Need help with understanding percent problem

[max]

Hello, please for give me if I am posting this under the wrong section. This may need to be under algebra.

First I want to say that I know how to solve the problem, but I want to understand solving it from a different point-of-view.

Mr. Cutler usually makes a 45% profit on every radio he sells. During a sale, he reduces his margin of profit to 40%, while his sales increase by 10%. What is the ratio of his new total profit to the original.

If I let the unknown equal the number of radios sold, this problem seems easier to solve. The answer is 44/45.
However, in my book it lets the unknown equal original price. When I try to solve it that way I can't figure how to come up with the "sales increase by 10%" part.

How should I think about that part?

If I let the original price be p.
The 45% profit he makes would be .45p.
During the sale the profit would be .40p. Would sales increase mean that profit increase by 10%? It seems almost like I need another variable.

 

[Loren]

Let n = number of radios sold originally.
Then n+.10n = 1.1n = number of radios sold after adjustment.
Let p[sub:2oydy87q]1[/sub:2oydy87q] = his original profit on each radio sold when profit margin is 45%.
Let p[sub:2oydy87q]2[/sub:2oydy87q] = his profit on each radio sold when profit margin is 40%.
Let c represent his cost for one radio.

p[sub:2oydy87q]1[/sub:2oydy87q] = n(.45c)
p[sub:2oydy87q]2[/sub:2oydy87q] = (n+.10n)(.40c) = 1.10n(.40c)
\(\displaystyle \frac{p_2}{p_1}=\frac{(1.1n)(.40c)}{n(.45c)}\)
Now, do the simplification and I think you have it.
It pays to name things completely.

 

[Denis]

Before: .45p
After: .40(1.10p)

After / Before = .45p / .44p = 45 / 44

 

[max]

Thanks for the replies, and thanks for showing a different way of solving the problem.
I'm afraid I might not have been to clear on my question.

Instead of solving for the unknown number of radios, solve for the original price of one radio.
When I try this, I don't understand how I can answer the part where it says "sales increase by 10%"
To me, to say that new profit on a radio = .40n(1.10n) seems like saying that new profit on one radio is increased by 10%.

If solving for original price of one radio, how can I understand how to finish the "sales increase by 10%" without using a new variable?

It does make perfect since to me though if I let unknown = number of radios sold.

 

[Denis]

What is the ratio of his new total profit to the original.

That's what the question is...what's that got to do with radio prices? Or number of radios?

 

[max]

Ok, my question may be stupid.

I just wanted to know how to look at the problem from another perspective.

If n = number of radios sold, a 10% increase in sales is 1.1n.
If 10 radios are sold then after a 10% increase in sales, 11 radios are sold.

If n = original price of a radio, then I don't understand what 1.1n means.
To me it sounds like an increase in the price of a radio even though in the problem it refers to an increase in sales.

My book solves this using n to be original price of a radio but doesn't give an explanation of what the 1.1n part means.

As I stated in my first post I know how to solve the problem. I just wanted to understand this looking from a different point-of-view.

 

[Deleted member 4993]

If n = number of radios sold, a 10% increase in sales is 1.1n.

If 10 radios are sold then after a 10% increase in sales, 11 radios are sold.

If n = original price of a radio, then I don't understand what 1.1n means.

which is it? n = number of radios sold or original price of a radio