SOLUTION: cost of articles is in ratio 3:2, on the first there is 40% loss and on the second there is 30% gain. What is the overall percentage of gain or loss?

Question 756515: cost of articles is in ratio 3:2, on the first there is 40% loss and on the second there is 30% gain. What is the overall percentage of gain or loss?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i calculate a 12% loss.
an easier way to figure it out is as follows:
since the cost is in a ratio of 3:2, then use any value for the cost that comes out to be in a 3:2 ratio.
assume the cost of the first product is 300 dollars and assume the cost of the second product is 200 dollars.
the sum of the costs for the first product and second product is currently set at 500 (300 + 200).
the first product takes a 40% loss and the second product takes a 30% gain.
300 - 300*.4 + 200 + 200*.3 = 300 - 120 + 200 + 60 which equals 440.
you went from 500 to 440 dollars.
that's a loss of 60 dollars which is equal to 12% of the original cost.

the harder way is to do it with variables rather than assuming some numbers.
you do the same thing only now you need more understanding of algebra.
let a represent the cost of the first product and let b represent the cost of the second product.

since the cost of the products are in the ratio of 3:2, then we can solve for a in terms of b or versa.

the ratio formula is 3/2 = a/b
that reads as:
3 is to 2 as a is to b.
we can cross multiply to get:
2a = 3b
divide both sides of this equation by 2 to get:
a = 3b/2

the sum of the costs of the products today is:
a + b which we can then translate to 3b/2 + b because a is equal to 3b/2 which means it can be replaced by 3b/2 in the equation.

3b/2 + b is the same as 3b/2 + 2b/2 which becomes 5b/2
5b/2 represents the cost of the 2 products today which is also the same as a + b
we get a + b = 5b/2 because we replaced a with its equivalent of 3b/2.

now to the revised cost.
a is reduced by 40% and b is increased by 30%.
a - .4a becomes .6a
b + .3b becomes 1.3b
the revised costs become:
.6a + 1.3b

since a is equal to 3b/2, we can replace a in the equation with 3b/2 to get:
the revised costs become:
.6(3b/2) + 1.3b
simplify this to get:
.9b + 1.3b which makes the revised costs equal to 2.2b.

the original costs are 5b/2 which translates to 2.5b
the revised costs are 2.2b.
the ratio of the revised cost to the original cost is equal to 2.2b / 2.5b which is equal to .88
this means the revised costs are equal to 88% of the original cost which means we took a loss of 12% because 88% - 100% is equal to -12%.

you can solve it by replacing the variables with assumed numbers since any number would do because all they have to be is in the correct ratio.

you cal also solve it directly by dealing with variables rather than numbers.
dealing with variables is a little more complex but the same logic is applied to both and both will arrive at the same answer if you did it correctly.

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