#### InducedFreedom

Hello everyone, I'm new to the forum, came here looking for a light at the end of the tunnel. I'm not a native speaker so please bear with me.

Problem: the whole thing is in Spanish but it pretty much says
We want to buy 3 different types of perfume (for men, women and kids) with a budget of 1,500,000.00. We know the cost of the kids perfume is 180,000.00 less than the sum of the other two (men & women) and we know the tax by product is 6% for men perfume, 12% for women perfume and 30% for kids perfume making the invoice 877,200.00 (taxes included). How much money are we going to spend on each perfume type (the high amount is due the currency we got in the country I live in).

I do not understand how to correlate these 3 equations to solve the matrix.
lets say:
X: men perfume
Y: women perfume
Z: kids perfume

then Z: (X+Y) - 180,000.00 (this equation would reflect the cost of each perfume).

X+Y+Z = 1,500,000.00 (this equation would reflect how much am I going to expend on each perfume type)

1.06X + 1.12Y + 1.3Z = 877,200.00 (this equation would reflect the amount am going to expend on each perfume with taxes)

I'm lost....

#### skipjack

Forum Staff
The English version is unclear. Can you post the original Spanish version in full?

#### InducedFreedom

Spanish Version:
Se adquieren tres tipos de perfumes. Para hombre, para mujer y para niÃ±os, con un presupuesto de 1,500,000.00. El costo del perfume para niÃ±os es de 180,000.00 menos que el del hombre y el de la mujer conjuntamente. Se sabe que el IVA por producto es asÃ­: 6% perfume para hombre, 12% perfume para mujer y 30% perfume para el de niÃ±os, lo que hace que la factura total con impuestos sea 877,200.00. Debes calcular la cantidad de dinero que se va a invertir en cada tipo de perfume. Resolver por el mÃ©todo de Cramer.

Translates to:
Three types of perfumes are acquired. For men, women and children, with a budget of 1,500,000.00. The cost of perfume for children is 180,000.00 less than that of men and that of women combined. It is known that TAX per product is as follows: 6% perfume for men, 12% perfume for women and 30% perfume for children, which makes the total invoice with taxes 877,200.00. You must calculate the amount of money that will be invested in each type of perfume. Solve by the Cramer method.

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#### skipjack

Forum Staff
It doesn't make sense to me. One can't form an equation that includes the budget amount of 1,500,000.00 if the total invoice is only 877,200.00.

#### Denis

Math Team
Your problem as stated is (I included 3 equations):

Three types of perfumes are acquired (for men, women and children)
at a total cost of 1,500,000

m + w + c = 1,500,000 [1]

The cost of perfume for children is 180,000 less than that of men
and that of women combined.

c = m + w - 180,000 [2]

TAX per product is as follows:
6% for men, 12% for women and 30% for children.
This makes the total invoice including taxes = 877,200

1.06m + 1.12w + 1.30c = 877,200 [3]

Calculate the amount of money that will be invested
in each type of perfume.
Clearly, [3] must be larger than 1500000

Making up an example:
[1]: m=530000, w=310000, c = 660000: m + w + c = 1500000

[2]: 660000 = 530000 + 310000 - 180000

[3]:
1.06m = 561800, 1.12w = 347200, 1.30c = 858000
561800 + 347200 + 858000 = 1767000

So total taxes in this example = 267000.

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