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 𝑥 denote the dollar amount spent by the teacher.
Let d1% be the discount allowed by company A. Now d1% means d1/100 of 𝑥. The actual dollar
𝑑1
𝑑1
amount after discount is 𝑥 − 𝑑1% 𝑜𝑓 𝑥 = 𝑥 − 100 𝑥 = (1 − 100) 𝑥
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 (𝑥 − 20). Since the discount rate is more in company A
than in company B, we have d2 > d1.The actual dollar amount after discount is 𝑥 − 𝑑2% 𝑜𝑓 (𝑥 −
𝑑2
𝑑2
𝑑2
20) = 𝑥 − 100 (𝑥 − 20) = (1 − 100) 𝑥 + 20 100 if 𝑥 > 20. If 𝑥 ≤ 20, the actual dollar amount spent
is $20. So we can write
𝑇ℎ𝑒 𝑎𝑐𝑡𝑢𝑎𝑙 𝑑𝑜𝑙𝑙𝑎𝑟 𝑎𝑚𝑜𝑢𝑛𝑡 𝑠𝑝𝑒𝑛𝑡 𝑎𝑓𝑡𝑒𝑟 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑖𝑛 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝐵
$20 𝑖𝑓 𝑥 ≤ 20
𝑑2
𝑑2 𝑖𝑓 𝑥 > 20
={
$ ((1 −
) 𝑥 + 20
)
100
100
If the dollar amount spent by the teacher is ≤ 20, 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 𝑥 > 20. That is the dollar amount spent by the teacher is more than
$20.
Let 𝑦 denote the actual amount charged by company A for a purchase of $𝑥 and let 𝑧 be the actual
amount charged by company B for a purchase of $𝑥.
Then we have
𝑦 = (1 −
𝑑1
) 𝑥 … … … … … … … 𝑒𝑞(1)
100
𝑧 = (1 −
𝑑2
𝑑2
) 𝑥 + 20
… … . . 𝑒𝑞(2)
100
100
When the two cost options are equivalent
𝑦 = 𝑧 … … … … … … … … … … … . (3)
To solve these equations we need numerical values for the discount rates 𝑑1 and 𝑑2.
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 𝑑1 = 10 and 𝑑2 = 20. The above three equations can be written as
𝑦 = (1 −
𝑧 = (1 −
10
) 𝑥 = (1 − 0.10)𝑥 = 0.90 𝑥 … … . . 𝑒𝑞(1)
100
20
20
) 𝑥 + 20 (
) = (1 − 0.20)𝑥 + 4 = 0.80𝑥 + 4 … 𝑒𝑞(2)
100
100
𝑦 = 𝑧 … … … … … … 𝑒𝑞(3)
That is
𝑦 = 0.90 𝑥 … . . 𝑒𝑞(1)
𝑧 = 0.80 𝑥 + 4 … … . 𝑒𝑞(2)
𝑦 = 𝑧 … … … . . 𝑒𝑞(3)
Substituting the expression for 𝑦 from equation (1) and the expression for 𝑧 in equation (2) into
equation (3), we obtain
0.90 𝑥 = 0.80 𝑥 + 4
Adding −0.80 𝑥 on both sides
0.90 𝑥 − 0.80 𝑥 = 0.80 𝑥 + 4 − 0.80 𝑥
0.10 𝑥 = 4
Multiplying both sides by 10
𝑥 = 40
Substituting the value of 𝑥 in equations (1) and (2), we get the values of 𝑦 𝑎𝑛𝑑 𝑧 as
𝑦 = 0.90 𝑥 = 0.90 × 40 = 36
𝑧 = 0.80 𝑥 + 4 = 0.80 × 40 + 4 = 32 + 4 = 36
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.
60
Company A
50
(40, 36)
Y= 0.90 x
40
10% discount for
every dollar
30
Company B
z= x if x <= 20
z = 0.80 x + 4 if x > 20
no discount for
purchases up to $20
and 20% dicount for
purchases in excess of
$20
20
10
y
z
0
0
10
20
30
40
50
60
70
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 𝑥 < 40, 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 𝑥 > 40, 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 𝑥 = 40, 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
𝑦 = 0.90 𝑥
The cost function of company B is given by
𝑧={
𝑥, 𝑖𝑓 𝑥 ≤ 20
0.80 𝑥 + 4 , 𝑖𝑓 𝑥 > 20
When 𝑥 < 40, we have 𝑦 < 𝑧. So company A offers better price for purchases less than $40.
When 𝑥 > 40, we have 𝑧 < 𝑦. So company B offers better price for purchases above $40.
When 𝑥 = 40, 𝑦 = 𝑧. 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.