Find Percentage to Reduce Cost to Zero over 24 Months

Question

I have a starting figure of 100. My end goal is to reduce this figure by 1/24 its original value over 24 months, so that the end result is 0.

During the first month, I reduce the starting figure by 4.1667% to result in 95.83.

Going forward, I no longer have access to the starting figure. How can I calculate the percentages needed to evenly reduce the starting figure to zero.

Thanks!!

Answer 1

You require the value to decrease in an arithmetic progression from $1$ to zero over the space of $24$ months. \begin{eqnarray*} 1, \frac{23}{24}, \frac{22}{24}, \frac{21}{24}, \cdots, \frac{2}{24}, \frac{1}{24},0. \end{eqnarray*} So it decrease by $ 100 \times \frac{1}{24}= 4.166 \cdots \% $ in the first month.

It decreses by $ 100 \times \frac{1}{23}= 4.347 \cdots \% $ in the second month.

It decreses by $ 100 \times \frac{1}{22}= 4.545 \cdots \% $ in the third month.

...

It decreses by $ 100 \times \frac{1}{2}= 50 \cdots \% $ in the penultimate month.

It decreses by $ 100 \times \frac{1}{1}= 100 \cdots \% $ in the last month.

Answer 2

You reduce by $$\left(4.1666...\times\frac{100}{x}\right)\%$$ at each stage, where $x$ is the amount you have at that stage. If you do this, you reduce the amount by $$\left(4.1666...\times\frac{100}{x}\right)\%\times x=4.1666...$$ which is what you wanted to do.

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So at the start, $x=100$, so you reduced by $4.1666...\%$ to obtain $95.8333...$. Next, reduce this value by $\left(4.1666...\times\frac{100}{95.8333...}\right)\%=4.3478...\%$, to obtain $91.666...$, as required. And so on.

Answer 3

At the month $n$ the starting value of $100$ is reduced to: $$ V_n=100-\frac{100}{24}\cdot n $$

so the percentage of reduction at this stage is $$ r\%=\frac{\frac{100}{24}}{100-\frac{100}{24}\cdot n}\cdot 100=\frac{100 }{24-n} $$