GMAT Quantitative Reasoning

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Questions 1 - 10

A coin is flipped four times. What is the probability of getting heads at least three times?

$\frac{5}{16}$

$\frac{1}{16}$

$\frac{1}{4}$

$\frac{11}{16}$

$\frac{3}{4}$

Explanation

Since this problem deals with a probability with two potential outcomes, it is a binomial distribution, and so the probability of an event is given as:

$P(n,k)=\frac{n!}{(n-k)!k!}p^k(1-p)^{n-k}$

Where is the number of events, is the number of "successes" (in this case, a "heads" outcome), and is the probability of success (in this case, fifty percent).

Per the question, we're looking for the probability of at least three heads; three head flips or four head flips would satisfy this:

$P(4,3)=\frac{4!}{(4-3)!3!}(\frac{1}{2})^{3}(\frac{1}{2})^{4-3}=4(\frac{1}{2})^{4}=\frac{1}{4}$

$P(4,4)=\frac{4!}{(4-4)!4!}(\frac{1}{2})^{4}(\frac{1}{2})^{4-4}=(\frac{1}{2})^{4}=\frac{1}{16}$

Thus the probability of three or more flips is:

$\frac{1}{4}+\frac{1}{16}=\frac{5}{16}$

What is the arc length for a sector with a central angle of $45^{\circ}$ if the radius of the circle is ?

$\frac{3\pi }{4}$

$\frac{\pi }{4}$

$\frac{3\pi }{2}$

$\frac{\pi }{2}$

$\pi$

Explanation

Using the formula for arc length, we can plug in the given angle and radius to calculate the length of the arc that subtends the central angle of the sector. The angle, however, must be in radians, so we make sure to convert degrees accordingly by multiplying the given angle by $\frac{\pi }{180}$ :

$S=r\theta$

$S=3(45^{\circ})(\frac{\pi }{180^{\circ}})=3(\frac{\pi }{4})=\frac{3\pi}{4}$

Let be a positive integer.

True or false: $i^{N} = 1$

Statement 1: is a prime number.

Statement 2: is a two-digit number ending in a 7.

EITHER statement ALONE is 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.

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.

Explanation

If is a positive integer, then $i^{N} = 1$ if and only if is a multiple of 4.

It follows that if $i^{N} = 1$ , cannot be a prime number. Also, every multiple of 4 is even, so as an even number, cannot end in 7. Contrapositively, if Statement 1 is true and is prime, or if Statement 2 is true and if ends in 7, it follows that is not a multiple of 4, and $i^{N} \ne 1$ .

What is the value of $\dpi{100} \small 2x+2y$ ?

Statement 1: $\dpi{100} \small x-3y=4$

Statement 2: $\dpi{100} \small x+y=4$

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

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

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Explanation

We know that we need 2 equations to solve for 2 variables, so it is tempting to say that both statements are needed. This is actually wrong! We aren't being asked for the individual values of x and y, instead we are being asked for the value of an expression.

$\dpi{100} \small 2x+2y$ is just $\dpi{100} \small 2\left ( x+y \right )$ , and statement 2 gives us the value of $\dpi{100} \small x+y$ . For data sufficiency questions, we don't actually have to solve the question, but if we wanted to, we would simply multiply statement 2 by 2.

$\dpi{100} \small 2x+2y=2\left ( x+y \right )=2$ * Statement 2 = $\dpi{100} \small 2\times 4=8$

A coin is flipped seven times. What is the probability of getting heads six or fewer times?

$\frac{127}{128}$

$\frac{255}{256}$

$\frac{1}{128}$

$\frac{1}{7}$

$\frac{6}{7}$

Explanation

Since this problem deals with a probability with two potential outcomes, it is a binomial distribution, and so the probability of an event is given as:

$P(n,k)=\frac{n!}{(n-k)!k!}p^k(1-p)^{n-k}$

Where is the number of events, is the number of "successes" (in this case, a "heads" outcome), and is the probability of success (in this case, fifty percent).

One approach is to calculate the probability of flipping no heads, one head, two heads, etc., all the way to six heads, and adding those probabilities together, but that would be time consuming. Rather, calculate the probability of flipping seven heads. The complement to that would then be the sum of all other flip probabilities, which is what the problem calls for:

$P(7,7)=\frac{7!}{(7-7)!7!}(\frac{1}{2})^7(1-\frac{1}{2})^{7-7}=(\frac{1}{2})^7=\frac{1}{128}$

Therefore, the probability of six or fewer heads is:

$1-\frac{1}{128}=\frac{127}{128}$

Let and be real numbers.

What is the product of and its complex conjugate?

Statement 1:

Statement 2:

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

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation

The complex conjugate of an imaginary number is , and

$(a + bi)(a - bi) = a^{2}+b^{2}$ .

Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of and that add up to 12:

Case 1:

Then , and the product of this number and its complex conjugate is $a^{2}+b^{2} = 6^{2}+6^{2} =36+36 = 72$ .

Case 2:

Then , and the product of this number and its complex conjugate is $a^{2}+b^{2} = 5^{2}+7^{2} =25+49= 74$ .

The two cases result in different products.

For a similar reason, Statement 2 alone provides insufficient information.

If both statements are assumed to be true, they form a system of equations that can be solved as follows:

$\underline{a - b = 16}$

Backsolve:

Since we know that and , then we know that the desired product is $a^{2}+b^{2} =14 ^{2}+(-2)^{2} =196+4 = 200$ .

A given circle has an area of $81\pi$ . What is the length of its diameter?

Not enough information provided

Explanation

The area of a circle is defined by the equation $A=\pi r^{2}$ , where is the length of the circle's radius. The radius, in turn, is defined by the equation $r=\frac{d}{2}$ , where is the length of the circle's diameter.

Given $A=81\pi$ , we can deduce that $r^{2}=81$ and therefore . Then, since $r=\frac{d}{2}$ , .

Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.

The circumscribed circle has circumference $20 \pi$ . Give the area of the inscribed circle.

$75 \pi$

$50 \pi$

$200 \pi$

$\frac{400 \pi }{3}$

$\frac{200 \pi }{3}$

Explanation

Examine the diagram below, which shows the hexagon, segments from its center to a vertex and the midpoint of a side, and the two circles.

Thingy

Note that the segment from the center of the hexagon to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two segments and half a side of the hexagon can be proved to form a 30-60-90 triangle.

The circumscribed circle has circumference $20 \pi$ , so its radius - and the length of the hypotenuse of the right triangle -

$20 \pi \div 2 \pi = 10$

By the 30-60-90 Theorem, the length of the shorter leg is half this, or 5. The length of the longer leg, which is the radius of the inscribed circle, is $\sqrt{3}$ times this, or $5 \sqrt{3}$ .

The area of the inscribed circle can now be calculated:

$A= \pi r^{2} = \pi \cdot \left ( 5 \sqrt{3 }\right ) ^{2}= \pi \cdot 25 \cdot 3 = 75 \pi$

What is the circumference of Circle ?

1.) The diameter of the circle is .

2.) The area of the circle is $25\pi$ .

Each statement alone is sufficient to solve the question.

Statement 1 is sufficient to solve the question, but Statement 2 is not sufficient to solve the question.

Both statements taken together are sufficient to solve the question.

Statement 2 is sufficient to solve the question, but Statement 1 is not sufficient to solve the question.

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

Explanation

We are asked to find the circumference of Circle and are given the diameter and the area. We also know that $C=2\pi r$ . Taking each statement individually:

1.) The diameter is and we know that the radius $r=\frac{d}{2}$ , so $C=2\pi r=2\pi\left(\frac{d}{2}\right)=\pi d=10\pi$ . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

2.) The area of Circle is $A=\pi r^{2}=25\pi$ , so we can determine that the radius . Since the circumference $C=2\pi r$ , Statement 2 is is sufficient to solve for the circumference of the circle by itself.