ALGEBRA 2 VENN DIARAM

Question

SALMON FISHING IS A BIG INDUSTRY IN OREGON, WHEN 24 STUDENT S VISITED THE STATE THEY HAD A CHANCE BOUTH TO CATCH AND TO EAT SALMON. 9 STUDENTS CAUGHT SALMON. THE NUMBERS WHO ATE BUT DID NOT CATCH SALMON WAS 4 TIMES THE NUMBER WHO CAUTGHT BUT DID NOT EAT SALMON. THE NUMBER WHO NEITHER CAUGHT NOR ATE SALMON WAS HALF THE NUMBER WHO DID BOTH

Answer 1

After setting up a Venn diagram (see below), the problem gives us two concrete numbers; the total number of students on the field trip (24) and the total number who fished (9). As a first pass, the 9 students who fished included both those who ate and didn't eat salmon. So in equation form, here's what we know:

1. x + 4x + y + 0.5y = 24 →. 5x + 1.5y = 24
2. x + y = 9

Given the fact that we now have two equations and two unknowns, we can solve for each variable and then complete the Venn diagram.

I favor the substitution method:

x = 9 - y

5(9-y) + 1.5y = 24 → 45 - 3.5y = 24 → 21 = 3.5y → 6

x = 9 - 6 → x = 3

Now referring back to the Venn diagram, we simply need to plug in the values for each group of students:

1. Students who fished for salmon but didn't eat any: x = 3
2. Students who ate salmon but didn't fish: 4x = 12
3. Students who did both: y = 6
4. Students who neither fished nor ate salmon: 0.5y = 3
5. 3 + 12 + 6 + 3 = 24

Answer 2

Here's a two-way table (if I can reproduce it here!)

Caught Didn't Catch Total

Ate y 4x

Didn't Eat x 0.5y ____

x + y = 9 4x + 0.5y = 15 24

Solve system of equations:

x = 9 - y

4(9 - y) + 0.5y = 15 → 36 - 4y + 0.5y = 15 → -3.5y = -21

y = 6 caught and ate

x = 3 caught but didn't eat

12 ate but didn't catch

3 neither caught nor ate

Answer 3

TOTAL = 24

A = Caught but did not eat

B = Caught and Ate

C = Ate but did not Catch

D = neither caught nor ate

Caught = A + B = 9

Caught but did not eat = A

Ate but did not catch = C = 4A

Both caught and ate = B

Neither caught not ate = D = (1/2)B

A + B + C + D = 24

9 + 4A + (1/2)B = 24

18 + 8A + B = 48

18 + 9 + 7A = 48

7A = 21

A = 3

B = 9-A = 6

C = 4A = 12

D = (1/2)B = 3

CHECK::

A + B + C + D = 3 + 6 + 12 + 3 = 24