Problem Solving Test Preparation for GMAT

Question 1

The ratio, by weight, of the four ingredients A, B, C, and D of a certain mixture is 4:7:8:12. The mixture will be changed so that the ratio of A to C is quadrupled and the ratio of A to D is decreased. The ratio of A to B will be held constant. If B will constitute 20% of the weight of the new mixture, by approximately what percent will the ratio of A to D be decreased?

A. 15%
B. 25%
C. 35%
D. 45%
E. 55%

Question 2

In XYZ Building, a flight of stairs connects each floor to the next, and each flight of stairs is separated from the next flight by a landing. Josie takes twice as long to climb a flight of stairs at a constant rate as she does to cross a landing at another constant rate. If it takes Josie 13.3 minutes to climb 13 flights of stairs and cross the landings between flights, not counting the landings at either end, how long will it take her to climb 27 flights and cross the intervening landings (again not counting landings at either end) at the same rate of travel?

A. 27.6 minutes
B. 27.8 minutes
C. 28.0 minutes
D. 28.2 minutes
E. 28.4 minutes

Question 3

A circle is inscribed within a regular hexagon in such a way that the circle touches all sides of the hexagon at exactly one point per side. Another circle is drawn to connect all the vertices of the hexagon. Expressed as a fraction, what is the ratio of the area of the smaller circle to the area of the larger circle?

A. v(2/3)
B. (v2)/3
C. (v3)/2
D. (v3)/4
E. 3/4

Question 4

A certain car can travel 48 kilometers on a liter of fuel. If the fuel tank’s contents decrease by 3.9 gallons over a period of 5.7 hours as the car moves at a constant speed, how fast is the car moving, in miles per hour? (1 gallon = 3.8 liters; 1 mile = 1.6 kilometers)

A. 52
B. 65
C. 78
D. 91
E. 104

Question 5

If x < y < z but x² > y² > z² > 0, which of the following must be positive?

A. x³ y⁴ z⁵
B. x³ y⁵ z⁴
C. x⁴ y³ z⁵
D. x⁴ y⁵ z³
E. x⁵ y⁴ z³

ANSWERS

1. D

First, make a quick table to represent the ratios in the original mixture:

One way forward is to make the A:C change, holding A:B and A:D constant. (We know that A:D decreases, but we don’t know by how much, so for now, pretend that the ratio stays constant.)

To quadruple the ratio A:C, we can either multiply A by 4 or divide C by 4. Since we want to leave A:B and A:D constant, it’s more efficient to divide C by 4. So if A:D weren’t changing, the new mixture would have these ratios:

Since A:D decreases, let’s add an unspecified amount to D. (We don’t want to mess with the A side of the ratio, since A is involved in other ratios.) Call that amount x.

Now we know that B will constitute 20% of the whole, so the ratio of B to the whole in the final mixture will be 20 to 100, or 1:5. If we look at the table, the ratio of B to the whole is 7 to 25 + x. We can equate these proportions:

1/5 = 7/(25 + x)
25 + x = 35
x = 10

So now we know that the final mixture has these proportions:

The new ratio of A to D is 4:22, or 2:11. As a fraction, this ratio is 2/11.

The original ratio of A to D is 4:12, or 1:3. As a fraction, this ratio is 1/3.

Finally, we are asked this: if 1/3 is decreased to 2/11, what is the percent decrease?

A fast way to compute this number is first to figure out the factor that you multiply 1/3 by to get 2/11. Call that factor y.

(1/3)y = 2/11
y = 6/11

So 2/11 is 6/11 of 1/3. As a percent, 6/11 is approximately 55% (as a decimal, 6/11 = 0.5454…). If the new number (2/11) is 55% of the old number (1/3), then the percent decrease is 100% – 55%, or 45%.

The correct answer is D.

2. C
Since it takes Josie twice as long to climb a flight of stairs as it does to cross a landing, let’s start with the smaller time. Let’s take the time to cross 1 landing and call it x. Then the “flight climbing time,” the time to climb 1 flight, is 2x.

Josie takes 13.3 minutes to climb 13 flights of stairs and 13 – 1 = 12 landings between them. (We “subtract one before we’re done” to count the landings between the flights.)

The relationship can now be written:

13.3 = time to climb 13 flights + time to cross 12 landings
13.3 = 13(2x) + 12x
13.3 = 26x + 12x = 38x
x = 13.3/38 = 133/380

380 can be factored easily: 380 = 38 × 10 = 2 × 19 × 2 × 5
133 is tricky; it equals 7 × 19.
So 133/380 reduces (with the 19’s canceling) to 7/20. That’s x.

Now, the quantity we want is the time to climb 27 flights and cross 27 – 1 = 26 landings.

27(2x) + 26x = ?
54x + 26x = ?
80x = ?
80(7/20) = ? Cancel out the 20, leaving 4 × 7 = 28 exactly.

The correct answer is C.

3. E

The first thing to recognize is that a regular hexagon has 6 equal sides and 6 equal internal angles of 120°, since the angles inside the hexagon must add up to (n – 2)×180 = 720°, or 120° per angle. This is a very symmetrical figure. The circle will touch each side exactly in the middle of the side, by symmetry, like so:

The outer circle will touch each vertex:

Now, to compare the areas of the two circles, we should compare their radii. The obvious place to draw radii is from the points of contact with the circles:

The triangle that’s been created is a 30-60-90 triangle. At point B, the radius is perpendicular to the side of the hexagon (which is tangent to the circle). The 120° angle of the hexagon is equally split by the longer radius, again by symmetry, creating a 60° angle at point A.

The ratios of the sides of a 30-60-90 triangle are 1: v3 : 2, with 2 as the longest side (the hypotenuse). The longer radius is the “2” side, while the shorter radius is the “v3” side.

Since the areas are proportional to the square of the radius (by A = nr^2), we know that the smaller area to the larger area would be sqrt{(3)^2} : 2^2, or 3 : 4. Expressed as a fraction, this ratio is 3/4.

Note that C is a trap answer: it expresses the ratios of the radii themselves.

The correct answer is E.

4. C
This problem is all about quick but careful unit conversion. One fast but safe way to convert units is to use conversion factors, which are essentially fractions with equivalent amounts on top and bottom (but different units). As you multiply conversion factors together, you cancel units, in the same way as you cancel common factors from numerators and denominators.

Start with kilometers per liter:
48 km / liter

You ultimately want to get to miles per hour, or miles / hour. So we need to eliminate kilometers from the numerator and replace it with miles. We can do so by multiplying by this conversion factor:
1 mile / 1.6 km

We get this:
(48 km / liter)(1 mile / 1.6 km) = 48/1.6 miles per liter

We can cancel numeric factors now, or we can wait. Since 1.6 is just a decimal point move away from 16, which goes into 48, let’s go ahead and cancel:
48/1.6 = 480/16 = 30 miles per liter

Now, let’s get rid of liters, converting to gallons in the denominator:
(30 miles / liter)(3.8 liters / 1 gallon) = (30 × 3.8) miles per gallon

Save the full computation for now, but a quick & easy move is to trade decimal points:
(30 × 3.8) miles per gallon = (3 × 38) miles per gallon

Finally, let’s “convert” gallons to hours. These two units measure very different things (volume and time), but we have an effective conversion between them: the car uses up 3.9 gallons over a period of 5.7 hours, so the rate of converting between gallons and hours for this trip is 3.9 gallons per 5.7 hours.

(3 × 38 miles / gallon)(3.9 gallons / 5.7 hours) = (3 × 38 × 3.9 / 5.7) miles per hour

Now we just need to simplify the fraction. Use decimal moves again:

3 × 38 × 3.9 / 5.7 = 3 × 38 × 39 / 57

38 = 2 × 19 and 57 = 3 × 19, so cancel out a common factor of 19:

3 × 38 × 39 / 57 = 3 × 2 × 39 / 3 = 2 × 39 = 78

The correct answer is C.

5. B

The first multi-part inequality that we are given, x < y < z, means that on a number line, x is to the left of y, which is to the left of z. There is no information about whether any of these quantities is positive or negative, though.

The second multi-part inequality that we are given, x^2 > y^2 > z^2 > 0, means that x is further from 0 than y, which is further from 0 than z (which itself is not 0). In other words, we know that |x| > |y| > |z|, and none of these variables equals 0. Again, we cannot infer anything from this information alone about the signs of the variables.

Finally, putting these two pieces of information together, we can conclude something about the signs of some of the variables. Take x. It is to the left of y, but it is also further from 0 than y. The only way for both of these facts to be true is for x to be negative (that is, to the left of 0 on the number line).

The same argument holds true for y itself, since it is to the left of z but also further from 0 than z. So we know that both x and y are negative.

However, we don’t know anything about the sign of z. It could be positive or negative (it just has to be the closest variable to 0).

Looking at the answer choices for a “must be positive” expression, we can plug in a minus sign (–) for x and y and a +/– for z.

The only positive-definite expression is x^3y^5z^4, or (minus)^3(minus)^5(plus or minus)^4 since this becomes a negative times a negative times a positive.

The correct answer is B.