

If you insist on the algebra (not recommended):

d1 = d2 + 30

When inserted into the question:
(d2 + 30)(r2) > (d2)(r1) ?
(d2)(r2) + 30(r2) > (d2)(r1) ?

Impossible to tell without knowing anything about r1 v. r2.

(2) r1 is 30 greater than r2.
Same logic here. We know that r1 is greater than r2, but we know nothing about the distances.

(1) & (2) Together:
Together, here is what we have conceptually:
(greater d)(lesser r) > (lesser d)(greater r)

We have one greater and one lesser element on each side. So what does this mean?

Because we don't know what proportion the +30 represents for either the distances or the rates, we don't know what those products will turn out to be. If d1 and d2 are 1,030 and 1,000 respectively, then that +30 is not much of a change. But if they are 31 and 1, then that +30 was a huge proportional difference! The same logic applies for our rates. We can't tell whether the relative difference of the rates is greater, or if the relative difference of the distances is greater, so we can't compare those products.

Or again, if you insist on algebra:

(d2 + 30)(r2) > (d2)(r2 + 30) ?
(d2)(r2) + 30(r2) > (d2)(r2) + 30(d2) ?
30(r2) > 30(d2) ?
r2 > d2 ?

We cannot answer this question without a comparison of r2 and d2. Insufficient.

The answer is E.

[ceilidh.erickson]

Compare this question with #43 in OG13/2015, which tests a similar concept.

If p1 and p2 are the populations and r1 and r2 are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?
(1) p1 > p2
(2) r2 > r1

Here, we can rephrase the question similarly as:
Is (p1)/(r1) > (p2)/(r2) ?
Cross-multiply: Is (p1)(r2) > (p2)(r1) ?

Basically the same question as the above. The statements, once again, are insufficient on their own:

1) p1 > p2

2) r2 > r1

When we put them together, though, we get this:

(greater p)(greater r) > (lesser p)(lesser r)

The product of the two greater terms will definitely be greater than the product of two lesser terms (because population and representatives have to be positive integers). So, the left side is definitely greater - sufficient.

Answer: C.

[ceilidh.erickson]

Another way to approach #350 would be to TEST VALUES:

Once again, used the rephrased target question:
Is (d1)(r2) > (d2)(r1) ?

(1) d1 is 30 greater than d2.

Because we have no information about the relative values of r1 and r2, this doesn't help. One rate could be extremely slow and the other extremely fast, or they could be the same. Insufficient.

(2) r1 is 30 greater than r2.

Because we have no information about the relative values of d1 and d2, this doesn't help. One distance could be extremely short and the other extremely long, or they could be the same. Insufficient.

(1) & (2) Together:

When testing values, start small:
d2 = 10 mi.
d1 = 40 mi.
r2 = 10 mph
r1 = 40 mph

Is (d1)(r2) > (d2)(r1) ?
(40)(10) > (10)(40) ? --> No, they're the same.

Now hold one thing constant, and vary the other. Let's keep the distances the same, but vary the rates:
d2 = 10
d1 = 40
r2 = 100
r1 = 130

Is (d1)(r2) > (d2)(r1) ?
(40)(100) > (10)(130) ?
4000 > 1300 ? ---> Yes.

We got 2 different answers for 2 different cases: insufficient.

[BTGmoderatorDC]

Another way to approach #350 would be to TEST VALUES:

Once again, used the rephrased target question:
Is (d1)(r2) > (d2)(r1) ?

(1) d1 is 30 greater than d2.

Because we have no information about the relative values of r1 and r2, this doesn't help. One rate could be extremely slow and the other extremely fast, or they could be the same. Insufficient.

(2) r1 is 30 greater than r2.

Because we have no information about the relative values of d1 and d2, this doesn't help. One distance could be extremely short and the other extremely long, or they could be the same. Insufficient.

(1) & (2) Together:

When testing values, start small:
d2 = 10 mi.
d1 = 40 mi.
r2 = 10 mph
r1 = 40 mph

Is (d1)(r2) > (d2)(r1) ?
(40)(10) > (10)(40) ? --> No, they're the same.

Now hold one thing constant, and vary the other. Let's keep the distances the same, but vary the rates:
d2 = 10
d1 = 40
r2 = 100
r1 = 130

Is (d1)(r2) > (d2)(r1) ?
(40)(100) > (10)(130) ?
4000 > 1300 ? ---> Yes.

We got 2 different answers for 2 different cases: insufficient.

Thanks a lot!

[BTGmoderatorDC]

Compare this question with #43 in OG13/2015, which tests a similar concept.

If p1 and p2 are the populations and r1 and r2 are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?
(1) p1 > p2
(2) r2 > r1

Here, we can rephrase the question similarly as:
Is (p1)/(r1) > (p2)/(r2) ?
Cross-multiply: Is (p1)(r2) > (p2)(r1) ?

Basically the same question as the above. The statements, once again, are insufficient on their own:

1) p1 > p2

2) r2 > r1

When we put them together, though, we get this:

(greater p)(greater r) > (lesser p)(lesser r)

The product of the two greater terms will definitely be greater than the product of two lesser terms (because population and representatives have to be positive integers). So, the left side is definitely greater - sufficient.

Answer: C.

Thanks a lot!

[BTGmoderatorDC]

This is DS #350 in OG 2017. lheiannie07, you have consistently neglected to post your sources. Please start doing so - it's illegal not to.

Since distance = rate x time, the number of seconds required to travel d1 feet at r1 feet per second can be expressed as (d1)/(r1), and
the number of seconds required to travel d2 feet at r2 feet per second can be expressed as (d2)/(r2).

We can transcribe the question as: Is (d1)/(r1) > (d2)/(r2) ?

Cross-multiply to simplify further (we're allowed to do so here because we know that distance and rate will always be positive, so we don't have to worry about negatives & flipping the sign of the inequality):

Is (d1)(r2) > (d2)(r1) ?

To answer this question, we would need to prove that the left side of the equation will definitively be greater (or less) than the right side. If d1 and r2 are both greater than d2 and r1, then the left side will definitely be bigger (or vice versa).

(1) d1 is 30 greater than d2
Ignoring the 30 for a moment, this simply tells us that d1 > d2. Since we know nothing about the rates, this is insufficient:
(greater d1)(?? r2) > (lesser d2)(?? r1) ... We can't tell which side will be greater.

If you insist on the algebra (not recommended):

d1 = d2 + 30

When inserted into the question:
(d2 + 30)(r2) > (d2)(r1) ?
(d2)(r2) + 30(r2) > (d2)(r1) ?

Impossible to tell without knowing anything about r1 v. r2.

(2) r1 is 30 greater than r2.
Same logic here. We know that r1 is greater than r2, but we know nothing about the distances.

(1) & (2) Together:
Together, here is what we have conceptually:
(greater d)(lesser r) > (lesser d)(greater r)

We have one greater and one lesser element on each side. So what does this mean?

Because we don't know what proportion the +30 represents for either the distances or the rates, we don't know what those products will turn out to be. If d1 and d2 are 1,030 and 1,000 respectively, then that +30 is not much of a change. But if they are 31 and 1, then that +30 was a huge proportional difference! The same logic applies for our rates. We can't tell whether the relative difference of the rates is greater, or if the relative difference of the distances is greater, so we can't compare those products.

Or again, if you insist on algebra:

(d2 + 30)(r2) > (d2)(r2 + 30) ?
(d2)(r2) + 30(r2) > (d2)(r2) + 30(d2) ?
30(r2) > 30(d2) ?
r2 > d2 ?

We cannot answer this question without a comparison of r2 and d2. Insufficient.

The answer is E.

Thanks a lot!