You have just started a sales job in a department store....

You have just started a sales job in a department store. Your pay consists of a base salary and a commission. The base salary is $$\$ 5,000$$. The scheme shown below is used to determine the commission rate. $$ \begin{array}{lr} \text { Sales Amount } & \text { Commission Rate } \\ \hline \$ 0.01-\$ 5,000 & 6 \text { percent } \\ \$ 5,000.01-\$ 10,000 & 8 \text { percent } \\ \$ 10,000.01 \text { and above } & 10 \text { percent } \end{array} $$ Note that this is a graduated rate. The rate for the first $$\$ 5,000$$ is at $6 \%$, the next $$\$ 5000$$ is at $8 \%$, and the rest is at $10 \%$. If your sales amounts to $$\$ 25,000$$, the commission is $5,000 * 6 \%+5,000 * 8 \%+15,000 * 10 \%=2,200$. Your goal is to earn $$\$ 30,000$$ a year. Write a program to find the minimum sales you have to generate in order to make $$\$ 30,000$$.

You have just started a sales job in a department store. Your pay consists of a base salary and a commission. The base salary is $$\$ 5,000$$. The scheme shown below is used to determine the commission rate.
$$
\begin{array}{lr}
\text { Sales Amount } & \text { Commission Rate } \\
\hline \$ 0.01-\$ 5,000 & 6 \text { percent } \\
\$ 5,000.01-\$ 10,000 & 8 \text { percent } \\
\$ 10,000.01 \text { and above } & 10 \text { percent }
\end{array}
$$
Note that this is a graduated rate. The rate for the first $$\$ 5,000$$ is at $6 \%$, the next $$\$ 5000$$ is at $8 \%$, and the rest is at $10 \%$. If your sales amounts to $$\$ 25,000$$, the commission is $5,000 * 6 \%+5,000 * 8 \%+15,000 * 10 \%=2,200$. Your goal is to earn $$\$ 30,000$$ a year. Write a program to find the minimum sales you have to generate in order to make $$\$ 30,000$$.

Key Concepts

Solving Equations with Piecewise Constraints

When working with piecewise functions, such as those in a graduated commission structure, one often needs to determine the level of input (like sales) required to achieve a desired outcome (such as a target income). This involves setting up and solving an equation that takes into account the varying rates at different intervals, ensuring that the solution adheres to the constraints defined by each tier.

Graduated Commission Structure

A graduated commission structure is a pay system where different portions of sales are rewarded at different rates based on specified thresholds. Instead of applying a single flat rate to all sales, the first portion of sales is calculated at a lower rate, then the next portion at a higher rate, increasing as thresholds are exceeded. This method incentivizes higher overall sales by offering increasing returns on sales beyond each tier.

Piecewise Functions

Piecewise functions are mathematical functions defined by different expressions over different intervals of the independent variable. In the context of commission schemes, each sales bracket has its own commission rate, and the overall commission is computed by summing the outputs of these individual pieces over the respective ranges of sales.

Tiered Rate System

A tiered rate system divides a quantity, such as sales or income, into segments or tiers, each of which is subject to a distinct rate or rule. This concept is commonly used in scenarios like tax systems and commission structures, allowing for marginal increases in benefit (or tax rate) as more of the quantity is accumulated.

Iterative and Algorithmic Methods

In cases where the mathematical equation derived from a piecewise function is not directly solvable through algebraic manipulation, iterative or algorithmic methods such as binary search or other numerical techniques are used. These methods incrementally search for and converge on a value that meets the target, ensuring that the calculated solution falls within the correct tiered intervals.

Transcript

00:02 Hello, for this problem we're trying to figure out how much money we need to make in sales to get $30 ,000 earnings.

00:10 So let's set up some variables.

00:13 We'll go ahead and say sales is equal to 0 .0 and we'll say our earnings is equal to i'm just going to start with a single penny and then what we want to do is just keep increasing earnings calculate how much or keep increasing sales figure out what that does to our earnings and as soon as we get our earnings of $30 ,000 we know that we're there.

00:42 So this is going to be a while loop.

00:44 So while earnings are less than $30 ,000, make sure we have all the zeros in there, i do.

00:57 And so what we're going to do now is first of all, we're going to increase.

01:03 Sales and i'm just going to increment sales by a penny at a time and so it's not going to be very efficient the best way to do this is just solve it as a math problem but we'll just do it penny at a time until we get bigger than 6 ,000 and then we have three different rates for the the commission so if the sales are less than are equal to 5 ,000 then the rate is going to be equal to 8%.

01:46 And then we'll do an lf or else if sales are less than or equal to 5 ,000.

01:59 No, no, that was 5 ,000.

02:01 This is 10 ,000.

02:07 Then the rate is equal to 10%.

02:13 And if it's more than that, so we can use an else statement, the rate is equal to 12%...

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