Mathematics Camp Algebra Question

Question

A mathematics camp is being held in Canberra for students from schools in the eastern states. New South Wales is to have 4 more students than the camp of Victoria, Queensland is to have half the number attending from New South Wales and Tasmania is to have one-third of the number from Victoria. The combined number of students from Queensland and Victoria is to be the same as the combined number from Tasmania and New South Wales. Find the number of students who will attend the camp from each state

Answer 1

Let's write a system of equations to model this questions.

 

 

VARIABLES

 

n = New South Wales students

v = Victoria students

t = Tasmania students

q = Queensland students

 

 

EQUATIONS

 

n = v + 4    (New South Wales has 4 more students than Victoria)

q = 1/2n    (Queensland has 1/2 the number from New South Wales)

t = 1/3v    (Tasmania has 1/3 the number from Victoria)

q + v = t + n    (Queensland and Victoria have the same number as Tasmania and New South Wales)

 

 

Now, let's make some substitutions. We'll rearrange the first three equations to try to find expressions for q, v, and t in terms of n.

 

n = v + 4

v = n - 4

 

q = 1/2n

 

t = 1/3v

t = 1/3(n - 4)

t = 1/3n - 4/3

 

 

Substitute those values into the fourth equation and solve for n:

 

q + v = t + n

1/2n + n - 4 = 1/3n - 4/3 + n

3/2n - 4 = 4/3n - 4/3

3/2n - 4/3n = -4/3 + 4

9/6n - 8/6n = -4/3 + 12/3

1/6n = 8/3

n = 6(8/3)

n = 16

 

The number of students from New South Wales is 16. Substitute that value back into the other equations to get:

 

v = n - 4

v = 16 - 4

v = 12

q = 1/2n

q = 1/2(16)

q = 8

t = 1/3(n - 4)
t = 1/3(16 - 4)

t = 1/3(12)

t = 4

 

There will be 12 students from Victoria, 8 students from Queensland, and 4 students from Tasmania.