Intercepts and Slopes

Question

For your birthday you were given a Transat Air gift card for $100. You can purchase any combination of regular tickets at $10 each and VIP tickets at $15 each.

A) If you buy 4 regular tickets, how many VIP tickets can you buy? 
B) If you buy 6 VIP tickets how many regular tickets can you buy? 
C) Let a variable represent each kind of ticket. Plot the above information on a set of axes. Draw the line and extend it so it touches both axes. 
D) List all the possible combinations of tickets that are represented on this line. 
E) Is it possible to use your gift card to buy a combination of tickets that is not on the line? Explain. 
F) Does every point on the line represent a combination of tickets? Explain. 
G) From your graph, what is the value of each intercept? What does each of these values represent?

Answer 1

Hi Asma

 

Let name by R the number of regular tickets and V the number of VIP tickets. The unit price for each R and V tickets are $10 and $15, respectively. Also, the maximum available money for buying R and V tickets is $100.

 

Then statement (A) can be written in math language as:  $100 - R*$10 = $100 - 4*($10) = $60 You have $60 for buying some V (VIP) tickets. How many?    Because we know that each V ticket costs $15 we have  ($60)/($15) = 4

Then we can buy only 4 V tickets.

 

Now the statement (B) can be written as: $100 - V*$15 = $100 - 6*($15) = $10  You have only $10 for buying R tickets. Due to each R ticket costs $10, you can only buy one (1) R ticket.

 

The statement (C) requires you consider the process of buying R and V tickets as following a linear trend. If you represent the X axis as Regular tickets (R) and Y axis as VIP tickets (V). We have the following line:

 

| V     

|

|'

|  '

|    '

|      '

|        '

|          '

|            '

|              '  

-------------------------------- R

 

 The line pass by the following points (4, 4)  and (1, 6)  Remember statements (A) and (B), if buy 4 R tickets you buy 4 V tickets (point (4,4). But, if you buy 1 R ticket you can buy 6 V tickets (point (1,6)).

 

The equation of this line is:  slope=  (6 - 4)/(1 - 4)  =  -2/3   then   V - 6 = (-2/3)*(R - 1)  or

                                                                                                 V = (-2/3)R +20/3

 

The statement (D) can be answered by varying the number R of tickets from 1 to 10 and using the line above, as follows:

 

R = 1   V =  6   You have $0 in your pocket

R = 2   V =  5   You have $5 in your pocket

R = 3   V =  4   You have $10  in your pocket (*)

R = 4   V =  4   You have $0 in your pocket

R = 5   V =  3   You have $5 in your pocket

R = 6   V =  2   You have $10 in your pocket (*)

R = 7   V =  2   You have $0 in your pocket

R = 8   V =  1   You have $5 in your pocket

R = 9   V =  0   You have $10 in your pocket (*)

R = 10  V = 0   You have $0 in your pocket

 

The response to statement (E) is yes. If you see the above list there are some cases indicated by (*) where you still have $10 in your pocket. It means that you are able to buy one extra R ticket in each case with (*). These cases (*) are not following the line.

 

From this line graph, if R =0 (intersect with V axis)  V = 20/3   or around V = 6 (it means that you can buy the maximum possible number of VIP tickets  (6*$15 = $90, you have only $5 in your pocket)

If V =0 (intersect with R axis)  R = 10  (you got the maximum number of Regular tickets)