The Chess Club, in order to fund its overseas tournaments, holds a bake sale each year. \u00a0The items at the sale cost $1.50 each. \u00a0To rent the hall, the club pays $240. \u00a0Find the break-even point.<\/p>\n Solution:<\/strong><\/p>\n Let P=profit Then P=1.50n – 240<\/p>\n Notice, of course, that profit doesn’t mean income; rather, it means income less expenses. \u00a0The only expense we are considering is the hall rental; the baked goods, we can assume, are donated.<\/p>\n At the break-even point, P=0.<\/p>\n Let’s look at a graph that models the situation. \u00a0To make some points to plot, we choose a few values of n, then plug each into our equation P=1.50n-240 to get the corresponding values for P. \u00a0For instance, let’s find P when n=100:<\/p>\n P=1.50(100) – 240<\/p>\n P=150-240<\/p>\n P=-90<\/p>\n Therefore, when n=100, P=-90. \u00a0By repeating that process for various values of n we arrive at the following table:<\/p>\n Each (n,P) from the table means a point on the graph. For instance, when n=100, P=-90; therefore, (100,-90) will be on our line. We plot the points from our table as follows:<\/p>\n You can see the break-even point: it’s where our line cuts the horizontal axis. Note that, in this case, the x axis has been renamed the “n” axis; similarly, the y axis is called the “P” axis. In word problems, textbooks often rename the variables, calling them letters that stand for elements in the specific problem.<\/p>\n In our case, of course, the break-even point is 160; when 160 baked goods are sold, the expenses are paid. Any additional sales are profit.<\/p>\n You can also find the break-even point by setting P to 0:<\/p>\n 0=1.50n – 240<\/p>\n Add 240 to both sides:<\/p>\n 240=1.50n<\/p>\n Now, divide both sides by 1.50:<\/p>\n 160=n<\/p>\n If desired, we could add (160,0) to our table of values.<\/p>\n Finding the break-even point can be a common question in math 10 and math 11. Its attractiveness is its real-world meaning:)<\/p>\n
\nLet n=number of baked goods sold<\/p>\n\n
\n n (number of baked goods sold) <\/td>\n P (Profit)<\/td>\n<\/tr>\n \n 0<\/td>\n -240<\/td>\n<\/tr>\n \n 100<\/td>\n -90<\/td>\n<\/tr>\n \n 200<\/td>\n 60<\/td>\n<\/tr>\n<\/table>\n ![](\"\/..\/breakevn.png\"\/)
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