sat practice test answers and explanations a

Section #2: View Explanations

1

Explanation for Correct Answer C.

Choice (C) is correct. If is more than then and The value of

is then

Explanation for Incorrect Answer A.

Choice (A) is not correct. This would be the value of if were less than However, is more than so and

Explanation for Incorrect Answer B.

Choice (B) is not correct. This is the value of The question asks for the value of

Explanation for Incorrect Answer D.

Choice (D) is not correct. This is the value of The question asks for the value of

Explanation for Incorrect Answer E.

Choice (E) is not correct. This is the value of The question asks for the value of

2


Choice (C) is correct. Let be the number in question. Since the result when is divided by

is equal to the result when is divided by it is true that Therefore, and


Choice (A) is not correct. When is divided by the result is when is divided by

the result is These two results are not equal.


Choice (B) is not correct. When is divided by the result is when is divided by

the result is These two results are not equal.


Choice (D) is not correct. When is divided by the result is when is divided by the

result is These two results are not equal.


Choice (E) is not correct. When is divided by the result is when is divided by the

result is These two results are not equal.

3

Explanation for Correct Answer E. Choice (E) is correct. The letter E CANNOT be folded along a vertical line so that its left half

would coincide with its right half. See the figure below.

Explanation for Incorrect Answer A. Choice (A) is not correct. The letter A can be folded along a vertical line so that its left half

would coincide with its right half, as indicated in the figure below. The question asked which letter CANNOT be so folded.

Explanation for Incorrect Answer B. Choice (B) is not correct. The letter I can be folded along a vertical line so that its left half


Explanation for Incorrect Answer C. Choice (C) is not correct. The letter O can be folded along a vertical line so that its left half



Choice (D) is not correct. The letter U can be folded along a vertical line so that its left half would coincide with its right half, as indicated in the figure below. The question asked which

letter CANNOT be so folded.

4

Explanation for Correct Answer A.

Choice (A) is correct. Since vertical angles have equal measure, it follows that

Substituting for and for in this equation gives Therefore,

Explanation for Incorrect Answer B. Choice (B) is not correct. This incorrect choice could be the result of visually estimating that

the angles marked and in the figure appear to have the same measure. However, the

figure is not drawn to scale, and neither nor is equal to

Explanation for Incorrect Answer C.

Choice (C) is not correct. The value of is but the value of is


Choice (D) is not correct. This is the value of not the value of


Choice (E) is not correct. This is the value of not the value of

5

Explanation for Correct Answer B.

Choice (B) is correct. Each of the five pairs satisfies the equation

is true; is true; is true;

is true; and is true.


Choice (A) is not correct. The first pair of numbers does not satisfy


Choice (C) is not correct. The first pair of numbers does not satisfy


Choice (D) is not correct. The first pair of numbers does not satisfy


Choice (E) is not correct. The first pair of numbers satisfies but the second pair

of numbers does not:

6


Choice (C) is correct. The ratio of David’s car expenses to his food expenses is

Since he spends per month on food, he spends dollars per month on his car.


Choice (A) is not correct. Since is equal to or of David’s monthly expenses, his

total monthly expenses are dollars. If his monthly car expenses were

equal to that would be of the total, or about However, David’s car

expenses are of his total expenses, not


Choice (B) is not correct. Since is equal to or of David’s monthly expenses, his

total monthly expenses are dollars. If his monthly car expenses were




Choice (D) is not correct. Since is equal to or of David’s monthly expenses,

his total monthly expenses are dollars. If his monthly car expenses were




Choice (E) is not correct. This is the amount that David spends per month on food, not on

his car. Since of his monthly expenses are for food and only are for his car, the dollar amounts cannot be the same.

7


Choice (B) is correct. Substituting for in the equation gives

It follows that Since and cannot be equal to the

equation is equivalent to


Choice (A) is not correct. This incorrect answer could be the result of substituting instead

of for


Choice (C) is not correct. This incorrect answer could be the result of substituting instead

of for


Choice (D) is not correct. This is the value of but the question asks for the value of


Choice (E) is not correct. If the value of were then would be equal to It would

follow that was equal to and then would be equal to This

could be true only if were but is a positive integer. Therefore,

8


Choice (A) is correct. The regular price of the refrigerator is At percent off the

regular price, the price would be or At percent off the

regular price, the price would be or an additional discount of

percent off this price would give a final price of or

Therefore, the savings from buying the refrigerator at percent off the regular price rather

than buying it on sale at percent off the regular price with an additional discount of

percent off the sale price is or


Choice (B) is not correct. This is percent of the reduction from the regular price

when the refrigerator is on sale at percent off the regular price. However, this is not what the question asks for.

Explanation for Incorrect Answer C. Choice (C) is not correct. This is the amount of money saved by buying the refrigerator at

percent off the regular price rather than buying it on sale at percent off the regular

price with an additional discount of percent off the sale price. However, the question asks

for the amount of money saved by buying the refrigerator at percent off the regular price

rather than buying it on sale at percent off the regular price with an additional discount of

percent off the sale price.


Choice (D) is not correct. If the refrigerator is discounted by percent off the regular price

with an additional percent discount off the sale price, is the amount of the additional discount. However, this is not what the question asks for.


Choice (E) is not correct. This would be the savings from buying the refrigerator at

percent off the regular price rather than buying it at percent off the regular price, with no

additional discount. However, there is an additional percent discount when the

refrigerator is on sale for percent off the regular price.

9

Explanation for Correct Answer E.

Choice (E) is correct. Since it follows that and

therefore,


Choice (A) is not correct. Since it follows that

and so not


Choice (B) is not correct. Since it follows that

and so not


Choice (C) is not correct. Since it follows that

Hence but the question asks for an expression equivalent to

not for one equivalent to


Choice (D) is not correct. Since it follows that

However, the question asks for an expression equivalent to not for one equivalent

to

10


Choice (C) is correct. If a triangle has one side of length and another of length then the

measure of the angle joining the sides of length and can have measure greater than

equal to or less then as shown below.

In each of these three cases, the area of the triangle is given by where is the height

of the triangle. The area of the triangle will be greatest when is greatest. In the case

where the angle joining the sides of length and has measure the value of is and

the area of the triangle is In the cases where the angle joining the sides of length and

has measure greater than or less than the value of is less than so the area of

the triangle is less than All triangles with one side of length and another of length

fall into one of these three categories, so the greatest possible area of such a triangle is


Choice (A) is not correct. A triangle with a side of length and another side of length can

have area but this is not the greatest possible area of such a triangle.


Choice (B) is not correct. A triangle with a side of length and another side of length can

have area but this is not the greatest possible area of such a triangle.


Choice (D) is not correct. The greatest possible area of a triangle with one side of length

and another side of length is not


Choice (E) is not correct. The greatest possible area of a triangle with one side of length

and another side of length is not

11


Choice (C) is correct. Since Garcia won the vote over Pérez by a ratio of to Garcia

received of the votes and Pérez received of the votes. There were a total of

votes cast for Garcia and Pérez. Thus, Pérez received of these votes. Therefore,

the number of votes cast for Pérez was or


Choice (A) is not correct. The ratio to divides the votes cast into parts, and of

is However, the question asks for the number of votes cast for Pérez,

which is of votes, not of


Choice (B) is not correct. Garcia received more votes than Pérez, but the question asks for the number of votes casts for Pérez.


Choice (D) is not correct. The number of votes cast for Garcia was but the question asks for the number of votes cast for Pérez.


Choice (E) is not correct. If Pérez had won by a ratio of to then the number of votes cast

for Pérez would have been However, Garcia won by a ratio of to so Pérez

received or votes.

12


Choice (C) is correct. The inequality is equivalent to or Thus, of

the positive integers that are less than or equal to the only ones that satisfy the

inequality are and Therefore, the probability in question is


Choice (A) is not correct. There are two integers less than or equal to that satisfy the

inequality so the probability in question cannot be


Choice (B) is not correct. The integers and both satisfy the inequality so the

probability in question is not


Choice (D) is not correct. The only positive integers less than that satisfy the inequality

are and If then Therefore, the probability in question is

equal to not


Choice (E) is not correct. The only positive integers less than that satisfy the inequality

are and If then Therefore, the probability in question is

equal to not

13

Explanation for Correct Answer D.

Choice (D) is correct. If is a number greater than then The amount by which is

greater than can be expressed as


Choice (A) is not correct. The only number greater than for which is In

general, if is a number greater than the difference between and need not be equal to


Choice (B) is not correct. The only number greater than for which is In general,

if is a number greater than the difference between and need not be equal to


Choice (C) is not correct. If is a number greater than for which then and

In general, if is a number greater than and then the difference between and

will not be


Choice (E) is not correct. If is a number greater than for which then

and However, is greater than Therefore, there is no number greater

than for which

14

Explanation for Correct Answer C. Choice (C) is correct. The segment connecting the center of the top base of the cylinder to a

point on the circumference of the other base is the hypotenuse of a right triangle whose legs are as follows: one leg is the altitude of the cylinder that connects the center of the top

base to the center of the bottom base, and the other leg is the radius of the bottom base drawn to the given point on the circumference. Thus, the legs of this right triangle have

lengths (height of the cylinder) and (half the diameter). Therefore, by the Pythagorean theorem, the distance from the center of one base to a point on the circumference of the

other base is

Explanation for Incorrect Answer A. Choice (A) is not correct. The distance in question is the hypotenuse of a right triangle with

legs of lengths and not the leg of a right triangle with other leg and hypotenuse


Choice (B) is not correct. Since is the height of the cylinder, the distance from the center of one base to the center of the other base is However, the question asks for the distance

from the center of one base to a point on the circumference of the other base.

Explanation for Incorrect Answer D. Choice (D) is not correct. The segment connecting the center of one base of the cylinder to a point on the circumference of the other base is the hypotenuse of a right triangle whose legs are the altitude of the cylinder and half the diameter of the base. Therefore, by the

Pythagorean theorem, the distance from the center of one base to a point on the

circumference of the other base is not


Choice (E) is not correct. The segment connecting the center of one base of the cylinder to a point on the circumference of the other base is the hypotenuse of a right triangle whose legs are the altitude of the cylinder and half the diameter of the base. Therefore, by the

Pythagorean theorem, the distance from the center of one base to a point on the

circumference of the other base is not

15


Choice (B) is correct. If and are integers such that and if

then and are factors of Since

the only cases that need to be considered are in which case

in which case and in which case However, the second set of

equations is the only one that has integer solutions ( and ), so is the

only possible value for Therefore, the correct answer is choice (B), II only.


Choice (A) is not correct. The only possible value of is If then It

would follow that and but the question states that and are integers.


Choice (C) is not correct. The value of can be but it cannot be If then

It would follow that and but the question states that and are integers.


Choice (D) is not correct. The value of can be but it cannot be If then

It would follow that and but the question states that and are

integers and that


Choice (E) is not correct. The value of can be but it cannot be or If then

it would follow that and but the question states that and are

integers. Also, if then it would follow that and but the

question states that and are integers and that

16

Explanation for Correct Answer D.

Choice (D) is correct. To travel along the streets from the firehouse to the building at a

fire truck can go blocks left and blocks up. This is a total of blocks. The fire truck can

also go block left, block up, block left, and block up; this is also a total of blocks. The

fire truck can also go block up, blocks left, and block up, which is likewise a total of

blocks. Any other route from the firehouse to the building at requires the fire truck to

travel more than blocks. Therefore, the minimum number of blocks the fire truck must

travel from the firehouse to reach the building at is which means the -distance of

the building at from the firehouse is

Explanation for Incorrect Answer A. Choice (A) is not correct. The minimum number of blocks a fire truck must travel from the

firehouse to reach the building at is so the -distance cannot be

Explanation for Incorrect Answer B. Choice (B) is not correct. The minimum number of blocks a fire truck must travel from the


Explanation for Incorrect Answer C. Choice (C) is not correct. The minimum number of blocks a fire truck must travel from the



Choice (E) is not correct. It is possible for a fire truck to travel blocks from the firehouse

to reach the building at but this is not the minimum number of blocks the fire truck must travel to accomplish this.

17


Choice (A) is correct. The -distance between point and point is because point is

blocks away from point Starting at point in order to arrive at point a fire truck must

arrive at a point that is blocks down and blocks to the left of point The blocks down

and blocks to the left can be traveled in any order: (1) down, down, left, left; (2) down, left, down, left; (3) down, left, left, down; (4) left, down, down, left; (5) left, down, left, down; or

(6) left, left, down, down. Therefore, there are six different routes that a fire truck can travel

the minimum -distance from to

Another way of counting the number of different -block routes from to is to see that

the blocks to the left can be chosen to be any of the blocks in a route from to

Therefore, the number of different routes is


Choice (B) is not correct. There are six different -block routes from to (1) down, down, left, left; (2) down, left, down, left; (3) down, left, left, down; (4) left, down, down, left; (5)

left, down, left, down; or (6) left, left, down, down.


Choice (C) is not correct. The fire truck can travel the blocks from to by traveling down

blocks and then to the left blocks, or by traveling to the left blocks and then down blocks (two routes). Or it can start with a block in one direction and then alternate directions

(two additional routes). But it can also travel from to by going down, left, left, down; or left, down, down, left (two more routes, for a total of six).


Choice (D) is not correct. There are six different -block routes from to (1) down, down, left, left; (2) down, left, down, left; (3) down, left, left, down; (4) left, down, down, left; (5)

left, down, left, down; or (6) left, left, down, down.


Choice (E) is not correct. The fire truck can travel the blocks from to by traveling down

blocks and then to the left blocks, or by traveling to the left blocks and down blocks. But there are four additional routes: down, left, down, left; down, left, left, down; left, down,

down, left; or left, down, left, down.

18


Choice (B) is correct. All of the buildings in the town that are an -distance of from the

firehouse ( ) lie on a square. The figure below shows all points that are an -distance of

from and the square on which they lie.

The vertices of the square are the points that are units away from along the vertical or

horizontal grid line passing through and the sides of the square have slope or


Choice (A) is not correct. In the -plane, the set of all points that are a distance of from a

given point do form a circle, but the -distance is not the same as the distance in the -plane.


Choice (C) is not correct. The buildings that are an -distance of from the firehouse lie on a square, as can be seen from the figure above. They do not all lie on the same right

isosceles triangle.


Choice (D) is not correct. The buildings that are an -distance of from the firehouse lie on a square, as can be seen from the figure above. They do not all lie on the same pair or

intersecting lines.


Choice (E) is not correct. The buildings that are an -distance of from the firehouse lie on a square, as can be seen from the figure above. They do not all lie on the same line.

19


Choice (C) is correct. If and are positive integers, then

It follows that


Choice (A) is not correct. The expression is equivalent to However,

the question asks for an expression equivalent to


Choice (B) is not correct. The expression is equivalent to

However, the question asks for an expression equivalent to


Choice (D) is not correct. The expression is equivalent to



Choice (E) is not correct. The expression is equivalent to


20


Choice (A) is correct. Since and are consecutive integers such that it

follows that The units digit of the product is so and must each be odd

numbers. If the units digit of were then the units digit of would be and the units digit

of would be not If the units digit of were then the units digit of would be and the

units digit of would be not If the units digit of were then the units digit of would

be and the units digit of would be not If the units digit of were then the units digit

of would be and the units digit of would be not If the units digit of is then the

units digit of is and the units digit of is Thus, it must be true that the units digit of is

Therefore, the units digit of is


Choice (B) is not correct. If the units digit of were then the units digit of would be

Then the units digit of would be not


Choice (C) is not correct. If the units digit of were then the units digit of would be and

the units digit of would be Then the units digit of would be not


Choice (D) is not correct. If the units digit of were then the units digit of would be and



Choice (E) is not correct. If the units digit of were then the units digit of would be and


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