We want to determine which of the relationships between the variables in the given tables is an inverse variation.
Table A
Let's consider the first table.
x
2
5
10
20
25
50
y
50
20
10
5
4
2
We see that as x increases, y decreases. This might be an inverse variation relationship. To verify that claim, recall that inverse variations can be expressed as an equation in three different ways.
xy= k, y=k/x, or x=k/yIn the above formulas, k is the constant of variation and k cannot be 0. Let's test to see if the product of x and y is constant for the given values.
x
2
5
10
20
25
50
y
50
20
10
5
4
2
xy
2* 50= 100
5* 20= 100
10 * 10= 100
20* 5= 100
25* 4= 100
50 * 2= 100
The product of each pair is 100. This means that xy= 100, and y varies inversely with x.
Table B
Let's take a look at the second table.
x
2
4
6
8
10
12
y
- 4
- 8
- 12
- 16
- 20
- 24
Once again, as x increases, y decreases. Let's see if the product of x and y is constant.
x
2
4
6
8
10
12
y
- 4
- 8
- 12
- 16
- 20
- 24
xy
2(- 4)=- 8
4(- 8)=32
6(- 12)=- 72
8(- 16)=- 128
10(- 20)=- 200
12(- 24)=- 288
The product of each pair is not constant. Therefore, the relationship between the variables is not an inverse variation.
Table C
x
1
2
3
4
5
6
y
5
10
15
20
25
30
We see that as x increases, so does y. Therefore, the relationship between the variables cannot be an inverse variation.
Table D
We can now take a look at the last table.
x
10
9
8
7
6
5
y
5
6
7
8
9
10
We see that as x decreases, y increases, which means that this could be an inverse variation. Let's see if the product of x and y is constant.
x
10
9
8
7
6
5
y
5
6
7
8
9
10
xy
10 * 5 = 50
9 * 6 = 54
8 * 7 = 56
7 * 8 = 56
6 * 9 = 54
5 * 10 = 50
The product of each pair is not constant. Therefore, the relationship between the variables is not an inverse variation. The correct answer is A.
