Over the year 2017 I was suppose to spend my income as follows
- 10% on emergencies
- 20% on investment
- 50% on running expenses and
- 20% on big purchases
but instead my actuals are now showing as
- 6% on emergencies
- 21% on investments
- 52% on running expenses and
- 21% on big purchases.
Now my total income in Rupees was ₹1,206,839 and a balance amount left in my banks is ₹18,667.
My question is in what percentage should I divide the balance amount left in my bank such that I am able to meet my initial plan of 10%,20%,50% and 20% respectively?
My 2nd part question is if I get a fresh new entry on income, in what ratio(or percentage) should I utilize it? How do I think of a perfect dynamic formula for myself.
Let $x$ be the amount of money you've spent, and let $y$ be your remaining funds.
let $\alpha_e$, $\alpha_i$, $\alpha_r$, and $\alpha_b$ be the percentages of your income so far spent in each category. Then the total amounts spent in each category are $\alpha_e x$, $\alpha_i x$, $\alpha_r x$, and $\alpha_b x$.
Let $\gamma_e$, $\gamma_i$, $\gamma_r$, and $\gamma_b$ be the percentage of the remaining funds that you will spend in each category. After spending your remaining funds, the total percentage spent in each category will be: $$\frac{\alpha_e x + \gamma_e y}{x+y}, \frac{\alpha_e x + \gamma_e y}{x+y}, \frac{\alpha_r x + \gamma_r y}{x+y}, \mbox{ and }\frac{\alpha_b x + \gamma_b y}{x+y} $$
You want these percentages to be $0.1$, $0.2$, $0.5$, and $0.2$, so you want to solve the system of equations:
$$\frac{\alpha_e x + \gamma_e y}{x+y}=0.1$$ $$\frac{\alpha_i x + \gamma_i y}{x+y}=0.2$$ $$\frac{\alpha_r x + \gamma_r y}{x+y}=0.5$$ $$\frac{\alpha_b x + \gamma_b y}{x+y}=0.2$$
subject to the constraints $\gamma_e + \gamma_i + \gamma_r + \gamma_b = 1$ and $\gamma > 0$ for each $\gamma$. Solving for the unknowns in each gives:
$$ \gamma_e = \frac{0.1(x+y)-\alpha_e x}{y} $$ $$ \gamma_i = \frac{0.2(x+y)-\alpha_i x}{y} $$ $$ \gamma_r = \frac{0.5(x+y)-\alpha_r x}{y} $$ $$ \gamma_b = \frac{0.2(x+y)-\alpha_b x}{y} $$
For the numbers you gave above, we have $x=1188172$, $y=18667$, $\alpha_e=0.06$, $\alpha_i=0.21$, $\alpha_r=0.52$, and $\alpha_b=0.21$. Plugging in gives: $$ \gamma_e=2.64 $$ $$ \gamma_i=-0.437 $$ $$ \gamma_r=-0.773 $$ $$ \gamma_b=-0.437 $$
In other words, you can't do it - the deficit in your emergency spending is too large to be made up with your remaining funds.
You can use this method to answer your second question by always letting $x$ be what you've spent so far and letting $y$ be however much you have left to spend.