We are asked to determine which of the given relationships represents a direct variation. Let's check each relationship to determine whether the ratio of y to x is constant.
Option A
In option A, we are given the following table.
Hours, x
1
2
3
4
Wages ($ ), y
10
22
36
50
We can calculate the ratio for each value of x by dividing the value of y by the value of x.
Hours, x
1
2
3
4
Wages ($ ), y
10
22
36
50
Ratio
10/1=10
22/2=11
36/3=12
50/4=25/2
A relationship is a direct variation when the ratio of y to x is constant. Since the ratios that we got are not the same, the relationship is not a direct variation.
Option B
Now let's take a look at the table from option B.
Hours, x
1
2
3
4
Wages ($ ), y
10
25
30
50
Like we did before, let's calculate each ratio by dividing the value of y by the value of x.
Hours, x
1
2
3
4
Wages ($ ), y
10
25
30
50
Ratio
10/1=10
25/2
30/3=10
50/4=25/2
In this case, the ratios are not the same, so the relationship is not a direct variation.
Option C
In option C, we are given the following table.
Hours, x
1
2
3
4
Wages ($ ), y
10
20
30
40
Let's find the ratios!
Hours, x
1
2
3
4
Wages ($ ), y
10
20
30
40
Ratio
10/1=10
20/2=10
30/3=10
40/4=10
As we can see, all of the ratios are equal to $10 per hour. This means that the relationship is a direct variation! Let's continue and check option D, just in case.
Option D
Finally, we will consider the table from option D.
Hours, x
1
2
3
4
Wages ($ ), y
6
16
30
48
Let's calculate the ratio for each value of x.
Hours, x
1
2
3
4
Wages ($ ), y
6
16
30
48
Ratio
6/1=6
16/2=8
30/3=10
48/4=12
In this case, the ratios are not the same, which means that the relationship is not a direct variation.
Summary
We found that only the relationship from option C is a direct variation. This means that the correct option is C.
