GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #1223 : Data Sufficiency Questions

 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:

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:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

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 #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: Neither denominator is a multiple of 5.

Statement 2: Both denominators are powers of 2.

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.

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

BOTH statements TOGETHER are insufficient 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.

Explanation:

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 #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: The denominator of  is a power of 7.

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

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 insufficient to answer the question. 

EITHER 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.

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

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

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 #243 : Arithmetic

 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. 

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

EITHER 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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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 #2 : 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:

EITHER statement ALONE is sufficient to answer the question.

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

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

Correct answer:

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

Explanation:

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 #245 : Arithmetic

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. 

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

Either statement is sufficient to answer the question. 

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

Both statements are needed to answer the question.

Correct answer:

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

Explanation:

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.

Example Question #1 : Dsq: Understanding Real Numbers

Is z a negative number?

(1) \dpi{100} \small 13z>14z

(2) \dpi{100} \small 4+z is positive

Possible Answers:

Statements (1) and (2) TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Correct answer:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Explanation:

 (1) Subtracting \dpi{100} \small 13z from both sides of \dpi{100} \small 13z>14z shows that  \dpi{100} \small 0>z, so  \dpi{100} \small z must be negative \dpi{100} \small \rightarrow SUFFICIENT

(2) Subtracting 4 from both sides of \dpi{100} \small 4+z>0 gives \dpi{100} \small z>-4. Therefore, for this statement, \dpi{100} \small z can be negative  but is not necessarily negative. \dpi{100} \small \rightarrow NOT SUFFICIENT.

Example Question #2 : Dsq: Understanding Real Numbers

If , , and  are points on the number line, what is the distance between  and ?

(1) The distance between  and  is 7.

(2) The distance between  and  is 12.

 

Possible Answers:

E. Statements (1) and (2) TOGETHER are NOT sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Correct answer:

E. Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation:

Looking at statements (1) and (2) separately, there is no information about the distance between  and . Looking at statements (1) and (2) together, there are two possibilities: 1)  and  are on different sides of , and the distance between them is 19; 2)  and  are on the same side of , and the distance between them is 5. Therefore, we cannot get the only possible answer to this question based on the two statements.

Example Question #3 : Dsq: Understanding Real Numbers

Data sufficiency question- do not actually solve the question

How many children are attending a party?

1. If 6 more children attend the party, there will be 24 children there.

2. If 10 children leave the party, there will be less than 12 children in attendance.

Possible Answers:

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

Each statement alone is sufficient

Both statements taken together are sufficient to ansewr the question, but neither statement alone is sufficient

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Statements 1 and 2 together are not sufficient, and additional information is needed to answer the question

Correct answer:

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

Explanation:

Statement 1 precisely tells how many children are attending the party while Statement 2 only provides a range.

Example Question #4 : Dsq: Understanding Real Numbers

A sandwich shop offers a discount in which customers pay $2.99 for each additional sandwich after purchasing their first sandwich. Emily bought her first sandwich for $4.95. If she purchased 5 additional sandwiches to take home, then the total amount she paid is equivalent to which of the following?

Possible Answers:

Correct answer:

Explanation:

The price of 6 sandwiches can be expressed as . Regrouping is needed since this is not in the answer choices. If $3.00 is used instead of $2.99, each of the five additional sandwiches is .01 too high and the total is .05 too high. Therefore, this amount needs to be subtracted from the $4.95.

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