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					Fruit Stand Lesson II
Today, we'll take another look at the supply and demand for fruit. Our goal will be to understand the significance of “equilibrium price.” Background Recall that the price of pineapples is determined by two groups of people: pineapple growers and buyers. We learned last time that both growers and buyers want the best possible outcomes for themselves. The growers are happiest when the price they sell their pineapples at is highest. The buyers are happiest when the price of pineapples is lowest. This creates a dynamic market exchange, where two opposing forces have to balance each other out. This "balancing" is what drives us to market equilibrium. The tables below present market data sorted into two rows:

Price per Pineapple Number of Pineapples the Grower Sells Price per Pineapple Number of Pineapples the Buyer Buys

1.5 0 0 8

2.1 1 1 6

2.7 2 1.5 5

3.5 3 2 4

3.9 4 2.5 3

4.5 5 3 2

5.1 6 4 0

Our Goal: Determine the final price of a pineapple in this free market exchange. Approach 1: Plotting the Supply and Demand lines from the given data points. In the same coordinate plane, plot the pairs of points from the grower and the buyer. Connect the dots when you're done. Note that we will have two axes: Price and Number of Pineapples Just like we use Y as the dependent variable and X as an independent variable, we'll use Price as our dependent variable and Number as our independent variable.

Price

Number of Pineapples

Note:

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Make sure you keep track of which coordinate points belong to which person. (Using two different colors would be a good way to do this!) Also, notice the approximate coordinate point where the two lines intersect. Approach 2: Graphing the Supply and Demand lines as equations. Background
Now, we'll graph the same lines as linear equations with slopes and intercepts. Slope-Intercept Form: y = mx + b Let's break this down and understand what each letter in the equation means.

y : this is the dependent variable, in our case the Price of a pineapple. x: this is the independent variable, in our case the Number of pineapples. m: this is the slope of the line. We think of slope as "rise over run" rise run

What does "rise over run" actually mean? It represents how much the line is changing in height divided by how much it's growing horizontally. Think of a ball rolling down a hill. The steeper the hill, the faster the ball rolls. In other words, slope is

Change in Y Change in X

Remember that when dealing with linear equations, "change" just means "subtraction". So "change in Y" means: final value of Y - beginning value of Y So in our case, the slope is

Change in Price Change in Number of Pineapples

b: this is the y-intercept of the line. In other words, this is the coordinate where our line meets the Y-axis.

y-intercept

So with this information, let’s find the equations for Supply (the Seller) and Demand (the Buyer)

Directions: 1. Break up into groups, and together as a team find the equations for the two lines 2

2. After you find the equations, you will be asked to find the intersection of the two lines. 3. Compare your group’s finds with the rest of the class!

Group Activity

Supply Line
Start with the Seller. From the given data, we see that the Supply line starts at the point (0, 1.5) and ends at the point (6, 5.1) Remember what these coordinate points mean! (0, 1.5) means (no pineapples when the price is 1.5) and (6, 5.1) means (6 pineapples when the price is 5.1) Step 1) Calculate the slope for the Supply Line

Change in Price = Change in Number of Pineapples =
Step 2) Find the Y-intercept for the Supply Line (ask yourself the question: "When the number of pineapples is 0, what is the corresponding price?"

Y-intercept =
Step 3) Plug in the slope and y-intercept you found into the Slope-Intercept formula.

Y = _____ X + ______

This is the Supply line.

Demand Line
Now let's look at the Buyer. From the data we were given, we see that the Demand line starts at the point (0,4) and ends at the point (8,0)

Step 4) Calculate the slope for the Demand Line Change in Price = Change in Number of Pineapples = ** Be careful with the subtractions here! **
Step 5) Find the Y-intercept for the Demand Line (ask yourself the question: "When the number of pineapples is 0, what is the corresponding price?"

Y-intercept =
Step 6) Plug in the slope and y-intercept you found into the Slope-Intercept formula.

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Y = _____ X + ______

This is the Demand line.

The Equilibrium Price (don't be afraid of fractions!)
We'll find this by setting the Supply equation equal to the Demand equation.

mD XD + bD

=

mS XS + bS

Solve for X (We'll call the X value you found X*) : Step 1) Subtract all the b's to one side and the X's to the other side. Your set up should look like this:

-0.5 X* + 4 = 0.6 X* + 1.5

Step 2) Once you have all the Xs on one side and the numbers without X on the other side, divide like this:

#1 = #2 X* and then #1 = X* = #* #2
Note: Leave your answer in terms of fractions! Be careful with division and addition of fractions. Step 3) Find the Y value by plugging in X* into either of the two equations, Demand or Supply. We'll call this Y value Y* If you choose the Demand line, for example, your set up should look like this:

Y* = -0.5 X* + 4

Y* = -0.5 #* + 4

Note: Use greatest common denominators to turn 4 into an appropriate fraction, so that you can add two fractions with the same denominators. Ask for help if you get lost! Check the equilibrium result you got by looking at the lines you plotted at the beginning. Are the coordinates of the intersection the same as X*, Y*? This point, (X*, Y*), should be the same as the one you found by solving the equations.

So what does this mean?
(X*, Y*) represents the equilibrium quantity and price of pineapples sold. At the price X*, the suppliers and buyers agree to exchange Y* pineapples.

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This same idea applies to the stock market, real estate market, and just about any other form of economic exchange. This is an important concept in economics and it's an algebra problem at heart!

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