Question

The table shows the relationship between the number of lemons purchased and their cost.



Number of Lemons (x) 2 5 9 12
Cost (y) $0.70 $1.75 $3.15 $4.20

Question
Which statement is true for this situation?

Answer options with 4 options
A.
It is a direct variation because values of y increase as the values of x increase.

B.
It is a direct variation because the relationship can be represented by y equals a x plus b, for a is greater than 0 and b equals 0.

C.
It is not a direct variation because the difference in the y values in the table vary.

D.
It is not a direct variation because the relationship can be represented by y equals a x plus b, for a is greater than 0 and b does not equal 0.

Answer 1

To determine whether the relationship between the number of lemons purchased (x) and their cost (y) represents a direct variation, we need to check if it can be expressed in the form \(y = kx\) (where \(k\) is a constant) or whether it can be expressed as \(y = ax + b\) with \(b = 0\).

From the provided data:

- For 2 lemons: \( y = 0.70\)
- For 5 lemons: \( y = 1.75\)
- For 9 lemons: \( y = 3.15\)
- For 12 lemons: \( y = 4.20\)

To find the cost per lemon (the constant of variation \(k\)), we can divide the cost by the number of lemons for each entry:

1. For 2 lemons: \( k = \frac{0.70}{2} = 0.35\)
2. For 5 lemons: \( k = \frac{1.75}{5} = 0.35\)
3. For 9 lemons: \( k = \frac{3.15}{9} = 0.35\)
4. For 12 lemons: \( k = \frac{4.20}{12} = 0.35\)

Since the value of \(k\) is constant (0.35), we can conclude that the relationship can be represented as \(y = 0.35x\), which implies that it is a direct variation (specifically, \(b = 0\)).

Now, we analyze the answer options:

A. **It is a direct variation because values of y increase as the values of x increase.**
(This is true, but this statement does not adequately capture the mathematical criteria for direct variation.)

B. **It is a direct variation because the relationship can be represented by y equals a x plus b, for a is greater than 0 and b equals 0.**
(This is the correct definition of direct variation.)

C. **It is not a direct variation because the difference in the y values in the table vary.**
(This is false; the values of y vary but in a linear fashion consistent with direct variation.)

D. **It is not a direct variation because the relationship can be represented by y equals a x plus b, for a is greater than 0 and b does not equal 0.**
(This is false; as established, \(b = 0\) in this scenario.)

The correct answer is **B**.