# Profitable company

Three businessman decide to open up their own company. They agree to distribute the yearly profits made in the same ratio as their initial investments. They invest R 50 000, R 75 000 and R25 000, respectively. The profit made by the company in the first year is R600 000.

(1) Express the ratio of the initial investments in simplest form.

Result

r = (Correct answer is: 2/3:1)
p1 =  200000
p2 =  300000
p3 =  100000

#### Solution:

$a=50000 \ \\ b=75000 \ \\ c=25000 \ \\ \ \\ 50000=2^4 \cdot 5^5 \ \\ 75000=2^3 \cdot 3 \cdot 5^5 \ \\ 25000=2^3 \cdot 5^5 \ \\ \text{GCD}(50000, 75000, 25000)=2^3 \cdot 5^5=25000 \ \\ \ \\ k=GCD(a,b,c)=GCD(50000,75000,25000)=25000 \ \\ \ \\ a_{0}=a/k=50000/25000=2 \ \\ b_{0}=b/k=75000/25000=3 \ \\ c_{0}=c/k=25000/25000=1 \ \\ \ \\ r=2:3:1$
$p=600000 \ \\ p_{1}=p \cdot \ \dfrac{ 2 }{ 2+3+1 }=600000 \cdot \ \dfrac{ 2 }{ 2+3+1 }=200000=2.000000\cdot 10^{ 5 }$
$p_{2}=p \cdot \ \dfrac{ 3 }{ 2+3+1 }=600000 \cdot \ \dfrac{ 3 }{ 2+3+1 }=300000=3.000000\cdot 10^{ 5 }$
$p_{3}=p \cdot \ \dfrac{ 1 }{ 2+3+1 }=600000 \cdot \ \dfrac{ 1 }{ 2+3+1 }=100000=1.000000\cdot 10^{ 5 }$

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