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Algebra: Applied Problems (OG 13)

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Algebra: Applied Problems (OG 13)

by gmattesttaker2 » Mon May 13, 2013 12:04 pm
Hello,

This is from OG 13 P. 90.

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


My approach:

Is d1/r1 > d2/r2?

1) d1 = d2 + 30

=> is (d2 + 30)/ r1 > d2/r2?

In-sufficient

2) r1 = r2 + 30

=> is d1/(r2 + 30) > d2/r2?

In-sufficient

From 1 and 2:

Is, ( d2 + 30 )/ ( r2 + 30 ) > d2/r2 ?

Let, d2 = 50 and r2 = 5

=> ( d2 + 30 )/ ( r2 + 30 ) not greater than d2/r2

I was just wondering for what values of d2 and r2 would (d2 + 30) / (r2 + 30) be greater than d2/r2?

Thanks a lot.

Best Regards,
Sri

P.S. OA: E
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by srcc25anu » Mon May 13, 2013 12:13 pm
If D2 < R2
if d2 = 1 and r2 = 2 then d2/r2 = 1/2 or 0.5 and d1 = d2+30 = 31 and r1 = r2+30 = 32 then d1/r1 is very close to 1 and more than d2/r2 which is 0.5

No constraint states that d2 cannot be less than r2.
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by gmattesttaker2 » Mon May 13, 2013 12:45 pm
srcc25anu wrote:If D2 < R2
if d2 = 1 and r2 = 2 then d2/r2 = 1/2 or 0.5 and d1 = d2+30 = 31 and r1 = r2+30 = 32 then d1/r1 is very close to 1 and more than d2/r2 which is 0.5

No constraint states that d2 cannot be less than r2.
Hello srcc25anu,

Thanks for your prompt reply. The example is clear now. I was just a bit confused with what you meant by "No constraint states that d2 cannot be less than r2"

Thanks again for your help.

Best Regards,
Sri
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by ceilidh.erickson » Fri May 17, 2013 7:45 am
Another way to think about this conceptually is this:

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 be positive, so we won't have to flip the sign of the inequality):

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

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

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

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, as srcc25anu pointed out, 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. [spoiler]Insufficient: E[/spoiler].



Compare this question with #43 in OG13, which tests a similar concept. Here, we can rephrase the question as:
Is p1/r1 > p2/r2 ?
Cross-multiply: Is (p1)(r2) > (p2)(r1) ?

Basically the same question as #90. 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.
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by ceilidh.erickson » Fri May 17, 2013 7:52 am
If you want to test values rather than (or in addition to) thinking conceptually, make sure to test EXTREME VALUES. This will be the quickest way to test for insufficiency. Your thought process should be, "let me test the smallest possible values for that distance, and then the largest possible values. Does that give me different answers to my question?"

Here is more about testing extremes: https://www.beatthegmat.com/questions-th ... tml#561400
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by aaggar7 » Tue May 21, 2013 6:43 pm
Hi,

Check these values:

Let D2=10,R2=10

So D1=10 + 30 = 40
R1 = 10 + 30 = 40
=> D1/R1 = 1 and D2/R2 = 1

Both are same.

Another set of values

D2 = 10 R2=2
So D1 = 40 and R1 = 32

D1/R1 = 40/32 which is smaller than D2/R2

Choose answer E
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