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GMAT Quantitative › Problem-Solving Questions

Questions 1 - 10

Cherry invested dollars in a fund that paid 6% annual interest, compounded monthly. Which of the following represents the value, in dollars, of Cherry’s investment plus interest at the end of 3 years?

$(1.005)^{36} * m$

$(1.005m)^{36}$

Explanation

The monthly rate is $\frac{6\ percent}{12}=0.5\ percent=0.005$

3 years = 36 months

According to the compound interest formula

$P=C\left ( 1+r \right )t$

and here , , , so we can plug into the formula and get the value

$m( 1+0.005)^{36} = (1.005)^{36} * m$

$\textup{QWERT}$ is a pentagon with two sets of congruent sides and one side that is longer than all the others.

The smallest pair of congruent sides are 5 inches long each.

The other two congruent sides are 1.5 times bigger than the smallest sides.

The last side is twice the length of the smallest sides.

What is the perimeter of $\textup{QWERT}$ ?

$35\ \textup{meters}$

$40\ \textup{meters}$

$30\ \textup{meters}$

$50\ \textup{meters}$

$45\ \textup{meters}$

Explanation

A pentagon is a 5 sided shape. We are given that two sides are 5 inches each.

Side 1 = 5inches

Side 2 = 5 inches

The next two sides are each 1.5 times bigger than the smallest two sides.

$\small 5*1.5=7.5$

Side 3 =Side 4= 7.5 inches

The last side is twice the size of the smallest side,

Side 5 =10 inches

Add them all up for our perimeter:

5+5+7.5+7.5+10=35 inches long

The arc $\widehat{AB}$ of a circle measures $120^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length . Give the length of the radius of the circle.

$\frac{\sqrt{3}}{6} x+ \frac{ \sqrt{3}}{3}$

$\frac{\sqrt{3}}{12} x+ \frac{ \sqrt{3}}{6}$

$\frac{\sqrt{3}}{4} x+ \frac{ \sqrt{3}}{2}$

$x \sqrt{3} + 2 \sqrt{3}$

Explanation

A circle can be divided into three congruent arcs that measure

$\frac{1}{3} \cdot 360 ^{\circ } = 120 ^{\circ }$ .

If the three (congruent) chords are constructed, the figure will be an equilateral triangle. The figure is below, along with the altitudes of the triangle:

Circle and triangle

Since , it follows by way of the 30-60-90 Triangle Theorem that

$BM = \frac{1}{2} AB = \frac{1}{2} (x+2) = \frac{1}{2} x+1$

and

$AM = \frac{ \sqrt{3}}{2} \cdot BM = \frac{ \sqrt{3}}{2} \cdot \left ( \frac{1}{2} x+1 \right ) = \frac{\sqrt{3}}{4} x+ \frac{ \sqrt{3}}{2}$

The three altitudes of an equilateral triangle split each other into segments that have ratio 2:1. Therefore,

$OA = \frac{2}{3} \cdot AM$

$=\frac{2}{3} \cdot \left ( \frac{\sqrt{3}}{4} x+ \frac{ \sqrt{3}}{2} \right )$

$= \frac{\sqrt{3}}{6} x+ \frac{ \sqrt{3}}{3}$

Rhombus

Note: Figure NOT drawn to scale.

The above figure is of a rhombus and one of its diagonals. What is equal to?

Not enough information is given to answer the question.

Explanation

The four sides of a rhombus are congruent, so a diagonal of the rhombus cuts it into two isosceles triangles. The two angles adjacent to the diagonal are congruent, so the third angle, the one marked, measures:

$\left (180 - 2 \times 24 \right )^{\circ } = \left (180 -48 \right )^{\circ } = 132^{\circ }$

In the -plane, point $(a,b)$ lies on a circle with center at the origin. The radius of the circle is 5. What is the value of $a^{2}+b^{2}$ ?

$25$

$5$

$10$

$16$

$32$

Explanation

$a$ and $b$ are the right-angle sides of a triangle, and the radius of the circle is the hypotenuse of the triangle. From the Pythagorean Theorem we would know that $a^{2}+b^{2}=r^{2}=5^{2}=25$ .

Transversal

Figure NOT drawn to scale.

Refer to the above figure.

True or false: $m \perp t$

Statement 1: $\angle 4$ is a right angle.

Statement 2: $\angle 3$ and $\angle 5$ are supplementary.

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

Statement 1 alone establishes by definition that $l\perp t$ , but does not establish any relationship between and .

By Statement 2 alone, since same-side interior angles are supplementary, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other. and $l\perp t$ , so it can be established that $m \perp t$ .

What is the area of a circle with a diameter of ?

$324\pi cm^2$

$81\pi cm^2$

$54\pi cm^2$

$36\pi cm^2$

$90\pi cm^2$

Explanation

The area of a circle is defined by $A=\pi r^{2}$ , where is the radius of the circle. We are provided with the diameter of the circle, which is twice the length of .

If , then $r=\frac{d}{2}=\frac{18cm}{2}=9cm$

Then, solving for :

$A=\pi r^{2}$

$A=\pi (9cm)^{2}$

$A=81\pi cm^2$

The arc $\widehat{AB}$ of a circle measures $90^{\circ }$ and has length . Give the length of the chord $\overline{AB}$ .

$\frac{2 \sqrt{2}}{\pi}x + \frac{4 \sqrt{2}}{\pi}$

$\frac{ \sqrt{2}}{\pi}x + \frac{ \sqrt{2}}{\pi}$

$\frac{ \pi \sqrt{2}}{4} x + \frac{ \pi \sqrt{2}}{2}$

$\frac{ \pi \sqrt{2}}{2} x + \pi \sqrt{2}$

$\pi x \sqrt{2}+ 2 \pi \sqrt{2}$

Explanation

The figure referenced is below.

Circle x

The arc is $\frac{90}{360} = \frac{1}{4}$ of the circle, so the circumference of the circle is

The radius is this circumference divided by $2 \pi$ , or

$\frac{ 4x+8}{2 \pi} = \frac{2}{\pi}x + \frac{4}{\pi}$ .

$\overline{AB}$ is, consequently, the hypotenuse of an isosceles right triangle with leg length $\frac{2}{\pi}x + \frac{4}{\pi}$ ; by the 45-45-90 Triangle Theorem, its length is $\sqrt{2}$ times this, or

$\left (\frac{2}{\pi}x + \frac{4}{\pi} \right ) \sqrt{2} = \frac{2 \sqrt{2}}{\pi}x + \frac{4 \sqrt{2}}{\pi}$

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A circle is inscribed in a square. The area of the circle is $2\pi$ . What is the area of the square?

$2\sqrt{2}$

$3\sqrt{2}$

Explanation

Since we know the area of the circle, we can tell that: $\pi \cdot r^{2}= 2\pi$ . Where is the radius of the circle.

The length of a side of the square will be since the diameter of the circle is the same length as the side length of the square.

Finally we can calculate the area of the square which will be $(2r)^{2}$ . $r=\sqrt{2}$ so the area will be , which is our final answer.

$\Delta ABC$ is an isosceles triangle with perimeter 43; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?

This triangle cannot exist.

Explanation

We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).

We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three. , , and include numbers that are not prime (21, 15, 9). This leaves us with only one possibility: