# SAT II Math II : Other Mathematical Relationships

## Example Questions

### Example Question #1 : Other Mathematical Relationships

Explanation:

In modulo 9 arithmetic, a number is congruent to the remainder of its division by 9.

Since

and

,

,

making "5" the correct response.

### Example Question #2 : Other Mathematical Relationships

varies directly as  and inversely as

If  and , then .

To the nearest whole number, evaluate  if  and .

Insufficient information is given to answer the question.

Explanation:

varies directly as  and inversely as . This means that for some constant of variation ,

Alternatively,

We can substitute the initial conditions for thevariables on the left side and the final conditions for those on the right side, then solve for :

### Example Question #151 : Sat Subject Test In Math Ii

varies directly as both  and the square of .

If  and , then

Evaluate  if  and .

Explanation:

varies directly as both  and the square of . This means that for some constant of variation ,

.

Alternatively stated,

.

We can substitute the initial conditions for the variables on the left side and the final conditions for those on the right side, then solve for :

### Example Question #152 : Sat Subject Test In Math Ii

Evaluate .

The system has no solution.

Explanation:

Rewrite the two equations by setting  and  and substituting:

The result is a two-by-two linear system in terms of  and :

This can be solved, among other ways, using Gaussian elimination on an augmented matrix:

Perform row operations until the left two columns show identity matrix . One possible sequence:

and . In the former equation, substitute  back for , and raise both sides to the power of 4: