A factory produces one type of widget. This month, the factory raised

The correct answers are:
X = 160
Y = 62.5

If the revenues are equal for 2 months, the equation becomes:
wp = ((x/100) of p) * ((y/100) of w)

Solving the equation , we get xy = 10000. The 2 values which multiply to 10000 are 160 and 62.5.
X is 160 since the price increased, and Y is 62.5 since the number of quantities sold decreased.

Correct me if I'm wrong. But if price p increases by x%, then new price is p(1+ (x/100)). So, the new price can not be (x/100)p as explained in some above solutions.

Is there any shortcut? Seems we need to try one by one of the answer choices?

Say.

x%-percentage raised original price
y%-percentage raised sold
say $1 = widget price, 10 = quantity sold
$1*10=$10 of revenue
so in order for the revenue to be the same
$1(x%)10(y%)=$10
x%*y%=1
62.5*160=1

sameerm22 yep, you are correct, if it were "increases by x%" then p(1+ (x/100)) but since it is "increases to x%" then x/100

Wow dave13 You have become an expert now

Revenue = Price x Quantity.

Before I get into my solution, let me take an example:

Let's say the factory doubles the price.
And after doubling the price, the revenue remains the same.

What would have happened to the Quantity?

If the price doubles, but the revenue remains the same, the quantity must be halved.

Since the overall revenue was the same, the increase in the price will correspond to a proportional decrease in the quantity. In maths terms, if the price became 2x, the quantity would have become 1/2x.

--

We can answer the question using this underlying understanding. 


1. "the factory raised the price of each widget to X% of the original price"

a. The price has been raised.
b. The price is not raised by X%. It is raised to X% of the original price. e.g. Say last month's price was $100, and this month's price is $110. Is X = 110 or 10? The price has been raised by 10%. It has been raised to 110% of the previous price. So, X = 110.

Since the price has been raised, X > 100.

2. "the factory only sold Y% as many widgets as last month, and the total revenue from the sale of widgets was equal for last month and this month."

a. The factory sold Y% as many widgets, not Y% less than last month, but Y%.
e.g. Let's say the quantity sold went down from 100 to 80. Would Y be 20 or 80? The factory sold 80% as many widgets as last month. Or, the factory sold 20% less widgets than last month. In this example, Y would be 80.
b. The total revenue was equal. The price per widget had increased. So, the quantity of widgets must have reduced.
Y < 100. 

X can only be 150, 160, or 180.
Y can only be 50, 62.5, or 75.

Now, how to deal with the remaining values.

I will start with the potential answers for X: 150, 160, 180 and check if the corresponding value of Y is there in the table. (We could start with Y as well and then check if the corresponding value of X exists in the table. I chose X because I want to avoid dealing with 62.5/100.)

i.
If the price became 150%, = 150/100 = 3/2 times, the quantity would have to be 2/3 times last month's quantity to keep the revenue the same.

2/3 times last month's quantity means 66.7% of the old quantity. (Knowing 1/3 = 33.3% helped me here.)
66.7% is not an option. I'll reject 150 for X.

ii.
If the price became 160% = 160/100 = 8/5 times, the quantity would have to be 5/8 times last month's quantity to keep the revenue the same.

5/8 times last month's quantity means 62.5% of the old quantity. (I used a calculator to calculate 5/8).
62.5 is present in the table. 

We could very well mark

X: 160
Y: 62.5

at this point and move on. We can't have two sets of values that would be consistent with the information provided. We've found one set of values that's consistent. This will be the only consistent pair.


iii.
Nevertheless, for practice, let me try X = 180.

If the price became 180% = 180/100 = 9/5 times, the quantity would have to be 5/9 times last month's quantity to keep the revenue the same.

5/9 times last month's quantity means 55.5% of the old quantity. (I knew that 1/9 = 11.11%. So, 5/9 must be 55.55%).
55.5 is not present in the table. Reject 180 for X.­­

Like The Fray said, where did I go wrong?
I used original price as 80 and original quantity as 200, therefore revenue= 16000.
Then, 16000/200 (150%), 16000/208 (160%), 16000/224(180%), only the first one, 16,000/200=80, which is 40% of the original quantity, which is not an option.
I am suspecting my values of 200, 208 and 224 are wrong due to the fundamental error of 'to __%'?
Please help.

­Mihir,

Not sure what you mean by "Then, 16000/200 (150%), 16000/208 (160%), 16000/224(180%), only the first one, 16,000/200=80, which is 40% of the original quantity, which is not an option."

If you elaborate, maybe I can help.

­Hi,
150 I derived by 80+150%of 80, 208 by doing 80+ 160% of 80 and so on..
Thank you!

MihirBathia

You are going wrong in the values.

First you should have taken them as 100 and 100. The solution ion would have become much simpler.

Even with your values, 80 increases to 150% of 80 or 120 and not BY 150%.

So, when 80 increases to 160% of itself, it becomes 128 and 16000/128 is 125.

Now 125 is 62.5% of 200. Thus, the answer is 62.5%

Solution:
Let us take both as 100 each. Why?
There are no restriction on values except x>100 and y<100.
Secondly you can substitute the new values as X and Y directly.

The revenue remains the same, so XY=100*100=10000
Check options: X=150, so Y=10000/150=200/3=66.7%

X=160, so Y=10000/160=62.5%…yes, there in option

Thus our answers are 160% and 62.5%

­Got it, thank you so much for the explaination! Made an error in understanding the language.

Please can you help me with why x has to be greater than 100. I see the explanation you gave over here that because the price increases x has to greater than 100. Can you shed some light on that. I don’t get it

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­Refer to this line "the factory raised the price of each widget to X% of the original price."

There is an increase, so "X"%  should be greater than 100. 

If the widget initially costs $100, and later the New price is raised to 160% of the original price.

New price = $(160/100) *100 = $160­

­
1. Let's say the original price was $20.

Now, let's say that the new price were also $20, what would the value of X be?

New price = X% of the original price

20 = X% of 20

X = ?

X = 100 in this case.

20 = 100% of 20.

With me thus far?


2. The question mentions: "the factory raised the price of each widget to X% of the original price".

The factory raised the price. So the new price is higher than the old one.

So if the original price was 20, the new price has to be greater than 20.

So, what can we say about the value of X now?

Let's say the new price is $25.

25 = X% of 20

X = 25/20 * 100
X = 125


Remember, the price was raised TO X% of the original price, and not BY X% of the original price. I've explain this further in the complete solution.

Can X be less than 100? e.g. 90?

New price = X% of original price

If X = 90,

New price = 90% of 20

New price = 18.

This new price is lower than the original price. We can't have that. So, X has to be greater than 100.

Tell me if something is still not clear.

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