"Eco Wildlife Preserve contains 5x" Algebra(5th ed.)Pg. 125

[mbb5s]

Manhattan GMAT Algebra (5th Ed.) Guide - Pg. 125, #7

Eco Wildlife Preserve contains 5x zebras and 2x lions, where x is a positive integer. If the lions succeed in killing z of the zebras, is the new ratio of zebras to lions less than 2 to 1?

(1) z > x
(2) z = 4

I understand how Statement (1) is sufficient. However, I do not understand why Statement (2) is NOT sufficient.

I worked through Statement (2) using the following method:
Using the rephrased inequality, as explained in the guide, you want to know if ((5x - z)/(2x)) < (2/1). If z = 4, as stated in Statement (2), then you want to know if ((5x-4)/(2x)) < (2/1). The inequality simplifies to x < 4, and the problem states that x is a positive integer > 0, so combined, 0 < x < 4. That means that x must equal 1, 2, or 3. If you substitute these 3 values for x back into the inequality ((5x-4)/(2x)) < (2/1), the left side of the inequality is less than the right side for each case. Using this method, you arrive at the conclusion that Statement (2) is also sufficient for concluding that the new ratio of zebras to lions is less than 2 to 1.

Where am I making a mistake?

Thanks!

[RonPurewal]

to prove the statement insufficient, just test cases.

if x = 1, then we start with 5 zebras and 2 lions, and end with 1 zebra and 2 lions. 1:2 is definitely less than 2:1.

if x = 1000, then we start with 5000 zebras and 2000 lions, and end with 4996 zebras and 2000 lions. that ratio is still a lot more than 2:1.

[RonPurewal]

the problem with your approach is that you forgot that the question is a question.

fixed version:

I worked through Statement (2) using the following method:
Using the rephrased inequality, as explained in the guide, you want to know, "Is ((5x - z)/(2x)) < (2/1)?". If z = 4, as stated in Statement (2), then you want to know, "Is ((5x-4)/(2x)) < (2/1)?". The inequality simplifies to "Is x < 4?", and the problem states that x is a positive integer > 0.

... so, not sufficient. (some positive integers are less than 4; the others aren't.)

[RonPurewal]

so, most important takeaway here:

If you're going to manipulate a yes/no question algebraically, ALWAYS copy "Is" and "?", in EVERY STEP.

[EloyJ759]

I have a small question. If I use algebra, I can easily demonstrate that statement 1 is sufficient. However, using smart numbers is confusing me. If I choose x=1 and z=3, I get a ratio r=1 (zebras/lions=2:1), so the answer to the question is "No, the zebra to lions ratio is not smaller than 2". If I choose x=1 and z=4, I get a ratio r=1/2 (zebras/lions=1/2), and the answer to the question is "Yes, the zebra to lions ratio is smaller than 2". Therefore, I get different answers to the main question "Is the zebra to lions ratio smaller than 2?" depending on the smart numbers I choose. Doesn't it mean statement 1 is not sufficient?

[RonPurewal]

I have a small question. If I use algebra, I can easily demonstrate that statement 1 is sufficient. However, using smart numbers is confusing me. If I choose x=1 and z=3, I get a ratio r=1 (zebras/lions=2:1), so the answer to the question is "No, the zebra to lions ratio is not smaller than 2".

you lost me there...

if you choose x = 1 and z = 3, then:
• you start with 5(1) = 5 zebras and 2(1) = 2 lions.
• z = 3 lions are eliminated.
• after the 3 lions are eliminated, there are 2 zebras and 2 lions.
• that's a ratio of 2:2 = 1:1.
• a ratio of 1:1 is definitely less than a ratio of 2:1.

how are you concluding that the ratio 1:1 is NOT less than the ratio 2:1?

[EloyJ759]

I have a small question. If I use algebra, I can easily demonstrate that statement 1 is sufficient. However, using smart numbers is confusing me. If I choose x=1 and z=3, I get a ratio r=1 (zebras/lions=2:1), so the answer to the question is "No, the zebra to lions ratio is not smaller than 2".

you lost me there...

if you choose x = 1 and z = 3, then:
• you start with 5(1) = 5 zebras and 2(1) = 2 lions.
• z = 3 lions are eliminated.
• after the 3 lions are eliminated, there are 2 zebras and 2 lions.
• that's a ratio of 2:2 = 1:1.
• a ratio of 1:1 is definitely less than a ratio of 2:1.

how are you concluding that the ratio 1:1 is NOT less than the ratio 2:1?

You're absoutely right...my mistake! Thanks for the help Ron!

[RonPurewal]

you're welcome.