Can't work out these percentages

Question

I have an amount of money. 10% of this money is currently shared between these 4 people in the following ratio:

Person 1 (1/3) 33.333% Person 2 (1/3) 33.333% Person 3 (1/6) 16.666% Person 4 (1/6) 16.666%

However, I now need to take an extra 1% of the total money making it 11% of the total shared between these people. The problem is all of this new 1% would be given to person 1 only, which would mean all the percentages need to be recalculated with person 1 getting the largest share.

How would I go about calculating this? Thanks

Answer 1

Let $\$T$ be the total amount of money, and let $\;\$2k,\;$ $\$2k,\;$ $\$k,\;$ $\$k\;$ be the amounts that Persons 1, 2, 3, 4 originally had, where $k$ is the proportionality constant. We are told that the 4 people originally had a combined amount of $\$(0.1)T.$ Therefore,

$$ 2k + 2k + k + k \; = \; 0.1T$$ $$ 6k = 0.1T$$ $$T = 60k$$

After Person 1 receives an additional $1\%$ of the total amount, the amounts of money the four people later have are $\;\$(2k + 0.01T),\;$ $\$2k,\;$ $\$k,\;$ $\$k.\;$ But since $2k + 0.01T = 2k + (0.01)(60k) = 2.6k,$ the amounts that the four people later have are

$$ \$2.6k, \;\; \$2k, \;\; \$k, \;\; \$k $$

Therefore, the share percentages the four people later have are

$$ \frac{2.6k}{6.6k} \times 100\%, \;\;\; \frac{2k}{6.6k} \times 100\%, \;\;\; \frac{k}{6.6k} \times 100\%, \;\;\; \frac{k}{6.6k} \times 100\% $$

or approximately

$$ 39.39\%, \;\; 30.30\%, \;\; 15.15\%, \;\; 15.15\% $$

Answer 2

$\frac{1}{10}$ of the total money is shared between $4$ people, leaving $\frac{9}{10}$, or $90\%$ left over. $1\%$ of the total money is $\frac{1}{100}$.

Adding this to the ten percent being shared, we get $$\frac{1}{10}+\frac{1}{100}=\frac{10+1}{100}=\frac{11}{100}=0.11=11\%$$ of the total money now being shared.

Person $(1)$ or $(2)$ was originally getting one third of one tenth of the total money$$\frac{1}{3}\cdot\frac{1}{10}=\frac{1}{30}=0.0\overline{33} = 3.\overline{33}\% $$ and person $(3)$ or $(4)$ was originally getting one sixth of one tenth of the total money $$\frac{1}{6}\cdot\frac{1}{10}=\frac{1}{60}=0.01\overline{66} = 1.\overline{66}\% $$

If person $(1)$ or $(2)$ gets the $1\%$, then they now have: $$\bigg(\frac{1}{30}+\frac{1}{100}\bigg)=\frac{13}{300}=0.04\overline{33}=4.\overline{33}\%$$ of the total money. Now, we should have that $\frac{13}{300}+\frac{1}{30}+\frac{1}{60}+\frac{1}{60}=\frac{11}{100}=11\%$ of the total money. Multiplying these by $\frac{100}{11}$ will give $$\frac{13}{33}+\frac{10}{33} +\frac{5}{33}+\frac{5}{33}=39.\overline{39}\%+30.\overline{30}\%+15.\overline{15}\% +15.\overline{15}\%=1$$ which is how much of the $11\%$ each person has, or how much percent each person would have if all of the money was only the $11\%$.

If instead person $(3)$ or $(4)$ got the one percent, then we repeat the process: $$\bigg(\frac{1}{60}+\frac{1}{100}\bigg)=\frac{4}{150}=0.02\overline{66}=2.\overline{66}\% $$

Now, we have $\frac{1}{30}+\frac{1}{30}+\frac{1}{60}+\frac{4}{150} =\frac{11}{100}=11\%$ of the total money. Multiplying again by $\frac{100}{11}$ gives $$\frac{10}{33}+\frac{10}{33}+\frac{5}{33}+\frac{40}{165}= 30.\overline{30}\%+30.\overline{30}\%+15.\overline{15}\%+24.\overline{24}\%=1$$