Problem Solving

Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store's revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?

A. 100p/(125 - p)
B. 150p/(250 - p)
C. 300p/(375 - p)
D. 400p/(500 - p)
E. 500p/(625 - p)

Let's use the INPUT-OUTPUT approach.

Let's say that Newspaper A accounted for 20% of all newspapers sold. In other words, p = 20
This means that Newspaper B accounted for 80% of all newspapers sold.

The question asks us to find the value of r, the percentage of newspaper revenue from Newspaper A.
To determine this, let's say that 100 newspapers we sold IN TOTAL.
This means that 20 Newspaper A's were sold and 80 Newspaper B's were sold.

REVENUE:
Newspaper A: 20 newspapers at $1 apiece = $20
Newspaper B: 80 newspapers at $1.25 apiece = $100
So, TOTAL revenue = $120

Since Newspaper A accounted for $20 of revenue, we can say that Newspaper A accounted for 16 2/3% of revenue. In other words, r = 16 2/3
Aside: We know this because $20/$120 = 1/6 = 16 2/3%

So, when we INPUT p = 20, the OUTPUT is r = 16 2/3.
We'll now plug p = 20 into each answer choice and see which one yields an output of = 16 2/3

A. 100(20)/(125 - 20) = 2000/105.
IMPORTANT: If we want, we can use long division to evaluate this fraction (and others), but we can save a lot of time by applying some number sense. Since 2000/100 = 20, we know that 2000/105 will be SLIGHTLY less than 20. So, we can be certain that 2000/105 does not equal 16 2/3. As such, we can ELIMINATE A.

B. 150(20)/(250 - 20) = 3000/230. We know that 3000/200 = 15, so 3000/230 will be less than 15. So, we can be certain that 3000/230 does not equal 16 2/3. As such, we can ELIMINATE B.

C. 300(20)/(375 - 20) = 6000/355. Hmmm, this one is a little harder to evaluate. So,we may need to resort to some long division (yuck!). Using long division, we get 6000/355 = 16.9.... ELIMINATE C.

D. 400(20)/(500 - 20) = 8000/480 = 800/48 = 100/6 = 50/3 = 16 2/3. perfect! KEEP

E. 500(20)/(625 - 20) = 10000/605 = 100/6.05. Notice that, above, we saw that 100/6 = 16 2/3. So, 100/6.05 will NOT equal 16 2/3. ELIMINATE E.

Answer: D

Cheers,
Brent

Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store's revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?

A. 100p/(125 - p)
B. 150p/(250 - p)
C. 300p/(375 - p)
D. 400p/(500 - p)
E. 500p/(625 - p)

If you're not sure how to proceed with this question, or if you're behind on time and you want to catch up, you can give yourself a 50-50 chance in about 10 seconds.

To do so, we'll see what happens when we use an EXTREME value for p.
Say p = 100
In other words, 100% of the newspapers sold were Newspaper A.
This means that 100% of the revenue is from Newspaper A.
In other words, when p = 100, then r = 100

At this point, we'll plug in 100 for p and see which one yields a value of 100.
Only answer choices B and D work.
B) 150(100)/(250-100) = 100 PERFECT
D) 400(100)/(500-100) = 100 PERFECT

Now take a guess (B or D) and move on.

Cheers,
Brent

(x+1)^2/(X-1)^2

If x is not = 0 or 1, and if x is replaced by 1/x everywhere in the expression above, then the resulting expression is equivalent to?

A) (x+1)^2/(x-1)^2
B) (x-1)^2/(x+1)^2
C) x^2+1/1-X^2
D) X^2-1/X^2+1
E) -(x-1)^2/-(X+1)^2

How to I calculate this quickly?


Looking for experts help on this - thank you!
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Solution:

Let's start by substituting 1/x for x. Then we'll find the common denominator for the resulting fractional expressions, and finally we'll simplify the resulting fraction. We have:



Even though our answer does not match any of the answer choices, our answer is really similar to answer choice A. In fact, they are the same. We can demonstrate that they are the same by factoring out a "-1" from the denominator of our fraction. So we can say:

1 - x = -(-1 + x) = -(x - 1)

Plugging this back into the fraction we have:



But, because this entire expression is being squared, the negative would turn into a positive, and the end result would be the same as:



The answer is A

Note: If this is difficult to see, let's do the same thing with an integer. Let's say we have 2. We could say the following:

(-2)^2 = (2)^2

4 = 4

Notice that because we squared the -2, it ended equaling 4. Although the expression in the problem above seems more complicated, the math behind what we did is exactly the same.