If r, s, and w are positive numbers such that x = 60r + 80s

[II]

If r, s, and w are positive numbers such that x = 60r + 80s and r + s = 1, is x < 70 ?

(1) r > 0.5

(2) r > s

I got this question correct by plugging-in numbers, but wanted to learn ALTERNATIVE APPROACHES to answer this.
Also what do you think the question is trying to test here ?

My approach:
Start with statement 1 -> let r = 0.6 and s must therefore be 0.4. Plug this back into 60r + 80s = 68. This answers YES to the question "is x < 70?".
Test with r=0.5 and s=0.5, we can see that x=70. So can safely say that if r>0.5 then x<70

Statement 2:
If r > s, then r must be greater than 0.5 (since r + s = 1). We can use the same logic as in statement 1 here. SUFFICIENT.

Answer D.

Thanks in advance.
II

[Ian Stewart]

The equation just represents a weighted average. If r > 0.5, the value will be closer to 60 than to 80.

Think about the equation you'd set up to solve this problem: peanuts cost $60/pound, and cashews cost $80/pound. If a mixture is r% peanuts and s% cashews, what is the cost per pound of the mixture? If you ignore the 100s we need to introduce when dealing with percentages, you should get the same equation as in the question. Clearly if the mixture is more than half peanuts, it will cost less than $70/pound. Each statement tells us that.

[ramyaravindran]

Not sure if I am incorrect in assuming r to be 0.9 and s to be 0.1. In this case x>72. So, why is the official Answer D. Should it not be E?. Can someone explain?

[Ian Stewart]

Not sure if I am incorrect in assuming r to be 0.9 and s to be 0.1. In this case x>72. So, why is the official Answer D. Should it not be E?. Can someone explain?

You might have made a small arithmetic mistake. If r = 0.9 and s = 0.1, then:

x = 60r + 80s = 60*0.9 + 80*0.1 = 54 + 8 = 62

[ramyaravindran]

Thanks for the explanation.

[II]

Also note: one of the most effective ways of solving GMAT inequality problems is to focus on the EXTREME VALUES of a given inequality. This is particularly helpful when solving problems that involve multiple inequalities.


Here we can plug-in extreme values for the r and s inequalities. We know that r must be greater than 0.5 and s must be less than 0.5:
- 60 * (greater than 0.5) = greater than 30
- 80 * (less than 0.5) = less than 40
- So (greater than 30) + (less than 40) = less than 70.

[rajaram19]

w = 60r + 80s ---1
r + s = 1 --2

3--->60r + 60s = 60 adding 1 + 3

w-20s = 60 since there is no 'r' here .Shud E be the answer??wat is the wrong thing i am doin?/plss explain someone

[adi_800]

w = 60r + 80s
w = 60r + 60s + 20s
r + s = 1
60r + 60s = 60
Hence, w = 60 + 20s.. Is w < 70 => Is 60 + 2s < 70 => 2s < 10 => Is s < 0.5 ?

Statement 1 -> r > 0.5 => s < 0.5 -> Ans to question is sufficient
Statement 2 -> r > s => Sufficient !!

[parul9]

It is D!
I followed adi_800's method!

[Brent@GMATPrepNow]

If r, s, and w are positive numbers such that x = 60r + 80s and r + s = 1, is x < 70 ?

(1) r > 0.5
(2) r > s

Given: r, s, and w are positive numbers such that w = 60r + 80s and r + s = 1

Target question: w < 70
STRATEGY: Always be on the lookout for opportunities to rephrase the target question. In many cases, a little work up front will make analyzing the statements much easier.

Since we're told that r + s = 1, we might recognize that we can manipulate the equation w = 60r + 80s to take advantage of this information.
We can write: w = (60r + 60s) + 20s
Then factor the first part to get: w = 60(r + s) + 20s
Substitute to get: w = 60(1) + 20s
In other words: w = 60 + 20s, which means the target question becomes: Is 60 + 20s < 70?
We can make things even easier by subtracting 60 from both sides to get: Is 20s < 10?
And we can divide both sides by 20 to get: Is s < 0.5?
REPHRASED target question Is s < 0.5?

Aside: Here’s a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: r > 0.5
We already know that r + s = 1
So, for example, if r = 0.5, then s = 0.5
Similarly, if r > 0.5, we can be certain that s < 0.5
In other words, the answer to the REPHRASED target question is YES, s is less than 0.5
Statement 1 is SUFFICIENT

Statement 2: r > s
If r + s = 1, we can subtract s from both sides to get: r = 1 - s
Now take statement 2 and replace r with 1 - s to get: 1 - s > s
From here, we can add s do both sides of the inequality to get: 1 > 2s
Divide both sides by 2 to get: 0.5 > s
Once again, the answer to the REPHRASED target question is YES, s is less than 0.5
Statement 2 is SUFFICIENT

Answer: D