Arithmetic

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GMAT Quantitative › Arithmetic

Questions 1 - 10

When assigning a score for the term, a professor takes the mean of all of a student's test scores.

Joe is trying for a score of 90 for the term. He has one test left to take. What is the minimum that Joe can score and achieve his goal?

Statement 1: He has a median score of 85 so far.

Statement 2: He has a mean score of 87 so far.

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.

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation

Knowing the median score is neither necessary nor helpful. What will be needed is the sum of the scores so far and the number of tests Joe has taken. But the number of tests taken is not given, and without this, there is no way to find the sum either.

On Monday, 40 people are asked to rate the quality of product A on a seven point scale (1=very poor, 2=poor.....6=very good, 7=excellent).

On Tuesday, a different group of 40 is asked to rate the quality of product B using the same seven point scale.

The results for product A:

7 votes for category 1 (very poor);

8 votes for category 2;

10 votes for 3;

6 vote for 4;

4 votes for 5;

3 votes for 6;

2 votes for 7;

The results for product B:

2 votes for category 1 (very poor);

3 votes for category 2;

4 votes for 3;

6 vote for 4;

10 votes for 5;

8 votes for 6;

7 votes for 7;

It appears that B is the superior product.

Which one of the following statements is true?

The median score for product A is less than the mean score for product A.

The median score of product A is greater than the median score of product B.

The mean score of product A is greater than the mean score of product B.

The median score of product A is greater than the mean score of product A.

The median score of product B is less than the mean score of product B.

Explanation

Median of A = 3

Mean of A = 3.2

Median of B = 5

Mean of B = 4.8

Some balls are placed in a large box; the balls include one ball marked "10", two balls marked "9", and so forth up to ten balls marked "1". A ball is drawn at random.

is an integer between 1 and 10 inclusive. True or false: the probability that the ball will have the number marked on it is greater than $\frac{1}{10}$ .

Statement 1: is a perfect square integer.

Statement 2: $N \ge 5$

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

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.

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

Explanation

The total number of balls in the box will be

Since

$1+ 2 + 3 + ...+ N = \frac{N (N + 1)}{2}$ ,

it follows that the number of balls is

$1+ 2 + 3 + ...+ 10= \frac{10 (10 + 1)}{2} = \frac{10 \cdot 11}{2} = 55$ .

The frequencies out of 55 of each outcome from 1 to 10, in order, is as follows:

$\left { 10, 9, 8, 7, 6, 5, 4, 3, 2, 1\right }$

Their respective probabilities are their frequencies divided by 55:

$\left { \frac{10}{55}, \frac{9}{55}, \frac{8}{55}, \frac{7}{55}, \frac{6}{55}, \frac{5}{55}, \frac{4}{55}, \frac{3}{55}, \frac{2}{55}, \frac{1}{55} \right }$ .

The probability that the ball will be marked "5" is

$\frac{6}{55} \approx 0.109 > 0.1 = \frac{1}{10}$ ;

therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than $\frac{1}{10}$ .

The probability that the ball will be marked "6" is

$\frac{5}{55} \approx 0.091 < 0.1 = \frac{1}{10}$ ;

therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than $\frac{1}{10}$ .

Therefore, it suffices to know whether the number on the ball is less than or equal to 5.

Statement 1 alone is insufficient, since there are two perfect square integers from 1 to 5 (1 and 4) and one perfect square integer from 6 to 10 (9). Statement 2 alone is insufficient, since it is not clear whether the number on the ball is 5 or a number greater than 5. However, from the two statements together, it can be inferred that , and that the probability of drawing a ball with this number is $\frac{2}{55} < \frac{1}{10}$ .

What is the mean of , , , , , and ?

Statement 1:

Statement 2:

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

Explanation

The mean of a data set requires you to know the sum of the elements and the number of elements; you know the latter, but neither statement alone provides any clues to the former.

However, if you know both, you can add both sides of the equations as follows:

$\underline{2a + 4b+6c + 5d + 7e + 3f = 8,000}$

Rewrite as:

and divide by 9:

Now you know the sum, so divide it by 6 to get the mean:

$2,000 \div 6 =333 \frac{1}{3}$ .

What is the mean of , , , , , and ?

Statement 1:

Statement 2:

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

Explanation

The mean of a data set requires you to know the sum of the elements and the number of elements; you know the latter, but neither statement alone provides any clues to the former.

However, if you know both, you can add both sides of the equations as follows:

$\underline{2a + 4b+6c + 5d + 7e + 3f = 8,000}$

Rewrite as:

and divide by 9:

Now you know the sum, so divide it by 6 to get the mean:

$2,000 \div 6 =333 \frac{1}{3}$ .

1. If the arithmetic mean of five different numbers is 50, how many of the numbers are greater than 50?

(1) None of the five numbers is greater than 100.

(2) Three of the five numbers are 24, 25 and 26, respectively.

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

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

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

EACH statement ALONE is sufficient.

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

Explanation

For statement (1), there are different combinations that satisfy the condition. For example, the five numbers can be $48,49,50,51,\ or\ 52,$ or the five numbers can be . Therefore, we cannot determine how many of the numbers are greater than by knowing the first statement.

For statement (2), even though we know three of them, the two unknown numbers can both be greater than , or one smaller and one greater. Thus statement (2) is not sufficient.

Putting the two statements together, we know that the sum of the two unknown numbers is

$50\cdot 5-\left ( 24+25+26 \right )=175$

Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.

A certain major league baseball player gets on base 25% of the time (once every 4 times at bat).

For any game where he comes to bat 5 times, what is the probability that he will get on base either 3 or 4 times? - Hint – add the probability of 3 to the probability of 4.

$\small .10$

$\small .15$

$\small .20$

$\small .25$

$\small .30$

Explanation

$\small n=5$

$\small p=.25$

$\small x=3$

$\small prob = .088$

$\small n=5$

$\small p=.25$

$\small x=4$

$\small prob = .015$

$\small .088 + .015 = .103$

Binomial Table

When assigning a score for the term, a professor takes the mean of all of a student's test scores.

Joe is trying for a score of 90 for the term. He has one test left to take. What is the minimum that Joe can score and achieve his goal?

Statement 1: He has a median score of 85 so far.

Statement 2: He has a mean score of 87 so far.

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.

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation

1. If the arithmetic mean of five different numbers is 50, how many of the numbers are greater than 50?

(1) None of the five numbers is greater than 100.

(2) Three of the five numbers are 24, 25 and 26, respectively.

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

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

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

EACH statement ALONE is sufficient.

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

Explanation

For statement (2), even though we know three of them, the two unknown numbers can both be greater than , or one smaller and one greater. Thus statement (2) is not sufficient.

Putting the two statements together, we know that the sum of the two unknown numbers is

$50\cdot 5-\left ( 24+25+26 \right )=175$

Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.

A bag contains red, yellow and green marbles. There are marbles total.

I) There are green marbles.

II) The number of yellow marbles is half of one less than the number of green marbles.

What are the odds of picking a red followed by a green followed by a yellow? Assume no replacement.

Both statements are needed to answer the question.

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

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

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.

Explanation

In order to calculate the probability the question asks for, we need to know the number of each color of marble.

I) Gives us the number of greens.

II) Gives us clues which will allow us to find the number of reds and yellows.

We need both statements to answer this question.

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