SAT Sample Test I Part C

 

1.

 

1 - 1/12

1 - 1/13

 

1.  Answer:  B

Plug in.

Column A:  1 - (1/12) = 11/12

Column B:  1 - (1/13) = 12/13

To compare 11/12 and 12/13 you can find a common denominator:

11/12 = 143/156 and 12/13 = 144/156 so 12/13 is large.

You can also convert to a decimal:

11/12 » .917 and 12/13 » .923

These calculations are time consuming, even with a calculator.

 

Shortcut:  Note that since the denominator 12 is smaller than the denominator 13, 1/12 is larger than 1/13.  Since we are subtracting a larger number in Column A, the answer will be smaller.

2.

Note:  Figure may not be drawn to scale.

 

AB+BC

AC

 

2.  Answer:  A

The lengths of two side of a triangle will always be longer than the length of the third side.

 

In the most extreme case, two sides may be very close to the length of the third side, but they will still be longer.

3. 

 

(x - 3)2

x2

 

3.  Answer:  D

Plug in numbers.   Start by using a number less than 3 and a number greater than 3.

For x = 0, (x - 3)2= 9 and x2 = 0, so Column A is larger.

For x = 5, (x - 3)2= 4 and x2 = 25, so Column B is larger.  Therefore, the answer depends on the choice of x. 

 

Extra:  Algebraically we are comparing (x - 3)2 to x2.

 

We can simplify both sides of the equality (or inequality, whatever the case may be).  Note that we can perform operations on both sides of the inequality as long as we don't multiply or divide both sides of the inequality by a negative number, since this will change the direction of the inequality sign.

 

Column A

Column B

(x - 3)2

x2

x2 - 6x + 9

x2

-6x + 9

0

9

6x

3

2x

3/2

x

 

Without knowing more about x, it is impossible to say whether x is larger or smaller than 3/2.  Therefore, the answer is D.

 4. 

 

The cost of 10 quarts of juice at $2.00 a quart

The cost of 5 liters of cola at $4.00 a liter

 

4.  Answer:  C

We are not comparing liquid volume but rather cost.  The cost of 10 quarts of juice at $2.00 a quart is $20.00 (10x$2.00) and the cost of 5 liters of cola at $4.00 a liter is also $20.00 (5x$4.00).

 5.  

 

 

5.  Answer:  D

If x and y are both greater than 0, Column A is equal to Column B.  For example, let x=2 and y=3.  Then Öx2y2 = Ö2242 =Ö36 = 6 and xy = (2)(3) = 6, so Column A and Column B are equal.

Similarly, if x and y are both less than zero, the two columns are equal.

However, if, for example, x=-2 and y=3, then

Öx2y2 = Ö(-2)242 =Ö(4)(9) = Ö36 = 6 and xy = (-2)(3) = -6.  Now Column A is greater than Column B, and the answer is D.

 

 

6.  Let x > 1

 

(x-4)(x-6)

(x-8)(x-3)

 

6.  Answer:  A

 

Plug in a small value and a large value of x.  Let x = 2. 

Column A is (2-4)(2-6) = (-2)(-4) = 8.

Column B is (-6)(-1) = 6.

Now let x = 10.

Column A is (6)(4) = 24.

Column B is (2)(7) = 14. 

In both cases Column A is larger.

 

Extra:  Use algebra.

We can simplify both sides of the equality (or inequality, whatever the case may be).  Note that we can perform operations on both sides of the inequality as long as we don't multiply or divide both sides of the inequality by a negative number, since this will change the direction of the inequality sign.

 

Column A

Column B

(x - 4)(x-6)

(x-8)(x-3)

x2 - 10x - 24

x2 - 11x - 24

-10x

-11x

-10

-11

 

By simplifying algebraically, we see that Column A is larger since -10>-11.

 

(Note that we were able to divide both sides by x since the problem stated that x is positive.   Otherwise, for negative values of x the direction of the inequality sign changes.) 

 

7.   

            r + 23,885 = 100,000

            s + 24,221 = 100,000

 

r

s

 

7.  Answer:  A

If r + 23,885 = 100,000 then r = 76,115.

If s + 24,221 = 100,000 then s = 75,779.

Therefore, r > s so Column A is larger.

 

Important shortcut:

Since the values of both r and s can be calculated it is certain that D is not the answer.  However, it is unnecessary to do any calculations.  Since the number added to r to get 100,000 is smaller than the number added to s to get 100,000, r must be larger.  That is, r is closer to 100,000.

8.

 

Volume of cylinder with base area 4 and height 10.

Volume of cylinder with base area 5 and height 8.

 

 

 

8.  Answer:  C

 

Note that the volume of a cylinder is

V = pr2h where the area of the base is pr2 and the height is h.  The volume of the cylinder in the expression on the left is 4 x 10 and the volume of the cylinder in the expression on the right is 8 x 5.

 

9.  Let b = 1/2. 

 

b2 + b3 + b4

b

 

 

 

9.  Answer:  B

 

= 1/4 + 1/8 + 1/16

 

= 4/16 + 2/16 + 1/16

 

= 7/16

 

Since b = 1/2 = 8/16 the expression on the right is larger and the answer is B.

10.  It is 100 miles from the center of town A to the center of town B and 100 miles from the center of town B to the center of town C.

 

Distance from the center of town A to the center of town C

100 miles

 

 

10.  Answer:  D

Draw a picture

 

 

In the first sketch the distance from A to C is 200 miles.  In the second sketch the distance is less than 100 miles.

11.

 Let x > 1 and let y < 1.

 

 

 

 

 

11.  Answer:  D

 

Plug in.

 

Let x = 2.  Then the expression on the left is 4(2)+4(4)=24.  Let y=0.  Then the expression on the right is 4(0)+0(0)=0.  It appears that the expression on the left will be bigger.  Try an extreme value of y, such as -10.  Then we will have 4(-10)+(-10)(-10)=60.  Therefore the choice of x and y will affect the calculations.

 

12.  A pair of fair, six-sided dice are rolled.

 

Probability of getting a 7

Probability of getting doubles

 

 

 

12.  Answer:  C

 

There are six ways of rolling a seven:  (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).  There are also six ways of rolling doubles:  (1,1), (2,2), (3,3), (4,4), (5,5), (6,6).  Therefore, the number of ways of each event happening is equal, and since the total number of outcomes is the same, the probability of each event happening is equal. 

 

Looking back:  There are 36 ways of rolling two dice (1,2), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), ...  If there are six ways of rolling a seven, the probability of rolling a 7 is 6/36 = 1/6.  Similarly, if there are six ways of rolling doubles, the probability of rolling a double is 6/36 = 1/6.  Therefore the probabilities are equal.

 

13.  Given the following set of data:

 

1, 1, 1, 1, 2, 3, 3, 4, 6, 8

 

The median added to the mode

The average (arithmetic mean)

 

 

13.  Answer:  A

 

The median is the middle number when the numbers are arranged in order, as they are in the problem.  Since no one number is in the middle, the median is the average of the two middle numbers:  (2+3)/2 = 2.5.  The mode is the most common number, which is 1.  Therefore the sum of the median and the mode is 2.5+1=3.5.

 

The mean is the sum of all the numbers divided by the number of numbers.

(1+1+1+1+2+3+3+4+6+8)¸1 = 30¸10 = 3

This is less than 3.5 so the answer is A.

14. 

 

The area of a square with side 3 inches.

The area of an equilateral triangle with side 3 inches.

 

14.  Answer:  A

The area of a square with side 3 inches is given by the formula

A = s2.  Therefore the area of the square is 9 square inches.

The area of an equilateral triangle is given by the formula

A = (Ö3/4)s2.  Therefore the area of the triangle is (Ö3/4)x9»3.9.  Therefore the square has the larger area.

 

Shortcut:  Draw a picture.

 

Note that the triangle and the square both have side s, and the triangle unquestionably has a smaller area.

15.  Let x be an integer greater than zero and not equal to 2.

 

 

1/(2-x)

1/(2-x2)

 

15.  Answer:  D

 

Solution:  One way to approach these problems is to examine whether there is a solution such that the expression on the left is equal to the expression on the right.  Since the only difference in the two terms is the x and the x2 term, the two expressions are identical if x=1.  Then, if x is any other number, such as 3, the two expressions will be different, and it is unnecessary to determine which one will be larger.

16.  If the measure of Ðy is 20°, find the value of x.

Do not grid the degree sign.

 

16.  Answer:  40

Since we have supplementary angles, y+4x=180°

Þ 20 + 4x = 180

Þ 4x = 160

Þ x = 40

 

17.  A certain medication requires 1/8 of an ounce for every 20 pounds of body weight.  How much medication should be given to a person who weighs 140 pounds?

 

17.  Answer:  7/8 or .875

This can be set up as a ratio.  Let m represent the amount of medicine required for a person weighing 140 pounds. 

Then (1/8) /20 = m/140

 

20m = (1/8)x140

20m = 17 1/2

m = 17 1/2 ¸ 20

= 35/2 x 1/20

= 7/8

 

Decimals can make computation simpler.  Note that 1/8 = .125.

.125/20 = m/140

20m = .125 x 140

20m = 17.5

m = 17.5/20

= .875

 

Shortcut: 

(1/8) /20 = m/140

 

Since 20x7=140 it is only necessary to multiply (1/8)x7 to find m.

(1/8)x7 /20x7 = m/140

 

18.  If (2)(2)(3)(3)(x) = 72 and (2)(2)(7)y = 84, then (x)(y) =

 

18.  Answer:  6

Solve for x and y:

(2)(2)(3)(3)(x) = 72

Þ 36x = 72

Þ x = 2

 (2)(2)(7)y = 84

Þ 28y = 84

Þ y = 3

Therefore, (x)(y) = (2)(3) = 6

 

19.  The ratio of men to women in a group was 3 to 1.  After 20 more men arrive the ratio of men to women is now 8 to 1.  How many women are in the group?

 

19.  Answer:  4

Let w represent the number of women in the group.  If the ratio of men to women in the group was originally 3 to 1 that means that there were three times as many men as women, so the number of men was 3w.  Later, the ratio of men to women climbs to 8 to 1 so there are now 8w men.  Since the difference in the number of men that were in the group earlier and the number of men that are in the group now is 20, we can write

3w + 20 = 8w

20 = 5w

4 = w.

 

Extra:  It helps to check the answer to understand how this problem works.

If there are 4 women, then there were 12 men when the ratio was 3 to 1.  After 20 more men arrived, there were 32 men.  The ratio of men to women is now 32 to 4, or 8 to 1.
20.  If 6 is added to (3/8) of a number the result is half of the number.  What is the number?

 

20.  Answer:  48

Let N be the number.

Then (3/8)N+6=(1/2)N

6=(1/2)N-(3/8)N

6=(1/8)N

48=N

 

21.  Given xy = 144.  If x and y are integers and if 3 < x < y, what is the greatest possible value of y?

 

21.  Answer:  36

To find the largest value of y we look for the smallest value of x.  Since x>3, the smallest value of x that divides into 144 is 4, so y=(144/4)=36.

 

22.  How many numbers less than 100 are the product of two positive consecutive even integers?

 

22.  Answer:  4

Examples of pairs of consecutive even numbers are (2, 4), (4, 8), etc.

List of the product of two consecutive even integers:

2x4=8<100

4x6=24<100

6x8=48<100

8x10=80<100

Since any other pairs of consecutive even numbers have a product larger than 100, we have found all the numbers less than 100 that are the product of two positive consecutive even numbers.

 

23.  If x2-y2=40 and x-y=20, then x+y=?

 

23.  Answer:  2

Note that x2-y2=(x+y)(x-y) so it x2-y2=40 and (x-y)=20 then

40=(20)(x+y)

40/20=(x+y)

2=x+y

 

24.  If the average of p and q is 6 and the difference between p and q is 8, then pq=?

 

Student generated response________

 

24.  Answer:  40

The average of p and q = 6 implies that

(p+q)/2 = 6

Þ p+q = 12

If the difference of p and q is 8, then p-q=8.

Now there are two equations and two unknowns:

              p + q = 12

              p - q = 8

Adding the equations together yields

2p = 20

Þ p = 10

If p +q = 12, then if p=10, q=2 and pq = 10x2 = 20.

 

25.  How much less is the area of a rectangle with width 2 and length 10 than the area of a square with the same perimeter?

 

25.  Answer:  16

We need to find the area of the rectangle and the area of the square.  First we need to find the perimeter of the rectangle in order to find the perimeter of the square.  When we have the perimeter of the square we can then find the side of the square, from which the area of the square can be calculated.

 

The area of a rectangle:  A = lw = 2x10 = 20

Perimeter of a rectangle:  P = 2(l)+2(w) = 2(2)+2(10) = 20

Perimeter of square:  P = 20 = 4s Þ s = 6

Area of the square:  A = s2 62 = 36

The difference in the area of the square and the area of the rectangle:  36-20=16.