# GMAT Math : Solving equations

## Example Questions

### Example Question #61 : Equations

and  are the volume and the radius of the same sphere.

Which of the following is a true statement?

varies directly as the eighteenth power of .

varies directly as the twenty-seventh power of

varies directly as the ninth power of .

varies directly as .

varies directly as the cube of .

varies directly as the twenty-seventh power of

Explanation:

The volume and radius of a sphere are related as follows:

.

By substitution,

If , then

,

and  varies directly as the twenty-seventh power of .

### Example Question #62 : Equations

and  are the surface area and the radius of the same sphere.

.

Which of the following is a true statement?

Explanation:

The surface area and radius of a sphere are related as follows:

Substituting:

If , then

,

and  varies directly as the fourth root of .

### Example Question #63 : Equations

Solve the following equation for .

Explanation:

To solve, simply isolate .

### Example Question #64 : Equations

Express  in terms of . You may assume all variables assume positive values.

Explanation:

, and , so, by two substitutions,

, so

### Example Question #65 : Equations

Express  in terms of .

Explanation:

, so

, so

Now substitute twice, and solve for :

### Example Question #66 : Equations

Express  in terms of .

Explanation:

Substituting twice:

### Example Question #67 : Equations

Express  in terms of .

Explanation:

Substitute twice:

### Example Question #68 : Equations

Which of the following is equal to  ?

Explanation:

and , so

Since , it follows that

### Example Question #69 : Equations

True or false: .

Statement 1:

Statement 2:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.x

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Substitute 5 for  in both statements, and it becomes immediately apparent that it is a solution of neither statement:

, a false statement.

, a false statement since the left quantity, having a zero denominator, is undefined.

Therefore, it follows from either statement alone that  is false.

### Example Question #70 : Equations

is a rational number. True or false:

Statement 1:

Statement 2:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone and solve for :

Either:

, in which case ,

or

, in which case .

Therefore, Statement 1 alone is inconclusive.

Assume Statement 2 alone and solve for :

The only solution to this equation is , so Statement 2 alone answers the question.

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