GMAT

Question 1

What are the 2 top tips for the GMAT?

Answer 1

• Spend extra time on the first 10 questions. They are absolutely crucial
• You are always better off making your best guess ⇒ the best guess can often be determined by which answer option that has the most in common with the rest (the most common answer theorem)

Question 2

What are the question types in the quantitative test and what share do they account for? (old test)

Answer 2

Data sufficiency questions = 40 %

Question 3

When are you trying to find a Yes/No answer for data sufficiency questions?

Answer 3

If the question asked starts with “is”, “are” or “does” you look for a yes/no answer rather than a number.

Question 4

What are integers?

Answer 4

Question 5

What are whole numbers?

Answer 5

Question 6

What are irrational numbers?

Answer 6

Numbers that canNOT be expressed as a fraction or ratio.

Examples: π and √2

Question 7

When do you know that the GMAT is testing you in imaginary numbers?

Answer 7

If they ask you to take any EVEN ROOT OF A NEGATIVE NUMBER.

√-100 or √-8 fx.

Question 8

What are the prime numbers up until 130

Answer 8

2, 3, 5, 7,

11, 13, 17, 19,

23, 29,

31, 37,

41, 43, 47,

53, 59,

61, 67,

71, 73, 79,

83, 89,

97,

101, 103, 107, 109,

113,

127.

Question 9

What does “absolute value” mean and how is it expressed?

Answer 9

A number’s absolute value is its distance from zero on the number line, and like any distance, its ALWAYS positive.

Absolute value of 3 is 3. Absolute value of -8 is 8.

Question 10

What is the order to work in maths?

Answer 10

PEMDAS

Parantheses

Exponents

Multiplication

Division

Addition

Subtraction

Question 11

What is (x + y)*(x + y)?

Answer 11

(x + y)^2

x^2 + 2xy + y^2

Question 12

What is (x - y)*(x - y)?

Answer 12

(x - y)^2

x^2 - 2xy + y^2

Question 13

What is (x + y)*(x - y)?

Answer 13

x^2 - y^2

Question 14

What is

Answer 14

Question 15

How do you multiply exponent together? fx: x^3 * x^5

Answer 15

X^3 * X^5

Simply add the exponents together: X^3 * X^5 = X^8

is the same as: (x*x*x) * (x*x*x*x*x) = x^3 * x^5 = X^8

Question 16

How do you divide exponents together? fx: y^3 / y^5

Answer 16

Simply subtract the denominator from the numerator.

y^3 / y^5 = (y3 - y5) = y^-2

Question 17

What is the value of one exponent raised to another exponent? such as: (x^6)^3

Answer 17

Just multiply the two exponents

(X^6)^3 = x^18

Question 18

What is 355^0?

Answer 18

1

ALL real numbers raised to the power of 0 = 1

Question 19

What is x^0?

Answer 19

1

ALL real numbers raised to the power of 0 = 1

Question 20

What is 3^-4?

Answer 20

which is = 1 / 3^4

which is 1 / 81

Question 21

What can be said about “any number greater than 1 raised to a negative exponent”?

Answer 21

It will be a number greater than 0 but less than 1.

Question 22

What is a number raised to a fractional power? such as X^⅙

Answer 22

ALL real numbers raised to the power of 0 = 1

X^⅙ =

Question 23

What is the result of 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^2, 2^9, 2^10

Answer 23

Question 24

What is the result of 3^2, 3^3, 3^4, 3^5?

Answer 24

Question 25

What is the result of 5^2, 5^3, 5^4?

Answer 25

Question 26

What is the result of 6^2, 6^3?

Answer 26

Question 27

What is the result of 7^2, 8^2, 9^2, 10^2, 11^2, 12^2, 13^2, 14^2, 15^2, 16^2?

Answer 27

7^2 = 49

8^2 = 64

9^2 = 81

10^2 = 100

11^2 = 121

12^2 = 144

13^2 = 169

14^2 = 196

15^2 = 225

16^2 = 256

Question 28

What is the difference between a finding the factors of a number and finding the prime factors? The processes?

Answer 28

Factors = all numbers can be used

Prime factors = only prime factors can be used

Question 29

What is basis points?

Answer 29

It is 1 of 100.

0.45 % = 45 basis points.

Question 30

What percent is 600 of 200?

Answer 30

300 %

but percentage increase from 200 to 600 is 200%

Question 31

What is the percentage increase from 300 to 600?

Answer 31

100 %

but 600 is 300 % of 200.

Question 32

How can you easily simplify a complex fraction? such as 210/2,772

Answer 32

Divide by 2 if possible, then 3, then 5, then 7

(210 / 2) / (2772 / 2) = 105 / 1386

(105 / 3) / (1386/3) = 35 / 462

(35 /7) / (462/7) = 5 / 66

Question 33

How do you know if a number is divisible by a prime number?

Answer 33

Take the number, fx. 2772

then write it out as 2+7+7+2 = 14

Any prime number going to 14 will work = either 2 or 7.

Question 34

What are the six main rules to remember in terms of exponents and roots?

Answer 34

Answer:

Question 35

Question:

Answer 35

Answer:

(3* 7) + (7* 3) = 21 +21 = 42 = D.

Makes sense, since you have 3 / (1 / 7), as 1/7 naturally goes into 1 seven times. And 3 is in the numerator, so it has to go up to three (which is three times as much as 1, i.e. 3 * 7 = 21).

The same logic can be applied for the other part.

Question 36

Question:

Answer 36

Answer:

64^k is the same as (4^3)^k

So following the rule of multiplying exponents when one exponent is raised to another, we simply have to figure out what:

3* k > 14

k > 14/3

k > 4,67

K must be B = 5.

Question 37

Question:

Answer 37

answer:

the prime numbers less than 10 are: 2, 3, 5 , 7

2*3 = 6 so cannot be A

Nothing can be multiplied to B as it has to be DISTINCT numbers = B

2*5 = 10

2*7 = 14

3*5 = 15

Question 38

What are the rules of the cube root table and how does the table look?

Answer 38

1. Take the last digit in a number: for example 6 in 636,056 and find that digit in the last digit column.
2. Look at the three first digits in a number and find the digit in the table where they do NOT exceed the cube. For example 636 in 636,056.

So cube root of 636,056 would be 86

Question 39

What are the solutions to the equation x^2 + -2x = 15?

Answer 39

First, re-order to separate 0.

x^2 - 2x - 15 = 0

Now factor (-15)

• 1 and 15 1 and -15
• 3 and 5 3 and -5

Question 40

Which pair can you manipulate to get to -2?

Answer 40

-1 + 15 = 14 so no

1 -15 = -14 so no

-3 + 5 = 2 so no

3 - 5 = -2 so YES

(x - 3) * (x + 5)

x + 3 = 0, then x = -3

x - 5 = 0, then x = 5

Question 41

What is the median of 105, 106, 107, 108, 109, 110, 111, 112, 113?

Answer 41

The middle value which in this range of 9 values will be the 5th value = 109.

Question 42

What is the median of 105, 108, 110, 112, 114, 115?

Answer 42

6 different numbers so the median is the average of the 3rd and 4th number.

(110 + 112) / 2 = 111

Question 43

What is the mode?

Answer 43

The most common element in a set

NOTE!!! A set can have more than one mode.

NOTE NOTE!!! A set can also NOT have a mode if no number occurs more than once

Question 44

What is the range?

Answer 44

The difference between the largest and the smallest number.

Question 45

How is the standard deviation calculated? and how is it useful to know?

Answer 45

The standard deviation is the square root of the result of the summation of the squares of the differences between the individual values of the set and the mean, divided by the number of items in the set.

You will not have to calculate this on the GMAT.

But… you might be asked which set of numbers has the largest standard deviation.

Question 46

What is the formula for calculating the third side of a right triangle?

Answer 46

A^2 + B^2 = C^2 (pythagoras)

Question 47

What are parallel lines?

Answer 47

Lines that if extended to infinity, would never intersect.

Question 48

How do you calculate the area of a triangle?

Answer 48

½ * Base * Height

Question 49

How do isosceles triangles look and what are their properties?

Answer 49

• Two sides of equal length
• The two angles opposite the sides of equal length are also of equal degree

Question 50

How do equilateral triangles look and what are their properties?

Answer 50

• All sides of equal length
• All angles have 60 degrees

Question 51

How do you calculate the area of an equilateral triangle?

Answer 51

(√3 / 4) * s^2

(1. 732 / 4) * s^2
0. 433 * s^2

Question 52

How do right triangles look and what are their properties?

Answer 52

• One angle is 90 degrees
• The third side (hypotenuse) can be calculated by A^2 + B^2 = C^2

Question 53

How do right isosceles triangles look and what are their properties?

Answer 53

• one angle with 90 degrees
• Two sides of equal length

Question 54

What is the 3-4-5 triangle type?

Answer 54

A common ratio used on the GMAT.

RIGHT triangles with the ratio of 3,4 and 5.

• A is 3, B is 4 then C = 5
• A is 6, B is 8, then C = 10
• A is 9, B is 12, then C = 15
• A is 12, B is 16, then C = 20

Question 55

What is the 5-12-13 triangle type?

Answer 55

A common RIGHT triangle ratio:

• A is 5, B is 12, then C = 13
• A is 10, B is 24, Then C = 26

Question 56

What is the 30 degrees, 60 degrees and 90 degrees triangle?

Answer 56

The 1 - √3 - 2

• If A is 1, B is √3, then c = 2
• If A is 5, B is 5*√3, then c = 10
• If A is 8, B is 8* √3, then c = 16

Question 57

What is the perimeter?

Answer 57

How far there is around a triangle, square etc.

The sum of the sides.

Question 58

How do a rhombus look and what are their properties?

Answer 58

A rhombus is a quadrilateral with sides of equal length and two sets of opposite angles with equal measures.

Question 59

How do a parallelogram look and what are their properties?

Answer 59

Two sets of parallel lines

The two sets have different length

Question 60

How do you calculate the area of a parallelogram?

Answer 60

If you recognize that it is simply a square and two triangles of equal size, the calculation simply become

AREA = BASE * HEIGHT

Question 61

How do trapezoids look and what are their properties?

Answer 61

One pair of parallel lines

The two other sides have equal length but are not parallel

Question 62

How do you calculate the area of a trapezoid?

Answer 62

Recognize that a trapezoid is a square with two equal triangles.

AREA = HEIGHT * ((A + C) / 2)

Thus we just take the height and the average of the length of the two parallel lines. Then it is essentially just as calculating a rectangle.

Question 63

How do you calculate the circumference?

Answer 63

The circumference is the distance around a circle (what we call a perimeter in a polygon).

It is simply:

Diameter * π

or: 2 * r * π

Question 64

What is the area of a circle?

Answer 64

A = r^2 * π

Question 65

How do you calculate the volume and surface area of a right circular cylinder?

Answer 65

Question 66

How do you calculate the volume and surface area of a square pyramid?

Answer 66

Question 67

How do you calculate the volume and the surface area of a cube?

Answer 67

Question 68

How do you calculate the volume and the surface area of a rectangular solid?

Answer 68

Question 69

How do you calculate the volume of a general cone/pyramid?

Answer 69

For a right circular cone, you would obviously find A by taking r^2*π

Question 70

How do you calculate the volume and surface area of a sphere?

Answer 70

Question 71

What is the result of “even*even”?

Answer 71

Even

4*4 = 16

Question 72

What is the result of “even*odd”?

Answer 72

Even

4*5 = 20

Question 73

What is the result of “odd*odd”?

Answer 73

odd

13*13 = 169

Question 74

What is the result of “even + even”?

Answer 74

Even

22 + 36 = 58

Question 75

What is the result of “odd + odd”?

Answer 75

Even

17 + 19 = 38

Question 76

What is the result of “odd + even”?

Answer 76

odd

21 + 22 = 43

Question 77

What is the result of “negative * negative”?

Answer 77

Positive

-4 * -5 = 20

Question 78

What is the result of “positive * negative”?

Answer 78

Negative

5 * -6 = -30

Question 79

What is the result of “positive + negative”?

Answer 79

It depends.

on which one is biggest.

Question 80

When an odd number is raised to an exponent, what will it result in?

Answer 80

It will still be an ODD number.

13*13 = 169.

No matter if it is raised to 2, 3, 4, 5, 6 etc., it will always be an odd number

Question 81

When an even number is raised to an exponent, what will it result in?

Answer 81

It will still be an EVEN number.

12*12 = 144.

No matter if it is raised to 2, 3, 4, 5, 6 etc., it will always be an even number

Question 82

What is the result of “non-integer * non-integer”?

Answer 82

It will always be a NON-INTEGER

0.5 * 0.5 = 0.25.

Question 83

What is the result of “integer * non-integer”?

Answer 83

It depends.

Question 84

What is the key component of combinatorial math and what is the formula used?

Answer 84

Factorials.

n! = n* (n - 1) * (n- 2) * - (n - 3) * ….

So

• 2! = 2* 1 = 2
• 3! = 3 * 2 * 1 = 6
• 4! = 4 * 3 * 2 * 1 = 24
• 5! = 5 * 4 ‘ 3 * 2 *1 = 120
• 6! = 6* 5 * 4* 3 * 2 ‘ 1 = 720
• 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040

Question 85

What is the formula for combinations and what does combinations refer to?

Answer 85

Combinations regards groupings where ORDER DOES NOT MATTER

Question 86

What is the formula for permutations and what does permutations refer to?

Answer 86

Permutation regards groupings where ORDER MATTERS.

Question 87

How do you multiply fractions? for example 4/9 x 3/2?

Answer 87

4/9 * 3/2

4*2 / 9*2 = 12/18 = 2/3

Question 88

What should you ask yourself for inference questions?

Answer 88

• If the statements in the passage are true, what else has be true?
• When having found potential answers, ask yourself “what if the opposite of this answer were true?”

ALWAYS Pick the moderate statement over the extreme.

Question 89

What are the three main ways in which an answer can be wrong for sentence correction questions?

Answer 89

It violates a rule of grammar

It is worded in an unclear way

It is worded in a “nonstandard” way; i.e. it sounds funny

Question 90

Which words should you look for to avoid being trapped by pronouns errors?

Answer 90

• He
• She
• It
• they
• them

Ask yourself whether

1. is it clear who the referent is?
2. is it the right pronoun for that referent?

Question 91

What are some of the most frequent quantity-words used in the GMAT?

Answer 91

Question 92

What are some of the most frequent countable/non-countable words used in the GMAT?

Answer 92

Countable words should be used when you can normally assign a whole number to it. Such as six cars.

Non-countable items are for example a little soup as you cannot quantify “a little soup”.

A cup of soup, of course, would be countable = 1.

Question 93

What is the most common error in sentence correction questions?

Answer 93

No error at all.

20 % has no errors at all.

Question 94

What are 3 points that should raise an alarm for sentence correction questions?

Answer 94

A sentence does not have to be wrong when it includes any of the following but it often will be:

• Any answer that includes “being”
• Passive verb forms when active verbs would work just as well
• Words ending in “-ing” when there are simpler choices

Question 95

If you have to give your best guess for sentence correction questions, which two rules should you follow?

Answer 95

1. Pick the shortest answer
2. Pick the “most common” answer.
   
   1. The one that has the most in common with the other options

Question 96

What are good examples of phrases in which the writer CONTINUES his/her line of thought?

Answer 96

Question 97

What are good examples of phrases in which the writer CHANGEs his/her line of thought?

Answer 97

Question 98

How should your essay outline ideally look?

Answer 98

• Paragraph 1: INTRODUCTION
  
  • State argument in a clear statement
  • Acknowledge that the opposite position has merits, but that for the following X reasons (summarize your points), my position is the correct one
• Paragraph 2: REASON 1, with supporting evidence/examples/facts
• Paragraph 3: REASON 2, with supporting evidence/examples/facts
• Paragraph 4: REASON 3, with supporting evidence/examples/facts
• Paragraph 5: CONCLUSION
  
  • For the X reasons previously stated, my argument is the correct one
  • The argument would have been more persuasive if it ______.

Question 99

What does bisect mean?

Answer 99

Divide (a line, angle, or shape) into TWO EXACTLY EQUAL parts.

XB bisects ABY only if it splits ABY into two exactly equal parts, meaning that there must be a 45 degree angle on both sides (given that ABY is a 90 degree angle).

Question 100

How can you derive the side of a square from its diagonal?

Answer 100

The side of a square is it’s diagonal divided by 2 (derived from pythagoras)

Question 101

How do you quickly add to fractions with different denominators?

Answer 101

Multiply the numerator of the first fraction with the denominator of the second fraction, which is equal to the numerator of the new fraction.

The same applies for the second part, multiply the numerator of the second fraction with the denominator of the first fraction, which is equal to the numerator of the new second fraction.

Lastly, you multiply the two denominators, which is equal to the new common denominator for the two new fractions.

Now you can just add them together normally.

Question 102

How do you quickly subtract two fractions with different denominators?

Answer 102

Same methods as how you add them, you just subtract instead when you get the common denominator.

Question 103

How do you multiply two fractions?

Answer 103

Question 104

How do you divide a number (or fraction) with a fraction?

Answer 104

Question 105

Answer 105

Question 106

Answer 106

Question 107

Answer 107

Question 108

Answer 108

Question 109

Answer 109

Question 110

What does “place value” refer to?

Answer 110

Question 111

What are inequalities in math?

Answer 111

Statements that involve “<” or “>”.

Question 112

What is a “perfect square”?

Answer 112

Question 113

What is a “perfect cube”?

Answer 113

Question 114

What is LCD and what is the LCD of two numbers?

Answer 114

Question 115

What are the results of (-3)^2 and -3^2 ?

Answer 115

(-3)^2 = -3 * -3 = 9

-3^2 = - 3*3 = - 9

Question 116

How can you simplify exponent expression so they are easier to work with?

Answer 116

For negative exponents, you can always divide that expression by one and change the negative exponent to a positive.

Also, you can always move exponents from the numerator to denominator or vice versa by changing the sign.

6. = 1 / 2^3 = 1 / 8
7. ) (3^3 * 1) / 1 = 9 / 1 = 9

Question 117

When do you make ADDITION and when do you make MULTIPLICATION of exponents?

Answer 117

Question 118

How do you add or subtract terms when the bases are different? (working with exponents)

Answer 118

Question 119

How can you rewrite the √5^12 ?

Answer 119

Question 120

How do you multiply and divide square roots?

Answer 120

Question 121

What should you do when you get to a result that is not an answer option when working with square root results such as √50 or √12?

Answer 121

Question 122

How do you ADD or SUBTRACT under the square root symbol?

Answer 122

Do NOT get confused or fall in the trap of using the rules from multiplication and division under the square root symbol.

Instead…

Question 123

What is 7^6 * 7^9?

Answer 123

7^15

when multiplying roots you can just add the roots together (when the base is the same)

Question 124

What is (a^3)^2?

Answer 124

A^6

when multiplying roots you can just add the roots together (when the base is the same)

Question 125

What is (3^2)^-3 ?

Answer 125

3^-6

Question 126

What is 5^5 / 5^3?

Answer 126

5^2 = 25

When dividing roots, you can subtract them as long as the base is the same

Question 127

What is 11^4 / 11^x ?

Answer 127

11^(4-x)

Question 128

what is (5^2)^x ?

Answer 128

25^x OR 5^2*x

Question 129

what is (5^6 * 5^4x) / 5^4 ?

Answer 129

We can simplify to:

5^2 * 5^4x

And then simplify to:

5^(4x + 2)

Question 130

What is y^7 * y^8 * y^-6?

Answer 130

y^7+8-6 = y^9

Question 131

What is y^4 / y^-3?

Answer 131

Y^(4-(-3) = y^7

Question 132

What is (3^2x * 3^6x) / 3^-3y?

Answer 132

3^8x / 3^-3y

3^8x + 3^3y OR …. 3^(8x + 3y)

Question 133

What is 27^⅓ + 9^½ + (3/9^0)

Answer 133

3 + 3 + 3/1 = 9

Question 134

What is -3^4?

Answer 134

-3 * -3 * -3 * -3 = -81

Question 135

Which of the following 5 options are equal to (⅖)^-3?

Answer 135

1. -(⅔)^3
2. (⅖)^⅓
3. (- ⅖ )^3
4. (5/2)^⅓
5. (5/2)^3

The answer is option 5 such that (⅖)^-3 = (5/2)^3.

You can shift around the numerator and denominator IF you also change the plus/minus sign in the root.

Question 136

What is 8^3 * 2^6?

Answer 136

(2^3)^3 * 2^6

2^9 * 2^6

2^9+6

2^15

Question 137

What is 25^4 * 125^3?

Answer 137

(5^2)^4 * (5^3)^3

5^8 * 5*9

5*17

Question 138

What is 9^-2 * 27^2?

Answer 138

(3^2)^-2 * (3^3)^2

3^-4 * 3^6

3^-4+6

3^2

9

Question 139

What is 6^3 + 3^3?

Answer 139

6^3 + 3^3

(2 * 3)^3 + 3^3

2^3 * 3*3 + 3*3

3^3 (2^3 + 1)

3^3 (9)

3^3 (3^2)

3^5

Question 140

What is (4^8 - 8^4) / (2^4 + 4^2)?

Answer 140

(4^8 - 8^4) / (2^4 + 4^2)

((2^2)^8 - (2^3)^4 ) / ( 2^4 + (2^2)^2 )

(2^16 - 2^12) / (2^4 + 2^4)

(2^12 (2^4 - 1)) / (2^4 (1+1))

(2^12 (15)) / (2^4 (2))

(2^12 (15)) / 2^5

2^7 * (15)

Question 141

What is √5 * √45?

Answer 141

√5*45

√225

15

Question 142

What is √3 * √27?

Answer 142

√3*27

√81

9

Question 143

what is √5000 / √50?

Answer 143

√5000/50

√100

10

Question 144

What is (√54 * √3) / √2 ?

Answer 144

√54 * 3 / √2

√168 / √2

√81

9

Question 145

What is √32?

Answer 145

Divide up 32…..

32 = 2 * 16

√16 * √2

4 * √2

Question 146

what is √180?

Answer 146

Divide up 180

2 * 90

3 * 60

4 * 45

5 * 36

The easiest is 5 * 36

√5 * √36

√5 * 6

Question 147

What is √135?

Answer 147

Divide up 135

3 * 45

6 * 22.5

9 * 15

√9 * √15

3 * √15

Question 148

What is √35^2 - 21^2?

Answer 148

Find a common ground…

35 = 5 * 7….. and 21 = 3 * 7

√7^2 (5^2 - 3^2)

√7^2 (25-9)

√7^2 (16)

√7^2 (4^2)

7 * 4

28

Question 149

What is √6 * (5^6 + 5^7)?

Answer 149

√6* (5^6(1+5))

√6 * (5^6 (6))

√6^2 * 5^6

6 * 5^3

6 * 125

750

Question 150

what is √(11^4 - 11^2)/ 30?

Answer 150

(11^2 (121 - 1)) / 30

(11^2 (120)) / 30

11^2 * 4

11^2 * 2^2

11 * 2

22

Question 151

What is √5^7 - 5^5 + 5^4?

Answer 151

√5^4(5^3 - 5^1 + 5^0)

√5^4 (125 - 5 + 1)

√5^4 (121)

√5^4 (11^2)

5^2 * 11

25 * 11

275

Question 152

What is the reciprocal?

Answer 152

The number (B) for any number (A) that makes A*B = 1.

5 = Reciprocal is ⅕ as 5*⅕ = 1

⅔ = reciprocal is 3/2 as ⅔ * 3/2 = 1

Question 153

what is the reciprocal of -⅚?

Answer 153

The reciprocal will be the opposite so switching:

6/-5

(-5 / 6) * (6 / -5) = 1

Question 154

How do you divide something by a fraction?

Answer 154

You simply multiply by that fraction’s reciprocal = you simply flip the numerator and denominator of that fraction.

Question 155

What must you be aware of if there is addition or subtraction in the fraction’s NUMERATOR?

Answer 155

You MUST divide with the ENTIRE denominator. Thus, you have to find common factors and simplify.

You can split up into two fractions

Question 156

What must you be aware of if there is addition or subtraction in the fraction’s DENOMINATOR?

Answer 156

You are NOT allowed to split up into two fractions in the same way. You ALWAYS have to divide the numerator by the ENTIRE denominator.

Question 157

What is a/12 - b/6 - b/4?

Answer 157

First, find a common denominator

a/12 - 2b/12 - 3b/12

next, add it together:

(a - 5b) / 12

Question 158

What is √3/2 - √2/3?

Answer 158

Question 159

Answer 159

Question 160

Answer 160

Question 161

Answer 161

Question 162

Answer 162

Question 163

Answer 163

Question 164

Answer 164

Question 165

Answer 165

Question 166

Answer 166

Question 167

Answer 167

Question 168

Answer 168

Question 169

Answer 169

Question 170

Answer 170

Question 171

Answer 171

Question 172

Answer 172

Question 173

Answer 173

Question 174

Answer 174

Question 175

What is 0.0004 * 0.032?

Answer 175

Calculate it as… 4 * 34 = 128.

Next… count the numbers AFTER the decimal.

0.0004 * 0.032 (7 numbers)

Now move the decimal point 7 to the left from 128.00

0.0000128

Question 176

What is 80 % of 75 % of 120?

Answer 176

8/10 * ¾ * 120

8/1 * ¾ * 12

4/1 * 3/1 * 12

4 * 3 * 12

72

Question 177

What is 72.12 * 10^-4?

Answer 177

Move the decimal 4 times to the left.

72.12

=

0.007212

Question 178

what is 12.6 / 0.3?

Answer 178

Multiply both the numerator and denominator by 10 to get

126 / 3

42

Question 179

what is 2.10 * 0.08?

Answer 179

Take it as

21 * 8 = 168

Now count the numbers AFTER the decimal points.

2.1 * 0.08 (3 numbers).

Move the decimal 3 to the left.

168.00 becomes 0.168.

Question 180

What is 0.49 / 0.07?

Answer 180

Multiply both the numerator and denominator by 100 to get

49 / 7

7

Question 181

What is 1.5 / ( ⅝ - 50 %)?

Answer 181

1.5 / (0.125)

Rewrite to fractions:

3/2 / ⅛

Use the reciprocal rule:

3/2 * 8/1 = 24/2 = 12

Question 182

What is x when 30 % less than x is 63?

Answer 182

0.7x = 63

7x = 630

630/7 = 90

Question 183

What percentage decrease is x IF 75 - x % is 54?

Answer 183

75 - x/100 *75 = 54

75-54 = 75x/100

21 = 75x/100

21 = 3x/4

7 = x/4

4*7 = 28 = x

28 %.

Question 184

10 % of 30 % of what number is 200 % of 6?

Answer 184

200 % of 6 = 12

10 % of 30 % = 3 % of x

3% of x = 12

100/3 * 12 = x

100 * 4 = x

x = 400.

Question 185

If there are 24 white marbles and 36 marbles in a bag, what is the ratio of blue to white marbles?

Answer 185

White = 24 = 2/5

Blue = 36 = 3/5

Total = 60 = 5/5

BLUE 3 to WHITE 2

3:2

Question 186

Arjun has a record of winning 3 tennis matches for every 2 he loses. If he played 30 matches, how many did he win?

Answer 186

Win 3 = 3:5

Lose 2 = 2:5

Total 5 = 5:5

If 30 matches, he would win ⅗ * 30 = 18 matches.

Question 187

What is x and y IF -2x + y = 1 AND 2(x - 3y) = 2

Answer 187

2(x - 3y) = 2

becomes

2x - 6y = 2

We can now eliminate x by adding the two equations together

2x - 6y = 2

• 2x +y = 1
• 5y = 3

y = -⅗

and x can be found substituting the number for y in to any of the equations.

• 2x + -⅗ = 1
• 2x = 1 ⅗
• 2x = 8/5
• x = ⅘

x = - ⅘

Question 188

If 4^y = 8^y+1, then 2^y is what?

Answer 188

4^y = 8^y+1

Find a common base…

(2^2)^y = (2^3)^y+1

when having a common base we can now simply set the exponents equal to each other and solve…

2y = 3(y+1)

2y = 3y + 3

y = -3

so… 2^y = 2^-3 = ⅛

Question 189

If (2x + 6(9-2x)) / (x-4) = -3, what is x?

Answer 189

First, multiply both sides with the denominator such that…

2x + 6(9 - 2x) = -3(x-4)

2x + 54 - 12x = -3x + 12

• 7x = -42
• x = -6

x = 6

Question 190

If a/c + b/3c = 1, what is “c” in terms of a and b?

Answer 190

a/c + b/3c = 1

First get a common denominator

3a / 3c + b/3c = 1

Now multiply the denominator to both sides

3a + b = 1*3c

3a + b = 3c

c = a + b/3

Question 191

What is the value of P when c = 100 and P = 300*c^2 - c?

Answer 191

P = 300 * 100^2 - 100

P = 300* 10,000 - 100

P = 3,000,000 - 100

P = 2,999,900

Question 192

What is ((√3x + 1) / 2) - 1 = 3. what is x?

Answer 192

multiply by 2 to remove the denominator.

√3x + 1 -2 = 6

√3x + 1 = 8

square both sides

3x + 1 = 64

3x = 63

x = 21

Question 193

What is x if 13 / (x + 13) = 1

Answer 193

13/(x + 13) = 1

13 = 1(x + 13)

13 = x + 13

x = 0

Question 194

2h - 4k = 0 and

k = h -3

Answer 194

2h - 4(h - 3) = 0

2h - 4h + 12 = 0

12 = 2h

h = 6

k = h - 3

k = 6 - 3 = 6

Question 195

What is b and g if…12b = 2g and 4g - 3b = 63?

Answer 195

12b = 2g

6b = g

4g - 3b = 63

substitute g with 6b…

4*6b -3b = 63

24b - 3b = 63

21b = 63

b = 3

then g will be…

6b = g

g = 6*3 = 18.

Question 196

If y - 2x - 1 = 0

and… x -3y - 1 = 0

what is the values of x and y?

Answer 196

First, rewrite the two equations for them to be as similar as possible…

y - 2x = 1

-3y + x = 1

now multiply the second equation with 2 to get an equal value of x for the two equations.

2(-3y + x) = 1*2

-6y + 2x = 2

Now add the two equations to eliminate x.

y - 2x = 1

• 6y + 2x = 2
• 5y = 3

y = - ⅗

2x = 1 + y

2x = 1 + 3/5

2x = 8/5

x = 4/5

Question 197

If 2p = (m - 5) / (n +2), what is m in terms of n and p?

Answer 197

We need to isolate m…

2p (n+2) = m - 5

2pn + 4p + 5 = m

Question 198

What is the value of z if… 2^x+y = √(z - 2) and… x = 2 - y

Answer 198

2^x+y can be re-written as we know that x = 2 - y

2^(2-y)+y

so…

2^(2-y)+y = √(z - 2)

2^2 = √(z - 2)

Now we square both sides…

(2^2)^2 = z - 2

16 = z - 2

z = 18.

Question 199

Rewrite x^2 + 14x + 33

Answer 199

To rewrite, we want it in the form…

(x ….)(x….).

to figure out what should be in the parentheses besides x, we have to look at the number “33” and figure out its factors AND find a set of factors that can yield 14 as well.

The factors of 33 are: 1 and 33. 3 and 11.

1 and 33 can never become 14.

3 and 11 can be added to become 14 so that is the set of factors we want.

(x + 3) (x + 11) = x^2 + 14x + 33

Question 200

what is x IF x^2 - 15x = - 26?

Answer 200

first, recognize that there most likely will be two different solutions as we have x^2.

x^2 - 15x = -26

x^2 -15x + 26 = 0

(x - 2) (x - 13) = 0

x - 2 = 0, then x = 2

x - 13 = 0, then x = 13.

X is either 2 or 13.

Question 201

What is x IF x^3 - 2x^2 = 3x?

Answer 201

First recognize that we might have three different solutions as x^3.

Now factor out x.

x^3 - 2x^2 = 3x

x^3 - 2x^2 - 3x

x (x^2 - 2x - 3)

x ( (x - 3) (x + 1)) = 0

x = 0, then x = 0

x - 3 = 0, then x = 3

x + 1 = 0, then x = -1

Question 202

What is ( x^2 + 7x + 12) / (x + 3) = ?

Answer 202

Rewrite the numerator

x^2 + 7x + 12

((x + 3) (x + 4)) / (x + 3)

We can remove (x + 3) from both numerator and denominator

We get…

(x + 4)

Question 203

what is y IF …. y^2 - 11y + 30 = 0?

Answer 203

y^2 - 11y + 30 = 0

(y - 5) (y -6) = 0

y - 5 = 0, then y = 5

y - 6 = 0, then y = 6

Question 204

what is x IF…. x^3 - 5x^2 + 4x = 0?

Answer 204

x^3 - 5x^2 + 4x = 0

x (x^2 - 5x + 4) = 0

x ( (x - 1)(x - 4) ) = 0

x = 0, then x = 0

x - 1 = 0, then x = 1

x- 4 = 0, then x = 4

Question 205

what can you reduce the following expression to? (a^2 - b^2) / (a - b)

Answer 205

(a^2 - b^2) / (a - b)

Write out the numerator…

((a + b) ( a - b)) / ( a- b)

Reduce…

(a + b).

Question 206

What can you reduce the following expression to? (2t - 1 + (2t-1)^2 ) / (2t - 1)

Answer 206

(2t - 1 + (2t-1)^2 ) / (2t - 1)

write out the numerator

(2t - 1 + ((2t-1)(2t-1))) / (2t - 1)

now factor out (2t - 1)…

(2t-1) (1 + (2t - 1)) / (2t-1)

Reduce…

(1 + (2t - 1))

1 + 2t - 1

2t…

Question 207

What can you reduce the following expression to? …. (3x^2 - 6x - 45) / (3x + 9)

Answer 207

(3x^2 - 6x - 45) / (3x + 9)

factor the numerator / rewrite…

((3x + 9)(x - 5)) / (3x+9)

Reduce…

(x - 5)

Question 208

What can you reduce the following expression to? (5ab + abc) / (abc^2 + 10abc + 25ab) ?

Answer 208

(5ab + abc) / (abc^2 + 10abc + 25ab) ?

First factor out “ab”.

(ab (5 + c)) / (ab (c^2 + 10c + 25))

reduce…

(5 + c) / (c^2 + 10c + 25)

rewrite the denominator…

(5 + c) / ((c + 5)(c + 5))

Reduce…

1 / (5 + c)…

Question 209

What is the solution to |−x - 4| > 8 ?

Answer 209

|−x - 4| > 8

• x -4 > 8 -(-x) - (-4) > 8
• x > 12 x + 4 > 8

x < -12 x > 4

Question 210

what is the solution to 2(x-1)^3 + 3 < 19 ??

Answer 210

2(x-1)^3 + 3 < 19

2(x-1)^3 < 16

(x-1)^3 < 8

x-1 < 2

Question 211

what is the solution to…. 2 |x + 0.32| = 7

Answer 211

2 |x + 0.32| = 7

x + 0.32 = 3.5

x + 0.32 = 3.5 -x - 0.32 = 3.5

x = 3.18 -x = 3.82

x = -3.82

Question 212

What is the solution to…. |x/4 + 3| = 0.5

Answer 212

x/4 + 3 = 0.5 -x/4 - 3 = 0.5

x/4 = -2.5 -x/4 = 3.5

x = -10 -x = 14

x = -14

Question 213

What is the solution to…. |x^3| < 64

Answer 213

x^3 < 64 -x^3 < 64

x < 4 -x < 4

x > -4

Question 214

A hose is placed into an empty pool and turned on at 2:00 pm. the pool, which holds 680 gallons of water, reaches its capacity at 5:24 pm. How many gallons of water per hour did the hose add to the pool?

Answer 214

680 gallons

Time: 5:24pm - 2:00pm = 3 hours and 24 minutes.

680g / 3:24h

680g / 204m = 3 ⅓

3 ⅓g/m * 60 = 200 gallons of water per hour

Question 215

40 students in a class of 200 got A’s on their test. 64 got B’s, 18 got D’s and 6 got F’s. If students can only get A, B, C, D or F as grades, what percent of the students got C’s?

Answer 215

a = 40

b = 64

c = ?

d = 18

e = 6

t = 200

40 + 64 + 18 + 6 + c = 200

c = 72

72 / 200 = 36 %.

Question 216

The temperature in Limerick is ¾ that in Cairo, where the temperature is 8/5 that in Halifax. If the temperature in Limerick is 66 degrees, what is the temperature in Halifax?

Answer 216

L = 66

L = ¾ C

C = 8/5 H

H = ?

L = ¾ (8/5H)

L = 24/20H

L = 6/5H

66 = 6/5H

66* ⅚ = H

H = 55

Question 217

At a convention of monsters, ⅖ have no horns, 1/7 have one horn, ⅓ have two horns, and the remaining 26 have three or more horns. How many monsters are attending the convention?

Answer 217

⅖ no horns

1/7 one horn

⅓ two horns

26 three or more horns

⅖ + 1/7 + ⅓

35/105 + 15/105 + 35/105 = 92/105

105/105 - 92/105 = 13/105

13/105*m = 26

m = 26 * 105/13

13m = 26*105

13m = 2730

m = 210

Question 218

Of all the homes on Gotham Street, ⅓ are termite-ridden, and ⅗ of these are collapsing. What fraction of the homes are termite-ridden, but NOT collapsing?

Answer 218

Ft = ⅓

Ft+c = ⅓ * 3/5 = 3/15

Ft-c = Ft - Ft+c

Ft-c = ⅓ - 3/15

Ft-c = 5/15 - 3/15 = 2/15

Question 219

A plane leaves Chicago in the morning and makes three flights before returning.The first flight has traveled twice as far as the second flight, and the second flight traveled three times as far as the third flight. If the third flight was 45 miles, how many miles was the first flight?

Answer 219

F1 = 2*F2

F2 = 3*F3

F3 = 45

F1 = ?

F1 = 2(3*F3)

F1 = 6*F3

F1 = 6*45

F1 = 270 miles.

Question 220

Arnaldo earns $11 for each ticket that he sells, and a bonus of $2 per ticket for each ticket he sells over 100. If Arnaldo was paid $2,400, how many tickets did he sell?

Answer 220

Q2 * P + Q2(P+2) = 2400

100*11 + Q2(11+2) = 2400

1100 + Q2(13) = 2400

Q2*13 = 1300

Q2 = 1300

Q = Q1 + Q2

Q = 100 + 100 = 200.

Question 221

Amar is 30 years younger than Lauri. In 5 years, Lauri will be three times as old as Amar. How old will Lauri be in 10 years?

Answer 221

A = L - 30, L = A + 30

L in + 10y?

L + 5 = 3A

L+5 = 3(L - 30 +5)

L+5 = 3L - 75

80 = 2L

40 = L

L + 10 = ?

40 + 10 = 50.

Question 222

What is the temperature in Fahrenheit when it is 30 degrees Celsius? C = 5/9(F - 32)

Answer 222

C = 30

C = 5/9(F - 32)

30 = 5/9(F-32)

270 = 5F - 160

430 = 5F

86 = F

Question 223

Joe’s car can travel 36 miles per gallon of fuel. Approximately how many kilometers can the car travel on 10 liters of fuel? (5 miles = approximately 8 kilometer; 1 gallon = approximately 4 liters).

Answer 223

36m / g * 8km /5m = 288km / 5gallons

288km/5gallons * 1gallon/4liters

288km / 20liters

14. 4 kilometers/l
15. 4 km/l * 10 liters = 144 kilometers.

Question 224

Svetlana is running a 10-kilometer race. She runs the first 5 kilometers of the race at a speed of 12 kilometers per hour. At what speed will she have to fun the last 5 kilometers of the race if she wants to complete the 10 kilometers in 55 minutes?

Answer 224

Distance = Rate * Time

5 = 12 * t

5 = 12t

t = 5/12

55 minutes * 1hour/60minutes = 55/60 = 11/12

11/12 - 5/12 = 6/12 = 1/2

The last 5 kilometers will be….

Distance = Rate * Time

5 = r * 0.5

r = 10 km/t.

Question 225

What is arc length?

Answer 225

A section of circle’s circumference.

Question 226

what is central angle?

Answer 226

The angle created by any two radii..

Fx. the angle determining what big a share a sector is of a whole circle

Question 227

What is sector area?

Answer 227

A “wedge” of the circle, composed of two radii and the arc connecting these two radii.

Question 228

What is the hypotenuse?

Answer 228

The longest side of a RIGHT triangle. The hypotenuse is opposite the right angle.

Question 229

What are the 5 pythagorean triplets?

Answer 229

8-15-17

5-12-13 and double up: 10-24-26

3-4-5 and double up: 6-8-10

Question 230

True or false? The point (4,14) is on the curve y = x^2 - 2.

Answer 230

The equation y = 3x + 4 is already in y = mx + b form.

For the point 4, 14. to be on the curve, you can plug in x = 4 into the equation.

y = 4^2 - 2

y = 16 -2

y = 14.

Which is the y-coordinate we were given so the answer is YES.

Question 231

A sector has a central angle of 270 degrees. If the sector has a radius of 4, what is the area of the sector?

Answer 231

270/360 = ¾ of a circle.

The circles Radius is 4.

So TOTALarea = 4^2*π = 16π

16π * ¾ = 12π.

Question 232

A sector has a radius of 8 and an area of 8π. What is the arc length of the sector?

Answer 232

SectorRadius = 8

SectorArea = 8π

TOTALarea = 8^2*π = 64π

8π / 64π = ⅛.

TOTALcircumference = D*π = (2*8)*π = 16π

SECTORarclength = 16π * ⅛ = 2π

Question 233

A sector has an arc length of π/2 and a central angle of 45 degrees. What is the radius of the sector?

Answer 233

SECTORarclength = π/2

SECTORcentralangle = 45

SECTORradius?

45/360 = ⅛. = the sector is ⅛ of the circle

So… π/2 is ⅛ of the circumference of the circle.

8π / 2 = 4π = TOTALcircumference

C = D * π

C = (2*r)*π

So the squareroot(4)*π = 2π.

SECTORradius = 2.

Question 234

Two sides of a triangle have lengths 4 and 8. Which of the following are possible side lengths of the third? a = 2, b = 4, c = 6, d = 8

Answer 234

8+4 = 12

8-4 = 4

4 < x < 12

only c = 6 and d = 8 works.

Question 235

Answer 235

diameter of the circle = 9.

Area of circle = r^2 * π

Area of circle = 4.5^2 * π = 20.25π

Question 236

Answer 236

A = 49

A = S*S

S = squareroot(49) = 7.

A^2 + B^2 = C^2

7^2 + 7^2 = 98

squareroot of 98 = C

98 can be broken down… square root of 98 = squarerootof( 2*7*7) = 7*squareroot(2)

Question 237

Does the point (3, -2) lie on the line y = 2x - 8?

Answer 237

Plug in x = 3 and see if it results in y = -2.

y = 2*3 - 8 = -2.

YES…

Question 238

For the line y = 4x +2, what is the y-coordinate when x = 3?

Answer 238

Plug in x = 3 and see…

y = 4*3 + 2 = 14.

Question 239

In the coordinate plane, a circle has center (2, -3) and passes through the point (5, 0). What is the area of the circle?

Answer 239

A = r^2 * π

A = (2 - 5)^2 + (-3 + 0)^2 * π = (9 + 9)π = 18π

Question 240

What is (0.0036*2.8) / ( (0.04)*(0.1)*(0.003))?

Answer 240

(0.0036*2.8) / ( (0.04)*(0.1)*(0.003))

(36*10^-4) * (28*10^-1) / ( (4*10^-2)*(1*10^-1)*(3*10^-3))

(36*28 / 4*1*3) * 10^(-4-1)-(-2-1-3)

36/3 * 28/4 * 10^-5-(-6)

12*7*10^1 = 840.

Question 241

What is a quotient and a remainder?

Answer 241

When x and y are positive integers, there exist unique integers q and r, called the quotient and remainder:

y = x*quotient + remainder, and 0 < r < x

Question 242

What is the quotient and remainder of y =29 when x =3?

Answer 242

y = x*quotient + remainder

29 = 9*3 + 2

Question 243

Is 0 positive or negative?

Answer 243

It is NEITHER positive or negative. It is just 0.

Question 244

Are 8/36 and 14/63 equivalent?

Answer 244

Yes…

they are EQUIVALENT as both represent 2/9

Question 245

What are mixed numbers?

Answer 245

Numbers that consist of both a whole number and a fraction.

fx… 7⅔

7⅔ = 23/3

Question 246

What is I x + y I equal to?

Answer 246

I x + y I < I x I + I y I

as… if x = 10, y = 2

10+2 = 10 +2 = 12 < 12

and…

if x = 10 and y = -2

then…

10 - 2 = 10 + 2 = 8 < 12

Question 247

What is the formula for calculating the PERCENTAGE increase/decrease?

Answer 247

Take…

(New number - Original number) / Original number

(30 - 24) / 24 = 6 / 24 = 0.25 = 25 %

Question 248

What is 0!, 1!, 2! and 3!?

Answer 248

0! = 1 (by definition 0! = 1)

1! = 1

2! = 1*2 = 2

3! = 1*2*3 = 6

Question 249

What are the permutations of A, B and C?

Answer 249

Essentially how many ways can you order A, B and C?

The answer is “3!”

3(n-1)(n-2) = 3*2*1 = 6

Question 250

If you have the letters, A, B, C, D and E…, how many combinations can you make when only choosing 2 items at a time?

Answer 250

n = 5

k = 2

5! / (2! (5-2)!)

5! / (2! * 3!)

120 / 2*6

120 / 12

10 combinations

Question 251

What is P(a) = 0.3 + P(b) = 0.42 IF 1) P(a) and P(b) are mutually exclusive? and 2) if P(a) and P(b) are independent?

Answer 251

P(a or b) = P(a) + P(b) (where A and B are mutually exclusive)

0.3 + 0.42 = 0.72 = 72 %

P(a or b) = P(a) + P(b) - P(a)*P(b) (where A and B are independent events)

0.3 + 0.4 - 0.3*0.42 = 0.7 - 0.126 = 0.574 = 57.4 %

Question 252

What are the solutions to x(x-3)(x^2+5) / x-4?

Answer 252

A fraction equals 0 if and only if its numerator equals 0.

Thus… x(x-3)(x^2+5) = 0

x=0, x-3=0 or x^2+5=0

x=0 or x=3 but there is no solution to x^2+5=0 as x^2 + 5 > 0 for all real numbers.

Thus, the two only solutions are x=0 and x=3.

Question 253

what is the formula used to find the roots of quadratic expressions and when is it relevant?

Answer 253

When you cannot factor a quadratic expression, you can always find its roots with the following formula..

Question 254

What is the rules regarding lengths of 45-45-90 degrees triangles and 30-60-90 triangles?

Answer 254

Question 255

If n = 33^43 + 43^33, what is the units digit of n?

Answer 255

Both numbers are raised to 3.. 33 and 43 so they will follow the sequence of 3,9,7,1,3,9,7,1… as 3^1 = 3, 3^2 = 9, 3^3 = 27 and 3^4 = 81.

thus…. the 43rd number will be 7 and the 33rd number will be 3.

7+3 = 10 = 0

Question 256

If x is a positive integer greater than 1, what is the value of x ?

2x is a common factor of 18 and 24.

x is a factor of 6.

Answer 256

The answer is A.

X > 1, then 2*x = > 4 AND has to be a common factor of 18 and 24.

The factors of 18 are… 1,2,3,6,9,18 and the factors of 24 are 1,2,3,4,6,8,12,24

The only common factor that is >4 is 6.

2x = 6 and then x = 3

2) X is a factor 3 AND greater than 1. It can be either 2, 3 or 6.

Question 257

If a triangle is circumscribed by a circle such that one of its sides is a diameter of the circle, what can be said about that triangle?

Answer 257

That it will be a right triangle.

Question 258

How do you calculate the slope of a line using two coordinates?

Answer 258

Question 259

On a 400-mile trip, Car X traveled half the distance at 40 miles per hour and the other half at 50 mph. What was the average speed of car x?

Answer 259

1. First determine the total traveling time

200 miles / 40 miles per hours = 5 hours

+

200 miles / 50 miles per hour = 4 hours

=

9 hours total

400 miles / 9 hours = 44 4/9 miles per hour.

Question 260

If machine X can produce 1,000 bolts in 4 hours and Machine Y can produce 1,000 bolts in 5 hours, in how many hours can Machines X and Y, working together at these constant rates, produce 1,000 bolts?

Answer 260

1/r + 1/s = 1/h

1/machine x + 1/machine y = 1/combined time

¼ + ⅕ = 1/h

5/20 + 4/20 = 1/h

9/20 = 1/h

9h = 20

h = 20/9

h = 2 2/9

Working together, Machines X and Y can produce 1,000 bolts in 2 hours and 2/9 hours.

Question 261

How many liter of a solution that is 15 % salt must be added to 5 liters of a solution that is 8 % salt so that the resulting solution is 10 % salt?

Answer 261

0.15n + 0.08(5) = 0.10(n+5) (multiply both sides by 10)

15n + 40 = 10n + 50

5n = 10

n = 2

So add 2 liters and we get a total of 5+2= 7 liters, which now has a salt percentage of 10.

Question 262

The price of an item is discounted by 20 % and then this reduced price is further discounted by an additional 30 %. These two discounts are equal to an overall discount of what percentage?

Answer 262

The discounted price is…

0.8p

After second discount…

0.7*0.8p = 0.56p.

1-0.56 = 0.44 = 44 % total discount.

Question 263

Each of 25 people is enrolled in history, mathematics or both. If 20 are enrolled in history and 18 are enrolled in mathematics, how many are enrolled in both history and mathematics?

Answer 263

History = 20-n

Both = n

Mathematics = 18-n

(20 - n) + n + (18 - n) = 25

38 - n = 25

-n = -13

n = 13

Question 264

In a certain production lot, 40 % of the toys are red and the remaining toys are green. Half of the toys are small and half are large. If 10 % of the toys are red and small, and 40 toys are green and large, how many of the toys are red and large?

Answer 264

Question 265

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p ?

Answer 265

Question 266

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine’s average speed for the entire trip?

Answer 266

Question 267

A worker carries jugs of liquid soap from a production line to a packing area, carrying 4 jugs per trip. If the jugs are packed into cartons that hold 7 jugs each, how many jugs are needed to fill the last partially filled carton after the worker has made 17 trips?

Answer 267

Carrying 4 jugs per trip, the worker carries a total of 4(17) = 68 jugs in 17 trips. At 7 jugs per carton, these jugs will completely fill 9 cartons with 5 jugs left over since (9)(7) + 5 = 68. To fill the 10th carton, 7 – 5 = 2 jugs are needed.

Question 268

How many prime numbers between 1 and 100 are factors of 7,150 ?

Answer 268

To find the number of prime numbers between 1 and 100 that are factors of 7,150, find the prime factorization of 7,150 using a method similar to the following:

7,150=10×715=(2×5)×(5×143)=2×5×5×(11×13)7,150=10×715=(2×5)×(5×143)=2×5×5×(11×13)

Thus, 7,150 has four prime factors: 2, 5, 11, and 13.

Question 269

Team A and Team B are competing against each other in a game of tug-of-war. Team A, consisting of 3 males and 3 females, decides to line up male, female, male, female, male, female. The lineup that Team A chooses will be one of how many different possible lineups?

Answer 269

Any of the 3 males can be first in the line, and any of the 3 females can be second. Either of the 2 remaining males can be next, followed by either of the 2 remaining females. The last 2 places in the line are filled with the only male left followed by the only female left. By the multiplication principle, there are 3 × 3 × 2 × 2 × 1 × 1 = 36 different lineups possible.

Question 270

In an electric circuit, two resistors with resistances x and y are connected in parallel. In this case, if r is the combined resistance of these two resistors, then the reciprocal of r is equal to the sum of the reciprocals of x and y. What is r in terms of x and y ?

Answer 270

Question 271

In Country C, the unemployment rate among construction workers dropped from 16 percent on September 1, 1992, to 9 percent on September 1, 1996. If the number of construction workers was 20 percent greater on September 1, 1996, than on September 1, 1992, what was the approximate percent change in the number of unemployed construction workers over this period?

Answer 271

Question 272

If n = 38 − 28, which of the following is NOT a factor of n ?

Answer 272

1. 97
2. 65
3. 35
4. 13
5. 5

Question 273

Car 1 starts from point A and heads for point B at 60 mph. Fifteen minutes later, car 2 leaves the same point A and head for point B at 75 mph. How long before car 2 overtakes car 1?

Answer 273

Question 274

Two planes leave the same point at 8 am. Plane 1 heads EAST at 600 mph and Plane 2 heads West at 450 mph. How long will they be 1400 miles apart? and how long did each plane travel?

Answer 274

Question 275

Two planes leave the same point at 8 am. Plane 1 heads EAST at 300 mph and Plane 2 heads West at 550 mph. How long will they be 3200 miles apart?

Answer 275

Question 276

Car 1 starts from point A and heads for point B at 40 mph. Twenty minutes earlier, car 2 left the same point A and headed for point B at 30 mph. How long before car 1 overtakes car 2?

Answer 276

Question 277

A boat can go 10 miles upstream in 30 minutes. The return trip downstream takes only 20 minutes. what is the speed of the boat in still water and the speed of the current in mph?

Answer 277

Question 278

What is “ - l -13 -(-17)l?

Answer 278

It will be…. - l-13 +17l

which is… - l-4l

Question 279

What is the sum of x, y, and z when… x+y=8, x+z=11 and y+z=7?

Answer 279

Write it up…

x + y = 8

x + z = 11

y + z = 7

2x + 2y + 2z = 26

so….

x + y + z = 13

Question 280

What are the solutions to…. 22 - ly+14l = 20?

Answer 280

22 - l y + 14 l = 20

that is… - ly + 14l = -2

which is.. l y + 14l = 2

and…

y + 14 = -2 or…. y + 14 = 2

y = - 16 or… y = - 12

Plug in to check…

l-16 + 14l = 2 AND l-12 + 14l = 2

Ok.

Question 281

What is the solutions to a, b and c when….

a + b = 10

b + c = 12

a + c = 16

Answer 281

We know that…

2a + 2b + 2c = 38

so…

a + b + c = 19

then…

19 - 10 = 9 = c

19 - 12 = 7 = a

19 - 16 = 3 = b

Question 282

If x = 4^20 + 4^21 + 4^22, what is the largest prime factor of x?

Answer 282

First factor out the common factor…

x = 4^20(1+4^1 + 4^2)

x = 4^20(1+4+16)

x = 4^20(21)

x = 4^20(3*7)

so the biggest prime factor is 7.

Question 283

What is (4/9)^(-3/2) ?

Answer 283

(4/9)^(-3/2)

remove the negative sign in the exponent by flipping

(9/4)^3/2

Take the square root of numerator and denominator and then exponent 3

3^3/2^3 = 27 / 8

Question 284

What is (4^y+4^y+4^y+4^y)(3^y+3^y+3^y)?

Answer 284

Factor…

4^y(1+1+1+1)*3^y(1+1+1)

4^y(4)*3^y(3)

4^y+1 * 3^y+1

(4*3)^y+1

12^y+1

Question 285

If m and n are positive integers, what is m when… (2^18)(5^m) = (20^n)?

Answer 285

2^18 * 5^m = (2*2*5)^n

2^18 * 5^m = 2^2n * 5^n

now just set the exponents equal to each other…

18 = 2n and… m = n

n = 9 = m = n

Question 286

What is the difference between x^2 = 16 and x = squareroot of 16?

Answer 286

x^2 = 16 has two solutions: -4 and 4.

Squareroot(16) only has one solution = 4. the positive solution.

Question 287

What is √63 + √28?

Answer 287

They can translate into…

√9*7 + √4*7

3*√7 + 2*√7

5*√7

Question 288

What is √150 - √96?

Answer 288

They must be rewritten….

√6 * 25 - √6 * 16

5*√6 - 4√6

1*√6

√6

Question 289

What is D when… d/4 + 8/d + 3 = 0?

Answer 289

First…

d + 4*8 / d = -3*4

d^2 + 32 = -12d

d^2 + 12d - 32 = 0

(d + 8) (d - 4) = 0

d = -8

d = 4

Question 290

Given that (p-3)^2 - 5 = 0, what is p?

Answer 290

(p - 3)^2 - 5 = 0

(p-3)^2 = 5

√(p-3)^2 = √5

l p-3 l = √5

p = 3 +/- √5

Question 291

What is… (3 - √7)(3 + √7)?

Answer 291

9 - 3*√7 + 3*√7 - 7 = 2

Question 292

Hugo lies on top of a building, throwing pennies straight down to the street below. The formula for the height, H, that a penny falls is H = V*t+ 5*t^2, where V is the origi­nal velocity of the penny (how fast Hugo throws it when it leaves his hand) and t is equal to the time it takes to hit the ground. The building is 60 meters high, and Hugo throws the penny down at an initial speed of 20 meters per second. How long does it take for the penny to hit the ground?

Answer 292

H = V*t + 5*t^2

60 = 20t + 5*t^2

5*t^2 + 20*t - 60 = 0

5(t^2 + 4t - 12) = 0

5/t+6)(t-2) = 0

t = -6 or t = 2.

T must be positive so t = 2

Question 293

What is z when z^2 - 10z + 25 = 9?

Answer 293

z^2 - 10z + 25 = 9

(z+5)^2 = 9

√(z+5)^2 = √9

z+5 = 3

z = 5 +/- 3

Question 294

If.. w ? F = (√F)^w, what is 4 ? (3 ? 16)?

Answer 294

First take the parantheses…

(3 ? 16) = (√16)^3 = 4^3 = 64

now… 4 ? 64

(√64)^4 = 8^4 = 4,096

Question 295

By which factor does t*b^4 increase, when we double b?

Answer 295

t*b^4 vs t*2b^4

tb^4

tb^4 = t*(2b(^4) = t(2b*2b*2b*2b) = t*16b^4 = 16tb^4

16 times bigger.

Question 296

If Sn = 15n - 7, what is S7 - S5?

Answer 296

S7 = 15*7 - 7 = 14*7 = 98

S5 = 15*5 - 7 = 75 - 7 = 68

98-68 = 30

Question 297

An = 2*An-1 - 4 and A6 = -4, what is A4?

Answer 297

We know A6 and the formula for calculating An so we plug in and first find A5 and then A4 working our way backwards…

A6 = 2*A5 - 4

-4 = 2*A5

0 = 2*A5

A5 = 0

Knowing that A5 = 0, we can find A4….

A5 = 2*A4 - 4

0 = 2*A4 - 4

4 = 2*A4

2 = A4

Question 298

If Sn = 3^n, what is the units digit of S65?

Answer 298

3, 9, 27, 81

Digit 65 will be a 3.

Question 299

If A @ B = 4A - B, what is (3 @ 2) @ 3?

Answer 299

Take the parentheses first…

(3 @ 2) = 4*3 - 2 = 10

Now…

10 @ 3 = 4*10 - 3 = 40 - 3 = 37

Question 300

If A€B = A^2 + B^2 + 2AB, what is a+b if A€B = 9?

Answer 300

We know that…

A^2 + B^2 + 2AB = 9

(A+B)(A+B) = 9 or… (-A-B)(-A-B) = 9

(A+B)^2 or (-A - B)^2

so… A+B = 3 or -3

Question 301

Life expectancy is defined by the formula (2SB)/G, where S=shoe size, B = avg. electric bill and G=GMAT score. If Melvin’s GMAT score is twice his electric bill, and his life expectancy is 50, what is his shoe size?

Answer 301

L = (2SB)/G

and we know that L=50 and that 2*B = G

so…

50 = (2*S*B) / (2*B)

50 = S

Shoe size = 50.

Question 302

The competitive edge of a baseball team is defined by the formula √(W/L), where w = #of wins and L = # of losses. This year, the GMAT all-stars had 3 times as many wins as last year and one-half as many losses as they had last year. By what factor did their competitive edge increase?

Answer 302

√(W/L) = √(1) last year

√(3W/0.5L) = √(6) this year

So essentially it increased by √6

Question 303

If An = 3 - 8n, what is A1?

Answer 303

Simple…

A1 = 3 - 8*1 = -5

Question 304

If An = 3 - 8n, what is A11-A9?

Answer 304

An = 3 - 8n.

then…

A11 = 3 - 8*11 = -85

A9 = 3 - 8*9 = -69

Difference is -16.

Question 305

If An = (An-1 * An-2) / 2, and A5 = -6, and A6 = -18, what is A3?

Answer 305

We can work our way backwards from the information given…

First, we find A4…

• 18 = (-6 * A4) / 2
• 36 = -6 * A4

A4 = -36 / -6 = 6

knowing A4, we can find A3….

A5 = (A4 * A3) / 2

• 6 = (6 * A3) / 2
• 12 = 6 * A3

A3 = -12 / 6 = -2

Question 306

If Sn = (4^n) + 3, what is the units digit of S100?

Answer 306

First we have to know the pattern for the roots of 4…

4^1 = 4

4^2 = 16

4^3 = 64

4^4 = 256

4^100 will have a units digit of 6.

So we just have to add 3.

6+3 = 9.

Question 307

The first term in an arithmetic sequence is -5 and the second term is -3, what is the 50th term?

Answer 307

We know that the change is +2.

we have the first two.. so 50-2 = 48..

48*2 = 96.

-3 + 96 = 93 will be the 50th term.

Question 308

If 1 > 1 - A*B > 0, which of the following must be true?

A/B > 0

A/B < 1

A*B < 1

Answer 308

First rewrite…

1 > 1 - A*B > 0 (subtract 1 from all)

0 > -A*B > -1 (multiple by -1)

0 < A*B < 1

so… A*B is between 0 and 1. thus they must both be positive and

A/B > 0 = true

A/B < 1, may be true but we dont know if a or b is biggest

A*B < 1 is true

Question 309

Is m*n < 10?

1. m < 2
2. n < 5

Answer 309

Each of the two information pieces is insufficient.

together they are still insufficient as it could be two negative numbers such as -2*-6 = 12.

Question 310

What is the solution to… l x+2 l < 5 ?

Answer 310

-5 < x < 5

“+2” by “-2 on both”

-7 < x < 3

Now -2 is the centerpoint.

Alternatively look at it this way..

x+2 < 5

x < 3

• x-2 < 5
• x < 7

x > -7

Question 311

What is the solutions to… l x-4 l < 3

Answer 311

-3 < x < 3

“-4” by adding 4 to both

1 < x < 7

4 is now the centerpoint.

alternatively..

x-4 < 3

x < 7

• x -(-4) < 3
• x < -1

x > 1

Question 312

what is the simplified version of… -3x + 7 < 2x + 32

Answer 312

• 3x + 7 < 2x + 32
• 25 < 5x
• 5 < x

Question 313

What can be said about G, if G^2 < G?

Answer 313

then G has to be:

0 < G < 1

Question 314

If 4x - 12 > x + 9 which of the following must be true?

1. x>6
2. x<7
3. x>7
4. x>8
5. x<8

Answer 314

4x - 12 > x + 9

3x > 21

x > 7

So …. x>6 is the only option for sure.

Question 315

If 0 < ab < ac, is a negative?

1. c < 0
2. b > c

Answer 315

A must be negative for negative C*A to be positive. 1 is sufficient.

if b > c, and a*b < a*c, then b must be a negative number that is less negative than c and a must be negative for both to turn positive numbers for the multiplications.

D is the answer. Each is sufficient.

Question 316

What should you immediately imply from x*y > 0

Answer 316

X and y are both positive OR both negative

Question 317

What should you immediately imply from x*y < 0

Answer 317

x and y have different signs

Question 318

What should you immediately imply from x^2 - x < 0

Answer 318

x^2 < x so… 0 < x < 1

Question 319

Is…. (-¾)^3 > -¾ ?

Answer 319

The answer is yes as ¾ < 1 and thus becomes closer to 0 when taken to any root.

Question 320

What can be said if… B^3 * A < 0 and A > 0?

Answer 320

We know that B must be a negative number as B^3 must be negative and anything raised to an uneven power will keep its sign.

Question 321

What is a when (b+a)/2a = 2, and a+b = 8.

Answer 321

Plug in a+b = 8.

8/2a = 2

8 = 4a

a = 2

Question 322

Eco wildlife preserve contains 5x zebras and 2x lions, where x is a positive integer. If the lions succeed in killing z of the zebras, is the new ratio of zebras to lions less than 2:1?

1. z > x
2. z = 4

Answer 322

What we are asked is….

(5x - z) / 2x < 2/1 ?

5x - z < 2*2x

5x - z < 4x

-z < -x

z > x is really what is asked.

1. z > x answers this and is sufficient
2. z = 4, we do not know x and cannot answer.

the answer is A.

Question 323

A retailer sells only radio and clocks. If she currently has 44 total items in inventory, how many of them are radios?

1. The retailer has more than 28 radios in inventory
2. the retailer has less than twice as many radios as clocks in inventory

Answer 323

1. tells us that R > 28 but is not sufficient on its own
2. tells us that C+R = 44 and… 1.5R = 44 so… 3R = 88 and R = <29,3333. Not sufficient.

Together we know that 28 < R < 29.333 so R must be 29.

Question 324

If x and y are nonnegative integers and x + y = 25, what is x?

1. 20x + 10y < 300
2. 20x + 10y > 280

Answer 324

So essentially just make x+y=25 to 10x+10y=250

Now use the information pieces

1. 20x + 10y < 300 - 10x + 10y = 250 = 10x < 50, so x < 5
2. 20x + 10y > 280 - 10x + 10y = 280 = 10x > 30, so x >3

together we can say that 3 < x < 5 s and x is an integer so it must be 4.

Question 325

If a/b = 16 and a/b^2 = 8, what is a*b?

Answer 325

start by simply dividing the first equation by the second, you will see that b =2

(a/b) / (a/b^2) = 16/8

(a/b) * (b^2/a) = (a*b^2)/(ab) = b = 16/8 = 2

So a/2 = 16, then a = 32.

and a*b = 32*2 = 64.

Question 326

What is x when… x^2 + 8x + 13 = 0?

Answer 326

The problem here is that there are no two integers for which the sum is 8 and the product is 13 so we cannot factor it.

Instead we must use the quadrateic formula..

x = (-8 +/- √(8^2 - 4*1*13)) / 2*1

x = (-8 +/- √(64 -52)) / (2)

x = 4 +/- (√12)/2

x = (-4 +√3, -4 - √3)

Question 327

What does the discriminant in the quadratic formula tell us?

Answer 327

The discriminant in the quadratic formula is…

√(b^2 - 4*a*c)

1. If √(b^2 - 4*a*c) > 0, there will be two solutions
2. If √(b^2 - 4*a*c) = 0, there will be one solution
3. If √(b^2 - 4*a*c) < 0, there will be no solution

Question 328

Which of the following has no solution for x?

1. x^2 - 8x - 11 = 0
2. x^2 + 8x + 11 = 0
3. x^2 + 7x + 11 = 0
4. x^2 -6x + 11 = 0
5. x^2 - 6x - 11 = 0

Answer 328

Here we must determine this from the discriminant from the quadratic formula, namely

√(b^2 - 4*a*c)

1. x^2 - 8x - 11 = 0 …… √(-8^2 - 4*1*-11) = √64-(-44) = √108 > 0 = two solutions
2. x^2 + 8x + 11 = 0… √(8^2 - 4*1*11) = √64-44 = √20 > 0 = two solutions
3. x^2 + 7x + 11 = 0 … √(7^2 - 4*1*11) = √49-44 = √5 > 0 = two solutions
4. x^2 -6x + 11 = 0…… √(-6^2 - 4*1*11) = √36-44 = √-8 < 0 = no solution
5. x^2 - 6x - 11 = 0 …… √(-6^2 - 4*1*-11) = √36-(-44) = √80 > 0 = two solutions

Question 329

What is 4 / (3-√2)?

Answer 329

When the denominator has a normal number AND a square root, you must use the conjugate…

for a + √b, the conjugate is given by a-√b

for a - √b, the conjugate is given by a+√b

(so the opposite sign)

thus…

4 / (3-√2) =

4*(3+√2) / (3-√2)(3+√2) =

(12 + 4√2) / (9+3*√2 - 3√2 - 2) =

(12 + 4*√2) / (7)

Question 330

If a*b = 12 and c/a +10 = 15, what is b*c?

Answer 330

a*b = 12 is also b = 12/a

c/a + 10 = 15 is also c + 10a = 15a which is c = 5a.

So… b*c is

12/a * 5a which is 60a/a = 60.

Question 331

If l x+1 l = l 3x -2 l, what can x be?

Answer 331

So we have two absolute value expressions…

Can be…

POS-NEG

POS-POS

NEG-NEG

NEG-POS

so either two different signs or both have the same sign. We have to calculate two options. One with same sign and one with different sign.

Lets start with different sign…

x+1 = -3x -(-2)

4x = 1

x = ¼

now both signs positive

x+1 = 3x -2

3 = 2x

x = 3/2

Question 332

What is the closest integer to x*y*z when we know that… x*y=2, x*z=8 and y*z=5?

Answer 332

We can multiply the 3 expressions together and we get…

x^2*y^2*z^2 = 2*5*8

x^2*y^2*z^2 = 80

squareroot(x^2*y^2*z^2) = squareroot(80)

x*y*z = close to 9.

Question 333

Which of the following has no solution for a?

• a^2 -6a + 7 = 0
• a^2 + 4a + 3 = 0
• a^2 - 4a + 5 = 0

Answer 333

We must use part of the quadratic root formula…

√(b^2 - (4*a*c))

• a^2 -6a + 7 = 0 …. √(-6^2 - (4*1*7)) = √36 - 28 = √8 > 0 (two solutions)
• a^2 + 4a + 3 = 0…. √(4^2 - (4*1*3)) = √16 - 12 = √4 > 0 (two solutions)
• a^2 - 4a + 5 = 0….. √(-4^2 - (4*1*5)) = √16 - 20 = √-4 < 0 (NO solutions)

Question 334

If f(x) = 2x^4 - x^2, what is the value of f(2√3)?

Answer 334

f(2√3) = 2(2√3)^4 - (2√3)^2

2*16*√81 - 4√9

32*9 - 4*3

270 + 18 - 12 = 276

Question 335

If g(x) = 3x + √x, what is the value of g(d^2 + 6d + 9)

Answer 335

g(d^2 + 6d + 9) = 3(d^2 + 6d + 9) + √(d^2 + 6d + 9)

3d^2 + 18d + 27 + √(d+3)^2

3d^2 + 18d + 27 + d + 3

3d^2 + 19d + 30

Question 336

If k(x) = 4x^3*a, and k(3) = 27, what is k(2)?

Answer 336

We have to use the information we know about k(3).

k(3) = 4*3^3*a = 27

4*a = 27/27

4*a = 1

a = 1/4

k(2) = 4*2^3*¼ = 2^3 = 8

Question 337

If f(x) = 2x^2 - 4 and g(x) = 2x, for what values of x will f(x) = g(x)?

Answer 337

Set the two equations equal to each other and solve…

2x^2 - 4 = 2x

2x^2 - 2x -4 = 0

2(x+1)(x-2) = 0

x = 2 or x = -1

Question 338

The velocity of a falling object in a vacuum is directly proportional to the amount of time the object has been falling. If after 5 seconds an object is falling at a speed of 90 miles per hour, how fast will it be falling after 12 seconds?

Answer 338

So first we take 90m/h and divide by 5 to get S.

90m/h / 5 = 18m/h per second, S

S(12) = 18*12 = 216m/h

Question 339

The “luminous flux,” or perceived brightness, of a light source is measured in lumens and is inversely proportional to the square of the distance from the light. If a light source produces 200 lumens at a distance of 3 meters, at what distance will the light source produce a luminous flux of 25 lumens?

Answer 339

We know that…

200*3^2 = 25 * D2^2

1800 = 25 * D2^2

1800/25 = D2^2

72 = D2^2

D2 = √72

Question 340

A strain of bacteria multiplies such that the ratio of its population in any two con­secutive minutes is constant. If the bacteria grows from a population of 5 million to a population of 40 million in one hour, by what factor does the population increase every 10 minutes?

Answer 340

We know that the 40m/5m = 8. So it increases by 8x in 60 minutes.

(60)√8 = (60)√2^3 = (20)√2 is for 1 minute.

we have 10 minutes so….

(20)√2^10 = √2.

After 2*10 minutes (20 minutes) it will have increased by √2*√2 = a factor of 2.

Question 341

If a and b are integers and -4 < a < 3 and -4 < b < 5, what is the maximum possible value for a*b?

Answer 341

Look at the extremes…

3*5 = 15

vs.

-4*-4 = 16

Question 342

Is m*n > -12?

1. m > -3
2. n > -4

Answer 342

1. m > -3 NS
2. n > -4 NS

Together we still don’t know as we can both take -3*10 and get -30 or… 4*5 and get 20.

Question 343

If 4/x < - ⅓ , what is the possible range of values for x?

Answer 343

4/x must be < -⅓

-12 < x < 0

Question 344

If 4/x < ⅓ , what is the possible range of values for x?

Answer 344

x > 12 OR…

x < 0

Question 345

Is x < y?

1. 1/x < 1/y
2. x/y < 0

Answer 345

1. 1/x < 1/y… from this x can either be a positive larger than y (+x > +y = ¼ < 1/2) OR a negative -x < y (1/-1) < ½
2. From this we know that one of the numbers must be negative but that alone is not sufficient

Together we know that one number must be negative - either x or y is negative AND that number must be x because the reciprocal of x (1/x < 1/y).

Thus we know that x (being negative) is < y (positive) = C.

Question 346

John is 20 years older than Brian. Twelve years ago, John was twice as old as brian. How old is Brian?

Answer 346

J = B + 20

J - 12 = 2(B-12)

B + 20 - 12 = 2B - 24

32 = B

Question 347

Caleb spends $72.50 on 50 hamburgers for the marching band. If single burgers cost $1.00 each and double burgers cost $1.50 each, how many double burgers did he buy?

Answer 347

Number of DoubleBurgers = D

D + S = 50

1*S + 1.5*D = 72.50

S = 50 - D

1(50-D) + 1.5*D = 72.5

• D + 1.5D = 72.5 - 50
  0. 5D = 22.5

D = 45

Question 348

United Telephone charges a base rate of $10.00 for service, plus an additional charge of $0.25 per minute. Atlantic Call charges a base rate of $12.00 for service, plus an additional charge of $0.20 per minute. For what number of minutes would the bills for each telephone company be the same?

Answer 348

10 + 0.25m = 12 + 0.20m

0. 25m-0.20m = 2
1. 05m = 2

m = 40

Question 349

Carina has 100 ounces of coffee divided into 5- and 10-ounce packages. If she has 2 more 5-ounce packages than 10-ounce packages, how many 10-ounce packages does she have?

Answer 349

5*A + 10*B = 100

A = B + 2

5(B+2) + 10B = 100

15B = 90

B = 6

A = 6+2 = 8

Question 350

Martin buys a pencil and a notebook for 80 cents. At the same store, Gloria buys a notebook and an eraser for $1.20, and Zachary buys a pencil and an eraser for 70 cents. How much would it cost to buy three pencils, three notebooks, and three erasers? (Assume that there is no volume discount.)

Answer 350

What is 3P + 3N + 3E?

P + N = $0.80

E + N = $1.20

P + E = $0.70

2P + 2N + 2E = $2.70

1.5(2P + 2N + 2E) = $2.70 * 1.5 = $4.05

Question 351

Andrew will be half as old as Larry in 3 years. Andrew will also be one-third as old as Jerome in 5 years. If Jerome is 15 years older than Larry, how old is Andrew?

Answer 351

Find A?

• Andrew will be half as old as larry in 3 years… 2(A+3) = (L+3)
• Andrew will also be one-third as old as Jerome in 5 years… 3(A+5) = J+5
• Jerome is 15 years older than Larry….. J = L + 15

2A + 6 = L + 3

3A + 15 = L + 20

difference is…

A + 9 = 17

A = 8

Question 352

A circus earned $150,000 in ticket revenue by selling 1,800 V.I.P. and Standard tickets. They sold 25% more Standard tickets than V.I.P. tickets. If the revenue from Standard tickets represents one-third of the total ticket revenue, what is the price of a V.I.P. ticket?

Answer 352

Find PRICE of VIP?

V + S = 1800

P1*V + P2 * S = 150,000

S = 5/4*V or… V = ⅘*S

⅘*S + S = 1800

S = 1800/1.8 = 1,000

Then… V = 1800 - 1000 = 800.

P1*800 + P2*1000 = 150,000.

S*P2 = ⅓ * 150,000

S*P2 = 50,000

then…

P2 = 50,000 / 1000 = 50.

P1*800 + 50*1000 = 150,000

P1*800 = 100,000

P1 = 100,000 / 800 = $125.

Question 353

Bookshelf holds both paperback and hardcover books. The ratio of paperback books to hardcover books is 22 to 3. How many paperback books are on the shelf?

1. The number of books on the shelf is between 202 and 247, inclusive.
2. If 18 paperback books were removed from the shelf and replaced with 18 hardcover books, the resulting ratio of paperback books to hardcover books on the shelf would be 4 to 1.

Answer 353

1. 22+3 = 25 so… 200, 225, 250 are possible solutions. Only 225 is within 202-247.
   
   1. 225/25 = 9…. so paperbook = 9*22 = 198 = SUF
2. P/H = 22/3… and (p-18)/(h+18) = 4/1 …. we have two equations and can solve so SUF
   
   1. (p-18) = 4((3p/22)+18)
   2. p-18 = 6p/11+ 72
   3. p-6p/11 = 90
   4. 11p/11 - 6p/11 = 90
   5. 5p/11 = 90
   6. p = (90*11)/5 = 198

Question 354

On the planet Flarp, 3 floops equal 5 fleeps, 4 fleeps equal 7 flaaps, and 2 flaaps equal 3 fliips. How many floops are equal to 35 fliips?

Answer 354

3FLOOPS = 5 FLEEPS

4FLEEPS = 7 FLAAPS

2 FLAAPS = 3 FLIIPS

35FLIIPS in FLOOPS is…

35/1 * ⅔ * 4/7 * ⅗ = (35*24) / (3*7*5) = 24/3 = 8

Question 355

Harvey runs a 30-mile course at a constant rate of 4 miles per hour. If Clyde runs the same track at a constant rate and completes the course in 90 fewer minutes, how fast did Clyde run?

Answer 355

Question 356

Imagine that two people are 14 miles apart and begin walking towards each other. Person A walks 3 miles per hour, and Person B walks 4 miles per hour. How long will it take them to reach each other?

Answer 356

They move closer to each other by 3+4mp/h = 7mph

so… 14i / 7 = 2 hours.

Question 357

Car X is 40 miles west of Car Y. Both cars are traveling east, and Car X is going 50% faster than Car Y. If both cars travel at a constant rate and it takes Car X 2 hours and 40 minutes to catch up to Car Y, how fast is Car Y going?

Answer 357

Question 358

If Lucy walks to work at a rate of 4 miles per hour, but she walks home by the same route at a rate of 6 miles per hour, what is Lucy’s average walking rate for the round trip?

Answer 358

Question 359

Machine A fills soda bottles at a constant rate of 60 bottles every 12 minutes and Machine B fills soda bottles at a constant rate of 120 bottles every 8 minutes. How many bottles can both machines working together at their respective rates fill in 25 minutes?

Answer 359

Question 360

Alejandro, working alone, can build a doghouse in 4 hours. Betty can build the same doghouse in 3 hours. If Betty and Carmelo, working together, can build the doghouse twice as fast as Alejandro, how long would it take Carmelo, working alone, to build the doghouse?

Answer 360

Carmelo alone?

Alejandro = 4 hours

Betty =3 hours

Betty + Carmelo = half the time of Alejandro

¼ A, ⅓ b, 1/c

⅓ + 1/c = 2 (¼)

1/c = ½ - ⅓ = 1/6

c = 6

Question 361

The population of a certain type of bacterium triples every 10 minutes. If the population of a colony 20 minutes ago was 100, in approximately how many min­utes from now will the bacteria population reach 24,000?

Answer 361

20 minutes ago = 100

10 minutes ago = 300

NOW = 900

in 10 minutes = 2700

in 20 minutes = 8100

in 30 minutes = 24,300

Question 362

The population of grasshoppers doubles in a particular field every year. Approxi­ mately how many years will it take the population to grow from 2,000 grasshoppers to 1,000,000 or more?

Answer 362

NOW = 2,000

1y = 4,000

2y = 8,000

3y = 16,000

4y = 32,000

5y = 64,000

6y = 128,000

7y = 256,000

8y = 512,000

9y = 1,024,000

Question 363

Two hoses are pouring water into an empty pool. Hose 1 alone would fill up the pool in 6 hours. Hose 2 alone would fill up the pool in 4 hours. How long would it take for both hoses to fill up two-thirds of the pool?

Answer 363

Question 364

An empty bucket being filled with paint at a constant rate takes 6 minutes to be filled to 7/10 of its capacity. How much more time will it take to fill the bucket to full capacity?

Answer 364

Question 365

Twelve identical machines, running continuously at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days?

Answer 365

Question 366

Al and Barb shared the driving on a certain trip. What fraction of the total distance did Al drive?

1. Al drove for 3/4 as much time as Barb did.
2. Al’s average driving speed for the entire trip was 4/5 of Barb’s average driving speed for the trip.

Answer 366

1. not sufficient as we only know Time (not rate)
2. not sufficient as we only know RATE (not time)

Together is sufficient as we know both time and rate.

Question 367

Sam earned a $2,000 commission on a big sale, raising his average commission by $100. If Sam’s new average commission is $900, how many sales has he made?

Answer 367

Question 368

A mixture of “lean” ground beef (10% fat) and “super-lean” ground beef (3% fat) has a total fat content of 8%. What is the ratio of “lean” ground beef to “super­ lean” ground beef?

Answer 368

Ratio of LEAN to SUPER LEAN?

10-8 = +2

3-8 = -5

Lean(+2) + SUPER-LEAN(-5) = 0 … (now just pick values for x and y such that the equation will equal 0).

5(+2) + 2(-5) = 0

10 - 10 = 0

Question 369

In a group of men and women with a ratio of 2:3, the men have an average age of 50 and the average age of the group is 56. What is the average age of women?

Answer 369

men 2 : women 3

overall = 56 years

men = 50 years

women = ?

2(56-50) + 3(w) = 0

-12 + 3w = 0

w = 4

women have a +4 differential and must be 56+4 = 60 years old.

Question 370

The average of 11 numbers is 10. when one number is eliminated, the average of the remaining numbers is 9.3. What is the eliminated number?

Answer 370

A = s/n

10 = s /11

s = 110

9.3 = s/(11-1)

s = 93

110 - 93 = 17

Question 371

Given the set of numbers {4,5,5,6,7,8,21}, how much higher is the mean than the median?

Answer 371

the mean is (4+5+5+6+7+8+21) /7 = 56 / 7 = 8

the median is 6 as it is the middle number.

8-6 =2

Question 372

A charitable association sold an average of 66 raffle tickets per member. Among the female members, the average was 70 raffle tickets. The male to female ratio of the association is 1:2. What was the average number of tickets sold by the male mem­bers of the association?

Answer 372

66 = s/n

female = 70 = s/n

Male 1, Female 2…. 1:2

female = 70-66 = 4

1(m) + 2(4) = 0

m = -8

66-8 = 58.

Question 373

Matt gets a $1,000 commission on a big sale. This commission alone raises his aver­ age commission by $150. If Matt’s new average commission is $400, how many sales has Matt made?

Answer 373

Question 374

If the average of x and y is 50,and the average of y and z is 80, what is the value of z-x ?

Answer 374

(x+y)/2 = 50

(y+z)/2 = 80

x+y = 100

and

y+z = 160

y+z-(x+y) = 160-100

z-x = 60

Question 375

On a particular exam, the boys in a history class averaged 86 points and the girls in the class averaged 80 points. If the overall class average was 82 points, what was the ratio of boys to girls in the class?

Answer 375

Find the ratio of boys to girls…

Boys = 86 = s/n

Girls = 80 = s/n

Overall = 82 = s/n

Boys… 86 - 82 = 4

Girls… 80 - 82 = -2

b(4) + g(-2) = 0

b = 1, g = 2,

1:2

Question 376

(9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

1. 14, 16
2. 9, 21
3. 15, 100

Answer 376

Note first that the mean = 15.

1. 14, 16 will NOT.. actually decrease SD
2. 9, 21 will increase SD
3. 15, 100 will increase SD substantially

2 and 3.

Question 377

What is the sum of all the positive integers up to 100 inclusive?

Answer 377

((100-1)+1)/2 = 101/2 = 50.5

100-1+ 1 = 100 numbers

50.5 * 100 = 5,050

Question 378

In a sequence of 8 consecutive integers, how much greater is the sum of the last four integers than the sum of the first four integers?

Answer 378

A, B, C, D, E, F, G, H

Every number of the 4 last are exactly 4 bigger than the same position of the first four.

(E+F+G+H) - (A+B+C+D) = 4*4 = 16 higher.

Question 379

How many terms are there in the set of consecutive integers from -18 to 33 inclusive?

Answer 379

33 + 18 + 1 = 52

Question 380

If the sum of the last 3 integers in a set of 7 consecutive integers is 258, what is the sum of the first 4 integers?

Answer 380

A, B, C, D, E, F, G

258/3 = 86.

thus.. F = 86.

81, 82, 83, 84, 85, 86(F), 87

The first four is A+B+C+D = 81+82+83+84 = 320+10 = 330.

Question 381

Of 28 people in a park, 12 are children and the rest are adults. 8 people have to leave at 3pm; the rest do not. If after 3pm, there are 6 children still in the park, how many adults are still in the park?

Answer 381

Question 382

Of 30 snakes at the reptile house, 10 have stripes, 21 are poisonous, and 5 have no stripes and are not poisonous. How many of the snakes have stripes AND are poi­sonous?

Answer 382

Question 383

Students are in clubs as follows: Science-20, Drama-30, and Band-12. No student is in all three clubs, but 8 are in both Science and Drama, 6 are in both Science and Band, and 4 are in Drama and Band. How many different students are in at least one of the three clubs?

Answer 383

ScienceT=20

DramaT=30

BandT = 12

in at least in club?

In all three = 0

in drama+science = 8

in science+band = 6

in band+drama=4

ScienceAlone = 20 - 8 -6 = 6

DramaAlone = 30 - 8 - 4 = 18

BandAlone = 12 - 6 -4 = 2

All together…. 6 + 18 + 2 + 0 + 8 +6 +4 = 44

Question 384

Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are 20 boys and 40 girls. If there are 6 happy boys and 4 sad girls, how many boys are neither happy nor sad?

Answer 384

Question 385

A car travels from Town A to Town B at an average speed of 40 miles per hour, and returns immediately along the same route at an average speed of 50 miles per hour. What is the average speed in miles per hour for the round-trip?

Answer 385

Question 386

a, b, and c are integers in the set {a, b, c, 51,85,72}. Is the median of the set greater than 70?

1. b > c > 69.
2. a < c < 71

Answer 386

1. b>c>69… then we know that (a, 51, c, 72, b, 85) and c is minimum 70. (70+72)/2 > 70. Sufficient
2. a<c></c>

</c>

Question 387

A store sells chairs and tables. If the price of 3 chairs and 1 table is 60% of the price of 1 chair and 3 tables, and the price of 1 table and 1 chair is $60, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)

1. 15
2. 20
3. 30
4. 40
5. 45

Answer 387

3*c + 1*t = 0.6(1*c+3*t)

1*c+1*t = 60

From this we know immediately that T is bigger than c.

We can thus rule out all other options than d=40 and e=45 for T.

Thus, we test one of them….

if t = 45, then c = 60-45 = 15

3*15 + 1*45 = 0.6(1*15+3’45)

90 = 0.6*150

true.

T = 45

Question 388

Boys and girls in a class are writing letters. There are twice as many girls as boys in the class, and each girl writes 3 more letters than each boy. If boys write 24 of the 90 total letters written by the class, how many letters does each boy write?

Answer 388

Question 389

Orange Computers is breaking up its conference attendees into groups. Each group must have exactly one person from Division A, two people from Division B, and three people from Division C. There are 20 people from Division A, 30 people from Division B, and 40 people from Division C at the conference. What is the smallest number of people who will not be able to be assigned to a group?

Answer 389

First figure out which group is the limiting factor…

Division A = 20/1 = 20 groups possible

Division B = 30/2 = 15 groups possible

Division C = 40/3 = 13 groups possible

So we get maximum 13 groups where the unassigned people will be…

Division A = 20-13*1 = 7 persons

Division B = 30-13*2 = 4 persons

Division C = 40-3*13 = 1 person

Minimum 12 unassigned employees.

Question 390

Velma has exactly one week to learn all 71 Japanese hiragana characters. If she can learn at most a dozen of them on any one day and will only have time to learn four of them on Friday, what is the least number of hiragana that Velma will have to learn on Saturday?

Answer 390

71 - 4 = 67 for six days

Max 12 a day = (6-1)*5 = 60.

67-60 = 7 minimum on Saturday

Question 391

A “Collector’s Coin Set” contains a one dollar coin, a fifty-cent coin, a quarter (= 25 cents), a dime (= 10 cents), a nickel (= 5 cents), and a penny (= 1 cent). The Coin Sets are sold for the combined face price of the currency. If Colin buys as many Coin Sets as he can with the $25 he has, how much change will Colin have left over?

Answer 391

1. 00
2. 50
3. 25
4. 10
5. 05
6. 01

= 1.91.

25 / 1.91….

19. 10 (10 sets)
20. 73 (3 sets)
21. 83 so 17 cents left.

Question 392

What can be said about the SUM of a set of consecutive integers in terms of divisibility?

Answer 392

• For any set of consecutive integers with an ODD number of items, the sum of all the integers is ALWAYS a multiple of (and thus divisible by) the number of items
• For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER a multiple of the number of items.

Question 393

If r, s, and t are consecutive positive multiples of 3, is rst divisible by 27,54, or both?

Answer 393

BOTH…

r, s and t are all multiples of 3. thus, the product of r*s*t must have THREE 3s as factors AND at least one of the numbers will be EVEN so the product will also have 2 as a factor.

Question 394

Is the sum of the integers from 54 to 153 inclusive divisible by 100?

Answer 394

NO

There are 100 integers from 54 to 153 (153-54+1 = 100).

For any EVEN number of consecutive integers, the sum of all the integers is NEVER a multiple of the number of integers.

thus, the sum of the integers from 54 to 153 will not be divisible by 100.

Question 395

A machinist’s salary at a factory increases by $2,000 at the end of each full year the machinist works. If the machinist’s salary for the fifth year is $39,000, what is the machinist’s average annual salary for his first 21 years at the factory?

Answer 395

$2,000 increments

y5 = $39,000

y1 = $39,000 - 4*2,000 = $31,000

t21 = $39,000 + 16*2,000 = $71,000

($71,000 + $31,000) / 2 = $51,000 average salary.

Question 396

Is the average of n consecutive integers equal to 1?

1. n is even.
2. If S is the sum of the n consecutive integers, then 0 < S < n.

Answer 396

1. It tells us that there is an EVEN number of consecutive integers and we know that the average of an even number of consecutive integers NEVER will be an integer, so this is actually SUFFICIENT although we know nothing about the numbers
2. Knowing A = s/n and that s<n></n>

</n>