# Simple % problem

Hello,

This will seem super easy to people with good maths brains but I am failing to make sense of it.

I am simply trying to figure out sales profit shown as percentage. An example: Manganese Alloy costs 35200 to produce and sells for 269100, resulting in a profit of 233900 I have tried 2 formulas to try determine what percent the profit is: || 233900 / 269100 x 100 = 86.9 but does 86.9 represent the profit? || 269100 / 35200 x 100 = 764.5

My great confusion comes in when I try, as a hypothetical test, to sell for the same cost as production (35200 sold for 35200). This is obviously a profit of zero but comes out as 100%, I know 35200 is 100% of 35200 but Im trying to derive the sales profit and not the value of one figure against another. As you can see Im getting myself twisted in a knot.

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This has nothing to do with Sage as a math tool ; it is a elementary school arithmetic applied to elementary school economics.

• the production cost (35200 in your example)
• sales price (269100 in your example.

Now for some definitions (there I may be wrong, not being versed in englis-langiuage economics ; take my word wot a grain of salt (or even a toin of salt...)) :

• profit = sales price - production cost.
• profit rate = profit/production cost.
• profit margin = profit/sales price.

To work the numerical results, Sage is about as necessary as a howitzer to kill a fly. Nevertheless :

Enter the data :

sage: production_cost = 32500
sage: sales_price = 269100


Work out the results :

sage: profit = sales_price - production_cost ; profit
236600
sage: profit_rate = profit/production_cost ; profit_rate
182/25


Note that Sage tends to give you "exact" results whenever possible ; in this case, a fraction. Since this is not easy to use for most people, we convert this fraction to a decimal expression by usong the method .n :

sage: profit_rate = (profit/production_cost).n() ; profit_rate
7.28000000000000


Meaning that your profit is 7.28 times your initial investment. Bloody unrealistic...

sage: profit_margin=(profit/sales_price).n() ; profit_margin
0.879227053140097


The profit margin is about 0.88, meaning that for each unit of your sales, 0.88 of this unit goes to your profit. Bloody unlikely again...

It is customary to express this margin (and your profit rate too, BTW) in so-called percentage, which is a fraction multiplied by 100 :

sage: profit_margin_pct=profit_margin*100 ; profit_margin_pct
87.9227053140097


meaning that for 100 units of sales, about 89 go tou your profit.

Important : Note that any competent school teacher would (rigtfully !!!) yell at me for not expressing my data and results with their units ; my physics professors in high school would have fainted... And here is acouple of important facts :

• costs and prices must have the same units for the addition/substraction to have a meaning ;
• rates and margin have no unit (they are "dimensionless numbers", important concept in economics and in physics, for different reasons : their value is independent of your unit system).

Now, I'd suggest you to get textbooks of elementary arithmetic and elementary economics, working them with a hand calculator (or even a slide rule, if you still can get one outside of a museum...). Sage will become useful later...

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