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GMAT Quantitative › Problem-Solving Questions

Questions 1 - 10

The points and form a line which passes through the center of circle Q. Both points are on circle Q.

To the nearest hundreth, what is the length of the radius of circle Q?

Explanation

To begin this problem, we need to recognize that the distance between points L and K is our diameter. Segment LK passes from one point on circle Q through the center, to another point on circle Q. Sounds like a diameter to me! Use distance formula to find the length of LK.

$d=\sqrt{(x-x')^2+(y-y')^2}$

Plug in our points and simplify:

$d=\sqrt{(9--4)^2+(3-5)^2}=\sqrt{13^2+2^2}=\sqrt{173}\approx13.15$

Now, don't be fooled into choosing 13.15. That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

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:

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

In $\Delta ABC$ , and . Which of the following values of makes $\Delta ABC$ a scalene triangle?

None of the other responses gives a correct answer.

Explanation

The three sides of a scalene triangle have different measures, so 15 can be eliminated.

By the Triangle Inequality, the sum of the lengths of the two smaller sides must exceed the length of the third side. Since , 8 violates this theorem; since , 22 does as well.

10 is a valid measure of the third side, since ; it makes all three segments of different length, so it is the correct choice.

Company B produces toy trucks for a shopping mall at a cost of $7.00 each for the first 500 trucks and $5.00 for each additional truck. If 600 trucks were produced by Company B and sold for $15.00 each, what was Company B’s gross profit?

$\$5000$

$\$0$

$\$9000$

$\$4000$

$\$14,000$

Explanation

First of all, we need to know that

$Gross\ Profit=Revenue-Total\ Cost$ .

There are 600 trucks produced. According to the question, the first 500 trucks cost $7.00 each. Therefore, the total cost of the first 500 trucks is $\$7.00\cdot 500=\$3500$ .

The other 100 trucks cost $5.00 each for a cost of $$5.00\cdot 100=$500$ .

Add these together to find the cost of the 600 trucks: $$3500+$500=\$4000$

The total profit is easier to calculate since the selling price doesn't change: $\$15.00\cdot 600=\$9000$

At this point we have both revenue and total cost, so the answer for gross profit is $\$9000-$4000=$5000$ .

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

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

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

$200 \pi$

$50 \pi$

$75 \pi$

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

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

The inscribed circle has circumference $20 \pi$ , so its radius - and the length of the longer leg of the right triangle - is

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

By the 30-60-90 Theorem, the length of the shorter leg is this length divided by $\sqrt{3}$ , or $\frac{10}{\sqrt{3}}$ ; the length of the hypotenuse, which is the radius of the circumscribed circle, is twice this, or $\frac{20}{\sqrt{3}}$ .

The area of the circumscribed circle can now be calculated:

$A= \pi r^{2} = \pi \cdot \left (\frac{20}{\sqrt{3}}\right ) ^{2}= \pi \cdot \frac{400}{3} = \frac{400 \pi }{3}$

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 $m \perp t$ .

Calculate the length of a chord in a circle with a radius of , given that the perpendicular distance from the center to the chord is .

Explanation

We are given the radius of the circle and the perpendicular distance from its center to the chord, which is all we need to calculate the length of the chord. Using the formula for chord length that involves these two quantities, we find the solution as follows, where is the chord length, is the perpendicular distance from the center of the circle to the chord, and is the radius:

$L=2\sqrt{r^2-d^2}=2\sqrt{5^2-3^3}=2\sqrt{25-9}=2\sqrt{16}=8$

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 .