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## Example Questions

### Example Question #1 : Dsq: Understanding Decimals

is a fraction in lowest terms. Is its decimal representation a terminating decimal or a repeating decimal?

Statement 1: is a power of 5.

Statement 2: is a multiple of 7.

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

**Correct answer:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

A fraction in lowest terms can be expressed as a terminating decimal if and only if its denominator has no prime factors other than 2 or 5; the numerator is irrelevant. Statement 1 is unhelpful since it only gives information about the numerator. If Statement 2 is assumed - that is, if we know the denominator has 7 as a prime factor - we know that the decimal representation of is repeating.

### Example Question #2 : Dsq: Understanding Decimals

and are fractions in lowest terms.

True or false: The result can be represented by an integer or a terminating decimal.

Statement 1: Both fractions have powers of 5 in their denominators.

Statement 2: Both fractions have powers of 3 in their numerators.

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Assume Statement 1 alone. A necessary and sufficient condition for a lowest-terms fraction to be equivalent to a terminating decimal is that its only prime factors are 2 and 5. Statement 1 says this about both fractions; the sum of terminating decimals is itself an integer or a terminating decimal.

Statement 2 alone provides insuffiicient information. Examine these examples:

Both fit the conditions of the main premise and Statement 2, but in only one case is the sum equal to a terminating decimal.

### Example Question #3 : Dsq: Understanding Decimals

and are fractions in lowest terms.

True or false: The result can be represented by an integer or a terminating decimal.

Statement 1: Both denominators are even.

Statement 2: Both denominators are multiples of 3.

**Possible Answers:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements to be true. A multiple of both 2 and 3 is also a multiple of 6, so both denominators are multiples of 6. Examine these two examples:

In both examples, the conditions of the problem, including both statements, are met, but in only one is the sum equivalent to a terminating decimal or integer.

### Example Question #3 : Dsq: Understanding Decimals

and are fractions in lowest terms.

True or false: The result can be represented by an integer or a terminating decimal.

Statement 1: Neither denominator is a multiple of 5.

Statement 2: Both denominators are powers of 2.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone. Examine these two examples:

Both fit the main condition and that of Statement 1, but in only one case is the sum equal to a terminating decimal or integer.

Assume Statement 2 alone. A necessary and sufficient condition for a lowest-terms fraction to be equivalent to a terminating decimal is that its only prime factors are 2 and 5. By Statement 2, both denominators have 2 as their only prime factor, so both can be represented by terminating decimals; their product must be terminating as well.

### Example Question #4 : Dsq: Understanding Decimals

and are fractions in lowest terms.

True or false: The result can be represented by an integer or a terminating decimal.

Statement 1: The denominator of is a power of 7.

Statement 2: The denominator of is a power of 3.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements. Examine these two examples; both fit the main condition and those given in both statements.

In only one case is the product equivalent to an integer or a terminating decimal.

### Example Question #6 : Dsq: Understanding Decimals

and are fractions in lowest terms.

True or false: The result can be represented by an integer or a terminating decimal.

Statement 1: The denominator of is three times the numerator of .

Statement 2: The denominator of is six times the numerator of .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. Rewrite the fractions as follows:

, where - that is, their lowest-terms representations.

By Statement 1, , so . Therefore,

Since is in lowest terms, is not a multiple of 3, so the 3 must remain in the denominator, even if the and the can be reduced. Since a fraction which, in lowest terms, has a denominator with any prime factor other than 2 or 5 cannot be represented by an integer or a terminating decimal, Statement 1 proves that has a repeating decimal as its equivalent.

A similar argument can be used to demonstrate that Statement 2 proves the same of .

### Example Question #5 : Dsq: Understanding Decimals

and are fractions in lowest terms.

True or false: The result can be represented by an integer or a terminating decimal.

Statement 1: The denominator of is five times the numerator of .

Statement 2: The denominator of is eight times the numerator of .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone. Examine these examples:

In both cases, the main conditon and that of Statement 1 are satisfied, but in only one case can the result be represented by a terminating decimal or integer.

A similar argument shows Statement 2 alone to be insufficient:

Assume both statements. Then rewrite the fractions as follows:

, where - that is, their lowest-terms representations.

By Statement 1, ; by Statement 2, . Therefore,

,

and their product is

,

a terminating decimal.

### Example Question #6 : Dsq: Understanding Decimals

Solve for in the following equation.

I) is equivalent to seventy nine hundredths.

II) The slope of the line parallel to this equation is .

**Possible Answers:**

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

**Correct answer:**

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

We have one equation and two unknowns. To find x, we need to find y.

I) Tells us that y is .79 therefore we can plug this value into our equation and solve for x.

Plugging in the y value from Statement I and using algebraic operations to solve for x.

Multiply first term.

Subtract 4.74 from both sides.

Divide by -13.

II) Is irrelevant.

Therefore Statement I) alone is sufficient to answer the question but Statement II) alone is not.

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