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.
(2) How much will each businessman receive by way of profit?

Result

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

Step-by-step explanation:

$$\begin{gathered} a=50000 \\ b=75000 \\ c=25000 \\ 50000=2^4\cdot 5^5 \\ 75000=2^3\cdot 3\cdot 5^5 \\ 25000=2^3\cdot 5^5 \\ \text{GCD}(50000,75000,25000)=2^3\cdot 5^5=25000 \\ k=GCD(a,b,c)=GCD(50000,75000,25000)=25000 \\ {a}_0=a\mathrm{/}k=50000\mathrm{/}25000=2 \\ {b}_0=b\mathrm{/}k=75000\mathrm{/}25000=3 \\ {c}_0=c\mathrm{/}k=25000\mathrm{/}25000=1 \\ r=2:3:1 \end{gathered}$$

$$\begin{gathered} p=600000 \\ {p}_1=p\cdot {\displaystyle \frac{2}{2+3+1}}=600000\cdot {\displaystyle \frac{2}{2+3+1}}=200000 \end{gathered}$$

$p_2=p\cdot {\displaystyle \frac{3}{2+3+1}}=600000\cdot {\displaystyle \frac{3}{2+3+1}}=300000$

$p_3=p\cdot {\displaystyle \frac{1}{2+3+1}}=600000\cdot {\displaystyle \frac{1}{2+3+1}}=100000$

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