Mathematical Literacy

Mathematical literacy is an individual's capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments, and to use and engage with mathematics in ways that meet the needs of that individual's life as a constructive, concerned, and reflective citizen".

Individuals encounter countless situations in day-to-day life that require a strong mathematical foundation in order to make informed decisions.

One real world situation

A teacher is looking for the best option in purchasing school supplies for a classroom. Company A is offering a discount for every dollar amount spent; Company B is offering a higher discount than Company A, but the discount is applicable only for every dollar amount spent above $20. Determine which company will offer a better price based upon the dollar amount the teacher spends on school supplies.

Two companies are there. Company A and Company B.

Company A is offering a discount for every dollar amount spent.

Company B is offering a higher discount than company A, but the discount is applicable only for every amount spent above $20.

We have to decide which company offer a better price.

The better price depends on the dollar amount spent by the teacher.

Let denote the dollar amount spent by the teacher.

Let d1% be the discount allowed by company A. Now d1% means d1/100 of . The actual dollar amount after discount is

Let d2% be the discount allowed by company B, which is applicable only to the amount spent in excess of $20. Now d2% means d2/100 of . Since the discount rate is more in company A than in company B, we have d2 > d1.The actual dollar amount after discount is if . If , the actual dollar amount spent is $20. So we can write

If the dollar amount spent by the teacher is , then clearly company A offers the better price as it offers a discount of d1% and company B offers no discount.

Now let us examine the case . That is the dollar amount spent by the teacher is more than $20.

Let denote the actual amount charged by company A for a purchase of and let be the actual amount charged by company B for a purchase of .

Then we have

When the two cost options are equivalent

To solve these equations we need numerical values for the discount rates and .

Suppose company A offers a discount of 10% for every dollar amount spent and company B offers a discount of 20% for every dollar amount spent in excess of $20.

Then and . The above three equations can be written as

That is

Substituting the expression for from equation (1) and the expression for in equation (2) into equation (3), we obtain

Adding on both sides

Multiplying both sides by 10

Substituting the value of in equations (1) and (2), we get the values of as

So when the teacher purchases for $40, the actual amount charged by both companies are equal to $36.

When she purchases for less than $40, company A offers a better price and when she purchases for more than $40, company B offers the better price.

Discussion (using graph)

Company A offers a discount of 10% for every dollar amount purchased. So the customer has to pay only 90% of the price. In the above graph the actual amount paid by the customer for different purchases are shown by the curve in green colour.

Company B offers no discount for purchases up to $20. But for purchases above $20, a discount of 20% for the amount exceeding $20 is given. In the above graph the actual amount spent by the customer for different purchases are shown by the curve in red colour.

For , the cost curve of company A (green colour) is below the cost curve of company B (red colour) and so company A offers the better price for purchases below $40.

For , the cost curve of company B (red colour) is below the cost curve of company A (green colour) and so company B offers the better price for purchases above $40.

At , both curves intersect. So at this point the offers of both the companies are equivalent.

So the best decision for the customer depends on the amount she is going to spend. If it is less than $40 she prefers company A. otherwise she prefers company B.

Discussion (using algebra)

The cost function of company A is given by

The cost function of company B is given by

When , we have . So company A offers better price for purchases less than $40.

When , we have . So company B offers better price for purchases above $40.

When , . At this point of purchase, the offers of the two companies are equivalent.

If every thing other than cost remains the same, company A is to be selected for small purchases and company b for large purchases.