207k views
0 votes
A school sports team contains 68 students. 33 do field events, 40 do track events, 23 do swimming, 14 do both field and track events. If 15 students do field events only and 10 do both swimming and track events, how many students do a) swimming only b)track events only c) all three events?

User Jjxtra
by
6.9k points

1 Answer

3 votes

Let's label the regions in the Venn diagram as follows:

F: students who do only field events

T: students who do only track events

S: students who do only swimming events

FT: students who do both field and track events

ST: students who do both swimming and track events

FST: students who do all three events

From the information given in the problem, we can fill in some of the values in the Venn diagram:

F + FT + 15 = 33 (33 students do field events, and 15 do field events only)

T + FT + 14 = 40 (40 students do track events, and 14 do both field and track events)

S + ST + 10 = 23 (23 students do swimming, and 10 do both swimming and track events)

F + T + S + 2FT + ST + FST = 68 (there are 68 students in total)

We can simplify these equations to:

F + FT = 18

T + FT = 26

S + ST = 13

F + T + S + 2FT + ST + FST = 68

To solve for the remaining unknowns, we need to use some algebra. We can start by solving for FT:

F + FT = 18

T + FT = 26

Adding the two equations, we get:

F + T + 2FT = 44

Rearranging, we get:

FT = (44 - F - T) / 2

Now we can substitute this expression for FT into the equation for the total number of students:

F + T + S + 2FT + ST + FST = 68

F + T + S + 2((44 - F - T) / 2) + ST + FST = 68

F + T + S + 44 - F - T + ST + FST = 68

Simplifying, we get:

S + ST + FST = 26

Now we have two equations involving S, ST, and FST:

S + ST = 13

S + ST + FST = 26

We can solve for S and ST by subtracting the first equation from the second:

S + ST + FST = 26

(S + ST) + FST = 26

Substituting S + ST = 13:

13 + FST = 26

FST = 13

So there are 13 students who do all three events. Now we can use the equation S + ST = 13 to solve for S:

S + ST = 13

ST = 10 (given in the problem)

S = 13 - 10 = 3

So there are 3 students who do swimming only. Similarly, we can use the equation T + FT = 26 to solve for T:

T + FT = 26

FT = (44 - F - T) / 2

Substituting F + FT = 18:

T + (18 - F) / 2 = 26

Multiplying both sides by 2:

2T + 18 - F = 52

User Jamadei
by
7.5k points