Understanding Inverse Variation in the Given Table

This question requires us to determine the missing values (A, B, and C) in a table that demonstrates an inverse variation. An inverse variation occurs when the product of two quantities remains constant. If we denote the two quantities as M and N, their relationship can be written as:

\( M \times N = k \)

Here, k represents the constant of variation.

Analyzing the Data Table for Inverse Variation

The problem presents the following data, which we can format to better understand the relationships:

We need to find the values of A, B, and C using the principle of inverse variation.

Step 1: Calculate the Constant of Variation (k)

To find the constant of variation, k, we use a pair of values where both M and N are provided. From the first column of the table, we have M = 15 and N = -4.

Using the formula for inverse variation:

\( k = M \times N \)

Substitute the known values:

\( k = 15 \times (-4) \)

\( k = -60 \)

Thus, the constant of variation for this table is -60.

Step 2: Determine the Value of A

We use the constant k = -60 to find A. According to the table, when M = -6, N = A.

Applying the inverse variation formula:

\( M \times N = k \)

\( (-6) \times A = -60 \)

To solve for A, we divide both sides by -6:

\( A = \frac{-60}{-6} \)

\( A = 10 \)

Step 3: Determine the Value of B

Similarly, we use k = -60 to find B. The table shows that when M = 2, N = B.

Using the formula:

\( M \times N = k \)

\( 2 \times B = -60 \)

To solve for B, we divide both sides by 2:

\( B = \frac{-60}{2} \)

\( B = -30 \)

Step 4: Determine the Value of C

Finally, we use k = -60 to find C. The table indicates that when M = C, N = 60.

Using the formula:

\( M \times N = k \)

\( C \times 60 = -60 \)

To solve for C, we divide both sides by 60:

\( C = \frac{-60}{60} \)

\( C = -1 \)

Summary of Calculated Values

We have found the following values for A, B, and C:

• A = 10
• B = -30
• C = -1

Therefore, the values of A, B, and C respectively are 10, -30, and -1.

Matching with Provided Options

Let's compare our results with the given options:

• Option 1: 10, -30, -1
• Option 2: 10, -1, 30
• Option 3: -30, 10, -1
• Option 4: -1, -30, 10

Our calculated values (A=10, B=-30, C=-1) match Option 1.