sat 6 full tests phần 8 doc

If the perimeter of a rectangle is four times the length of the rectangle, then the width of the rectangle is what percent of the length?. In a certain rectangle, the length is three tim

Solve the equation: 0.66X = 4 – 0.34X  X = 4

X

3 = 3 x 2 = 6

8 If X + 2Y = 24 and Y – 3X = 10, what is the value of X?

From the second equation, Y = 3X + 10, replace this with Y in the first equation: X + 2(3X + 10) = 24  X + 6X + 20 = 24  7X = 4  X=4/7

9 If X + Y = 15 and X – Y = -5, what is the value of X/Y?

Add the equations to get: 2X = 10  X = 5

Y = 15 – X = 10

X/Y = 5/10 = ½

10

The triangle in the figure above is not drawn to scale

If the measurement of angle 4 is 115.5o, what is the measurement of angles 1 and 2 (in degrees) ?

Notice that angles 3 and 4 are vertical angles and thus equal

The sum of the angles in the triangle is 180o and therefore we can write the following connection: angle 1 + angle 2 + 115.5o = 180o  the sum

of angle 1 and 2 is equal to (180 – 115.5 = 64.5o)

11 ABC is an isosceles triangle, AB = BC If the measurement of angle ABC is between 102 and 105, what is the value of the measurement of angle BCA minus CAB?

Draw a sketch of the triangle

Since the triangle is an isosceles, angles BCA = CAB and therefore the answer to the question is always zero no matter what the third angle is

12 If the sum of two numbers is 6 and their difference is 2, what is the square of their product?

Let X and Y be the two numbers

X + Y = 6 and Y – X = 2 are the two equations

 Y = 4 and X = 2

Their product is 8 and the square of their product is 82 = 64

Chapter 2:

1 If the product of two numbers is 9 and their difference is 0, what is their sum?

Let X and Y be the numbers

We can write the following equations: XY = 9 and Y – X = 0

Y = X  X2 = 9  X = Y = 3

The sum of the numbers is 3 + 3 = 6

2 If 645 = 22X, what is the value of X?

Rewrite the expression: 645 = 22X  (26)5 = 22X  230 = 22X

Compare the powers: 2X = 30  X = 15

3 If 1253 = 5Y, what is the value of

2

3Y

?

Rewrite the expression: 1253 = 5Y  (53)3 = 5Y  59 = 5Y

Compare the powers: Y = 9

The value of 3Y/2 = 27/2 = 13.5

4 If the perimeter of a rectangle is four times the length of the rectangle, then the width of the rectangle is what percent of the length?

Let W be the width and L the length of the rectangle

The perimeter of the rectangle is 2W + 2L

We can write the following connection: 2W + 2L = 4L  W = L

(square)

Therefore the width is 100% of the length and so the answer is 100

5 In a certain rectangle, the length is three times the width and the

perimeter is equal to 64 What is the value of the length of the rectangle?

We can write the following connections: L = 3W and 2L + 2W = 64 Replace L with 3W and write: 2(3W) + 2W = 64  8W = 64  W=8

L = 3 x 8 = 24

6 There are 50 blue balls and 120 red balls in a jar containing 170 balls only If only blue balls are to be added to the jar so that the probability of randomly picking a blue ball from the jar becomes 1/2, how many blue balls must be added to the jar?

Let X be the number of blue balls that must be added

We want the portion of the blue balls to be half of the entire amount of balls in the jar and therefore 50 + X (the new number of blue balls) divided be 170 + X (the entire number of balls) should be ½

70 170

2 100 2

The probability of drawing the first red marble is (15)/(45) = 1/3

The probability of drawing the second red marble is (14)/(44) = 7/22 The joint probability is the multiplication of the probabilities, and

therefore the answer is

66

7 22

7 3

While throwing a dice there are 6 results: 1, 2, 3, 4, 5 and 6

Only three results are over 3: 4, 5 and 6 and therefore the probability is 3 out of 6 or ½

9 The average (arithmetic mean) of 6 positive integers is 110 The value

of two of the integers is 24 and 28 and the other integers are greater than

6

33 32 31 28 24

10

If the sum of 4 consecutive numbers is 220, what is the average

(arithmetic mean) of the first and the last among those numbers?

Let x, x+1, x+2 and x+3 be the four numbers

We can write the equation: x + x + 1 + x + 2 + x + 3 = 220

 4x + 6 = 220  x = 52

The average arithmetic mean of the first and the last numbers is

(52 + 55)/2 = 53.5

11 What is the time elapsed from 12:12 to 23:43, in minutes?

Start from 12:12, add 11 hours to reach 23:12

Add 31 more minutes to reach 23:43

Altogether, its 11 hours and 31 minutes

In minutes its: 11 x 60 + 31 = 691 minutes

12 What is the angle between the large and the small hand of the clock at 12:30, in degrees?

At 12:30, the angle is not 180o since the hour hand (the small hand) rotated a bit clockwise Every hour the small hand of the clock moves 30oand so in 30 minutes, it moved 15o

The angle between the hands of the clock is (180 – 15 = 165) degrees

It might go easier if you draw a sketch of a clock

Chapter 3:

1 If X > 6 and X3X2.5XY = X8, what is the value of X?

These questions are only solved by comparing the powers of both sides,

in our case of X

X3X2.5XY = X5.5 + Y = X8  5.5 + Y = 8  Y = 2.5

2 If 2X+2 = 4X-1, what is the value of X?

These questions are only solved by comparing the powers of both sides

5 Nikki and Mike bought a new house for $120,000

Their families paid 42% of the price and the rest was divided equally and annually across six years How many thousand of dollars did Nikki and Mike pay each year?

Their families paid 42% of $120,000 and so all they had to pay

themselves is

(100% - 42% = 58%) of $120,000

0.58 x 120,000 = $69,600

Each year they would pay a sixth of that amount, thus (69,600/6 =

11,600) and so the answer is 11.6 thousands of dollars

6 A new computer costs a thousand dollars including tax If Travis paid for three quarters of his new computer every month for a year, how much did he spent each month assuming that the payments were equal?

Travis paid for only 75% of his computer, thus $750

He paid that price in 12 equal payments, each ($750 / 12 = $62.5) and so the final answer is 62.5

7

The following figure is of a parallelogram

What is the value of X + Y + Z ?

Look at the upper left triangle, the sum of the angles there should equal

180o and so X + Y + 115 = 180  X + Y = 65o

Since the shape is a parallelogram, Z = 115o and so X + Y + Z = 180

8

An isosceles triangle was attached to a rectangle

If X = 3.5 and Y = 1.5, what is the perimeter of the figure above?

The perimeter of the figure above is made from two sides of the triangle and three more sides of the rectangle, thus X + X + X + Y + Y = 3X + 2Y, which is also equal to 3(3.5) + 2(1.5) = 13.5

9 If X  Y  100 and Z2 X4 Y4 Z2 X5 Y7, what is one possible value for Y2?

Simplify the expression: Z2 X4 Y4 Z2 X5 Y7  1  X 1 Y3 

1 = (XY)Y2  Y2 = 1/(XY) = 1/100  Y2 = 0.01

10 If A A



 5 52

125 , what is one possible value for A?

Compare the powers of 5 in each side

< 54-A  3+A < 4-A  2A < 1 

A < 0.5 and so one possible value would be anything smaller than 0.5, for example 0.25 or 1/6

11 The volume of a cylinder is 3 cubic feet

The radius was increased by three times and the height was increased by

2 times, what is the new volume of the cylinder in cubic feet?

o

115

o X

o Y

o

Z

X

Y

When the radius is increased by 3, the area of the base of the cylinder increases by 9 times The new volume of the cylinder is 9 x 2 = 18 times greater than before and so the new volume is 3 x 18 = 54 cubic feet

12

What is the volume of water needed in order to fill a third of the box in the figure above?

Calculate the volume of the box: V = 12 x 4 x 7 = 336

A third of that volume is equal to 336/2 = 112

7

Chapter 4:

1 A, B and C are digits between 0 to 9

CA and AB are double digit numbers and ABA is a three digit number

What is the value of ABA?

In the tens digit, we can see that A + B = A and thus B=0

The sum of two double digit numbers is a three digit number and so its hundreds digit must be one, thus A=1

The number is therefore 101

2 A and B are digits between 0 to 9

A2 = 4B (4B is a double digit number)

?



 B

A

The only number squared with a tens digit of 4 is 7 (72 = 49)

And so A=7 and B=9

A x B = 7 x 9 = 63

3 X and Y are two digits between 0 and 9 When 36 is multiplied by another double digit number, the result is 3XY

What is one possible value for Y?

Start with an easy number, 36 times 10 = 360 add 36 to get 396

And therefore one answer is 0 and the other can be 6

4 If A and B are positive integers, A < 34 – B and A > 17, what is the greatest possible value of (2A – B)?

Since A + B < 34, take A as 32 and B as 1 and this way A will be the largest and B the smallest (2A + B) would be equal to (2 x 32 - 1) = 63

5 If 5 6 30Y , what is the value of Y?

Compare the powers from both sides of the equation

In the figure above, B is the center of the circle and A is the center of the square If the radius of the circle is 2, what is the area of the square?

The radius of the circle is actually half the diagonal of the square and so the diagonal is equal to 2 2

The ratio between the sides of a square to its diagonal is 1: 2 and so each of the sides of the square are equal to 2

The area of the square is simply 2 x 2 = 4

7 A chocolate box contains only white, sweet and bitter chocolate in the following ratio: 2:3:4 respectively The sweet chocolate is either with or with out walnuts, and 4 times as many sweet chocolate are with walnuts than with out If a chocolate is chosen at random, what is the probability that it would be a sweet chocolate with walnuts?

In this question it is smart to plug in numbers

Say that there are 90 chocolates in the box

According to the ratio, there are (3/9) x (90) = 30 sweet chocolates Since there are four times as many chocolates with walnuts as there are with out there are (4/5 x 30 = 24) walnut chocolate and (1/4 x 30 = 6) with out

The probability of pulling a sweet chocolate with walnuts is 24/90 = 4/15

or 0.266 or 0.267

8 The area of a certain rectangle (which is not a square) is 25 inches

squared, what is one possible length of its smaller side?

If the rectangle was a square each side would be 5 and so 5 x 5 = 25 Since the rectangle is not a square, the larger side is bigger than 5 and the smaller side is smaller than 5

The acceptable answers are 4, 3, 2 and 1

9 What is the area of a square if the sum of the diagonals is 24

We know (using the Pythagoras principle) that the ratio between the sides

of a square to its diagonal is 1:1: 2 and therefore in this specific square, each side is (12/ 2) and the area is 72

2

144 2

The value would be (7 – 3) = 4

12 If 6 < A < 20 and -12 < B < -5, what is the greatest possible value of

AB in absolute value?

Since B is negative and A is positive, AB will also be negative

If we take the most negative B and the largest A, the result would be the greatest in absolute value And so A = 20 and B = -12  AB = -240 and

so the answer is 240

Chapter 5:

1 If (X4 + 2X2 + X)(X2 + X) = aX6+bX5+cX4+dX3+eX2, what is the value

of

(c – a)?

Open the parenthesis to get: X6+X5+2X4+3X3+X2, and therefore

a=1 and c=2 and so (c – a) = 1

2 If (X2 + X)(X2 + X) = aX4+2X3+cX2+dX, what is the value of

a + b + c + d ?

Open the parenthesis to get: X4+2X3+X2 and so a=1, b=2, c=1 and d=0

a + b + c + d = 1 + 2 + 1 = 4

3

In the figure above, (x,y) is the coordinate of a point found in the middle

of the side of the square 2y-x =

Since the figure is of a square y=5

The coordinate is of a point in the middle of the side and therefore x=2.5 2y-x = 10 – 2.5 = 7.5

4

In the figure above, (x,y) is the intersection point of the two lines

What is the value of y/x?

In order to find the intersection point compare the functions of the line: 3x + 4 = -x +8  4x = 4  x=1 and y = 7

y/x = 7/1 = 7

y

x

) ,

( y x

) 0 , 5 (

4

3 

 x y

8





 x y y

x

5 A coin is marked with the number 8 on one side and the number 9 on the other What is the probability of receiving an odd number on the second toss?

The probability every single toss is ½ for 8 and ½ for 9

Therefore the probability of receiving an odd number on the second toss

is 0.5 or ½

6 A jar contains 4 red balls, 3 green balls and 7 blue balls

What is the probability of not drawing out a white ball?

Since all the balls are not white, there is not chance of pulling a white ball and the probability is therefore 1

This question checks if you know that 1 is the highest probability

One side of the rectangle is 5.5 and the larger side is 5.5 x 2 = 11

9 Every month Paul works 60 hours for 30 pounds per hour Due to cutbacks his wage was decreased to 20 pounds per hour How many additional hours would Paul have to work in order to make the same amount of money each month?

Every month Paul made 60 x 30 = 1,800 pounds

If he still wants to make the same amount of money he should work

B A

Solve this one from the end

Take a fraction where the numerator is half the denominator, for example 1/2 This fraction is not good; they both need to be over 4 and so this time take 5/10 If you subtract 3 from the denominator and the numerator, the fraction would be 2/7, which does fulfill the requirements

Therefore the original fraction is 2/7

11 Every hour Ana sneezes 5 sneezes more than Reese, and each one of them sneezes an equal number of times every hour If during a whole day both girls sneezed 360 times, how many times did Reese sneeze each hour?

Reese sneezed X times per hour and so Ana sneezed X+5 sneezes

In one hour, they sneezed X + X + 5= 2X + 5

In a whole day (24 hours), they sneezed 24(2X + 5) = 48X + 120

48X + 120 = 360  48X = 240  X=5 and this is the answer

12 When Tim was 8 he was two times older than Rick during that time

If today Tim is 14 years of age, how old is Rick today?

During the time that Tim was 8, Rick was half his age, thus 4

If today Tim is 14 years old (6 years later), Rick is also 6 years older, thus

10 and so the right answer is 10

Chapter 6:

1 If the average of three different positive integers is 120, what is the

smallest possible value of the median among the three numbers?

All the numbers are integers greater than 0

The smallest possible value of the numbers is 1 and the smallest possible value of the median is therefore 2, and that is the answer

The three numbers is 1, 2 and 357 the median is 2

2 If X2 = 2XY – Y2 + 14.5, what is the value of (X-Y)2?

The expression X2 = 2XY – Y2 + 14.5 can be written as

X2 – 2XY + Y2 = 14.5  (X-Y)2 = 14.5 and so the answer is simply 14.5

3 If 17XY = 34X + 51X, what is the value of (3.5)Y?

Simplify the expression: 17XY = 34X + 51X  17XY = 85X 

Divide by 17X  Y = 5 and therefore 3.5Y = 3.5 x 5 = 17.5

4

Note: Figure not drawn to scale

In the figure above, what is the value of A in degrees?

A, 17o and 97o are all vertical angles to the inner angles of the triangle and therefore equal The sum of the angles in the triangle is 180o and so

we can write the following equation: 180 = 97 + 17 + A  A = 180 – 97 – 17 = 66 degrees

5 The bank gave Elaine a loan with an interest of 8% on the original amount per month If Elaine loaned $1,525, how much interest will she pay over a period of three months (in dollars)?

Every month, there is an 8% interest on 1,525, which is 0.08 x 1525 =

$122 per month Elaine will pay 3 x 122 = $366 to the bank

6 If in a certain jar there are X+1 black marbles and 2X+2 white marbles, what is the probability of randomly pulling a white ball?