EXCERPTS FROM UIL, PSIA & TMSCA MATHEMATICS MAGIC ADDITION AND SUBTRACTION OF FRACTIONS Traditional students are taught to add and subtract fractions by first finding the least common denominator. The method that will be presented is much simpler to learn.

Example A : 59

+ 310

=

Step #1 : Multiply the denominator of the fraction on the right With the numerator of the fraction on the left. 10 x 5 = 50 Step #2 : Multiply the denominator of the fraction on the left with The numerator of the fraction on the right. 9 x 3 = 27 Step #3 : The sum of the results from Steps #1 and #2 will be the numerator of the answer. 50 + 27 = 77 Step #4 : Multiply the denominators of the two fractions. 9 x 10 = 90 Step #5 : Determine if the resulting fraction can be simplified.

77

90

FINDING THE LCM OF TWO NUMBERS The following method of finding the LCM of two numbers has not appeared in a mathematics book that I am aware of, yet I am confident that it is less confusing then the method that makes use of “prime factorization”. Beginning with its first introduction to students, the following method should be an integral part of every mathematics book. Step #1 : Students should be made aware that the product of the GCF and LCM of two numbers is equal to the product of the two numbers. Examples should be provided to make this obvious. Step #2 : Using the statement made in Step #1, it should be noted that the following is true : LCM = (product of numbers) ÷ GCF Example A : Find the LCM of 12 and 20.

Solution : LCM = (12 s 20) ÷ (GCF of 12 and 20) = (12 x 20) ÷ 4 At this point, students should be told that the easiest way to solve the problem is to divide 4 into one of the two numbers and multiply the result by the other number. Note : (12 ÷ 4) x 20 = 3 x 20 = 60 CHANGING FROM BASE 10 TO ANOTHER BASE Example A : 45 base 10 = __________ base 6. Step #1 : The first digit of the answer (from right to left) is eq ual to the remainder when you divide the given number by the base.

45 ÷ 6 = 7, remainder 3. Write down the 3.

Step #2 : Divide the quotient in Step #1 by the base. The remainder Is the next digit of the answer.

7 ÷ 6 = 1, remainder 1. Write down the 1.

Step #3 : Continue dividing the quotient of the previous step by the base, always writing down the remainder until the final digit of the answer is found.

1 ÷ 6 = 0, remainder 1. Write down the 1.

Answer : 113 Example B : 73 base 10 = __________ base 9. Step #1 : The first digit of the answer (from right to left) is eq ual to the remainder when you divide the given number by the base.

73 ÷ 9 = 8, remainder 1. Write down the 1.

Step #2 : Divide the quotient in Step #1 by the base. The remainder Is the next digit of the answer.

8 ÷ 9 = 0, remainder 8. Write down the 8.

Answer : 81

CHANGING FROM A BASE TO BASE 10

234 base 5 equals __________ base 10.

Step #1 : Multiply the first digit to the left by the base

2(5) = 10 Step #2 : Add the middle digit with the result from Step #1. 4 + 10 = 13

Step #3 : Multiply the result from Step #2 by the base. 13(5) = 65 Step #4 : Add the digit to the right by the result from Step #3.

4 + 65 = 69

Example A : 213 base 8 equals __________ base 10. Step #1 : 2(8) = 16 Step #2 : 1 + 16 = 17 Step #3 : 17(8) = 136 Step #4 : 3 + 136 = 139 Example B : 321 base 4 equals __________ base 10. Step #1 : 3(4) = 12 Step #2 : 2 + 12 = 14 Step #3 : 14(4) = 56 Step #4 : 56 + 1 = 57

SAMPLE PROBLEMS

1. 72 - 8 ÷ 4 = _________. When expressions have more than one operation, we have to follow rules for the order of operations. (1) First do all operations that lie inside parentheses. (2) Next, do any work with exponents or radicals. (3) Working from left to right, do all multiplication and division. (4) Finally, working from left to right, do all addition and subtraction. Solution : 72 - (8 ÷ 4) = 72 - 2 = 70 Example A : 45 - (12 + 18) = _________. Solution : 45 - 30 = 15 Example B : 4 x 3 + 5 x 2 = __________. Solution : (4 x 3) + (5 x 2) = 12 + 10 = 22 10. 21 x 17 + 21 x 13 = __________. Solution : 21(17 + 13) = 21(30) = 630 Example A : 31 x 24 + 31 x 16 = __________. 31(24 + 16) = 31(40) = 1240 11. 1 + 2 + 3 + 4 + … + 19 = __________. When adding the positive integers from 1 to n, the sum is equal to

n(n +1)

2.

Solution : 19(19 +1)2

= 1920

2

!"#

$%&

= 19(10) = 190

Example A : 1 + 2 + 3 + 4 + … + 24 = __________.

Solution : 24(24 +1)2

= 24

2

!"#

$%&25 = 12(25) = 300

12. 1 + 3 + 5 + . . . + 19 = __________.

Rule : 1 + 3 + 5 + . . . + k = k +1

2

!"#

$%&2

Solution : 19 +12

!"#

$%&2

= 10 2 = 100

Example A : 1 + 3 + 5 + . . . + 13 = __________.

Solution : 13+12

!"#

$%&2

= 7 2 = 49

13. 2 + 4 + 6 + 8 + . . . + 44 = __________.

2 + 4 + 6 + . . . + k = k(k + 2)4

Solution : 44(44 + 2)4

= 44

4

!"#

$%&44 + 2)( ) = 11(46) = 506

Example A : 2 + 4 + 6 + . . . + 20 = __________.

Solution : 20(20 + 2)4

= 20

4

!"#

$%&(20 + 2) = 5(22) = 110

Example B : 2 + 4 + 6 + . . . + 30 = __________.

Solution : 30(30 + 2)4

= 30(32)4

= 3032

4

!"#

$%&

= 30(8) = 240

15. 3 13

% = _________ (fraction).

Memorization of percents is recommended to make this easier to solve. Step #1 : Convert the mixed number into an improper fraction.

103

Step #2 : Add two zeroes to the denominator, then reduce to lowest terms.

10300

= 13

Example A : 8 13

% = _________ (fraction).

Step #1 : 253

Step #2 : 25300

= 112

16. If 6 pears cost $1.21, then 2 dozen pears cost $_________. Note : 2 dozen pears = 24 pears. To find the cost of 24 pears, multiply the cost of 6 pears by 4. $1.21(4) = $4.84

Example A : If a dozen Valentines cost $8.76, then 4 Valentines cost $_________.

Solution : The cost of 4 Valentines will be 412

= 13

the cost of a dozen Valentines.

$8.763

= $2.92

22. CLVI = _________ (Arabic Numeral) Note : M = 1000, D = 500, C = 100, L = 50, X = 10, V = 5 and I = 1 Solution : CLVI = 100 + 50 + 5 + 1 = 156 Example A : XLIII = __________ (Arabic Numeral). Solution : (50 - 10) + 3 = 40 + 3 = 43 35. The range of 1, 2, 1, 3, 1, 4, and 0 is __________. The range of a list of numbers is the difference between the smallest and largest Numbers. Solution : 4 - 0 = 4 Example A : The range of 5, 9, 12, 23, 14, and 2 is __________. Solution : 23 - 2 = 21

40. What percent of 40 is 32? __________.

Solution : 3240

(100) = 45

(100) = 4(20) = 80

Example A : What percent of 25 is 15? _________%.

Solution : 1525

(100) = 35

(100) = 3(20) = 60

42. Which is smaller, 811

or 1013

?

Solution : Cross multiply (from denominator to numerator). The smaller number will be the fraction with the smaller product.

8(13) = 104 (104 is associated with the fraction 811

.)

11(10) = 110 (110 is associated with the fraction 1013

.)

Since 104 is less than 110, the smaller fraction is 811

.

59. (23 x 5 + 4) ÷ 7 has a remainder of __________. Solution : The remainder when 23 is divided by 7 is 2. The remainder when 5 is Divided by 7 is 5. The remainder when 4 is divided by 7 is 4. (23 x 5 + 4) ÷ 7 has a remainder that can be found by first doing the following : 2 x 5 + 4 = 14, then find the remainder when 14 is divided by 7. The answer is 0. Example A : (32 x 13 + 45) ÷ 6 has a remainder of __________. Solution : Find the remainder of each number inside of the parentheses. The remainder when 32 is divided by 6 is 2. The remainder when 13 is divided by 6 is 1. The remainder when 45 is divided by 6 is 3, Substitute the remainders for numbers inside the parentheses. 2 x 1 + 3 = 2 + 3 = 5 ; The answer is 5. Example B : (29 2 + 48) ÷ 9 has a remainder of __________. Solution : Find the remainder of each number inside of the parentheses. 29 divided by 9 has a remainder of 2. 48 divided by 9 has a remainder of 3.

Substitute the remainders for numbers inside the parentheses. 2 2 + 3 = 4 + 3 = 7 ; The answer is 7. 61. The product of the GCF and the LCM of 24 and 30 is __________.

The product of the GCF and the LCM of any two numbers is equal to the product of the two numbers.

Solution : 24(30) = 720 Example A : The product of the GCF and the LCM of 14 and 21 is __________. Solution : 14(21) = 294 Example B : The product of the GCF and the LCM of 15 and 20 is __________. Solution : 15(20) = 300

65. If 4x7

= 20, then x = _________.

Solution : 74

(20) = 7(5) = 35

Example A : If 2x3

= 12, then x = _________.

Solution : 32(12) = 3(6) = 18

78. The set {M, A, T, H} has _________ subsets. Note : An n-element set has 2 n subsets. Solution : 2 4 = 16 Example A : The set {T, W, O} has _________ subsets. Solution : 2 3 = 8 79. How many elements in the power set of {3, 8, 1}? ________. Note : The power set of a given set is a set whose elements are the subsets of the set. Rule : 2 n , where n is the number of elements in the set Solution : 2 3 = 8

Example A : How many elements in the power set of a 5-element set? ________. Solution : 2 5 = 32 84. .414141... = __________ (proper fraction). Write a fraction whose numerator is the repeated digits and whose denominator consists of as many 9's as there are repeating digits.

Solution : 4199

Example A : .666... = __________ (proper fraction).

Solution : 69

= 23

Example B : .023023... = __________ (proper fraction).

Solution : 23999

Example C : .185185... = __________ (proper fraction).

Solution : 185999

Note : You should be aware of the fact that 999 = 27 x 37. When reducing a fraction whose denominator is 999, you should determine if the numerator is divisible by 3, 9, 27 or 37.

185999

= 5(37)

27(37) = 5

27

88. The next term in the sequence 3, 5, 8, 13, 21, 34, … is _________. Note : This is a Fibonacci style sequence. The sum of the 1st and 2nd terms is equal to the third term. The sum of the 2nd and 3rd terms is equal to the 4th term. The sum of the 3rd and 4th terms is equal to the 5th term, and so on. Solution : 21 + 34 = 55 Example A : The next term in the sequence 1, 6, 7, 13, 20, 33, … is _________. Solution : 20 + 33 = 53 89. The 13th term of 2, 5, 8, 11, 14, … is __________. Step #1 : Find what number is being added to find each term in the pattern.

They are adding 3. Step #2 : Subtract 1 from the term you are looking for and multiply by the result of Step#1. (13 - 1) x 3 = 12 x 3 = 36 Step#3 : Add the result of Step #2 to the first term in the patteren. 36 + 2 = 38 Example A : The 12th term of 3, 10, 17, 24, 31, … is _________. Step #1 : Note that they are adding 7 to find each term. Step #2 : (12 - 1) x 7 = 11 x 7 = 77 Step #3 : 77 + 3 = 90 93. If 8x - 2 = 6x + 28, then x = __________. Solution : 8x - 2 = 6x + 28 8x - 6x = 28 + 2 2x = 30 x = 15 98. The smallest leg of a right triangle with integral sides is 7”. The hypotenuse is __________”. Note : The given leg is an odd prime number. Step #1 : Square the number. 7 2 = 49 Step #2 : Find two consecutive integers whose sum is equal to the result of Step #1. 49 = 24 + 25 Step #3 : The resulting Pythagorean Triplet is 7, 24, 25, The hypotenuse is 25. Example A : The smallest leg of a right triangle with integral sides is 11”. The longest leg is __________”. Solution : Step #1 : 11 2 = 121 Step #2 : 121 = 60 + 61 Step #3 : 11, 60, 61, The long leg is 60. 100. The sides of a right triangle are integers. If one leg is 6, then the other leg is ________. If the given leg is even, do the following : Step #1 : Divide the leg in half and square it.

6

2

!"#

$%&2

= 3 2 = 9

Step #2 : Find the integers on either side of the result from Step #1. 8 and 10 are the integers on either side of 9. Step #3 : Select the smaller integer obtained in Step #2. Answer : 8 Example A : The sides of a right triangle are integers. If one leg is 10, then the other leg is ________.

Step #1 : 102

!"#

$%&2

= 5 2 = 25

Step #2 : 24 and 26 are the integers on either side of 25. Step #3 : Answer : 24 111. The measure of an interior angle of a regular decagon is __________ degrees. Note : An interior angle and an exterior angle are supplementary (the sum of their measures is 180°). If n is the number of sides of a regular

polygon, then 360°n

is equal to the number of degrees of the

exterior angle. To find the interior angle, subtract the exterior angle from 180°.

Solution : 180° - 360°10

= 180° - 36° = 144°

Example A : The measure of an interior angle of a regular hexagon is __________ degrees.

Solution : 180° - 360°8

= 180° - 45° = 135°

115. The 12th triangular number is __________.

The nth triangular number is n(n +1)2

. The first few triangular numbers are

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

Solution : 12(12 +1)2

= 122

(13) = 6(13) = 78

Example A : The 15th triangular number is __________.

Solution : 15(15 +1)2

= 15(162

) 15(8) = 120

118. When the sides of a square are tripled, by what factor is the area multiplied? ________. Solution : Since you want to find out by what factor the area will be multiplied and area is square units, the factor is 3 2 = 9 Example A : When the radius of a sphere is tripled, by what factor is the volume multiplied? _________. Solution : Since you want to find out by what factor the volume will be multiplied and volume is cubic units, the factor is 3 3 = 27

164. 7! ÷ 5! = __________.

Solution : 7! ÷ 5! = 7!5!

= (7)(6)(5)(4)(3)...

(5)(4)(3)... = (7)(6) = 42

124. The discriminant of 3x 2 - 2x + 1 = 0 is _________.

Note : If ax 2 + bx + c = 0, the discriminant is b 2 - 4ac. Solution : (- 2) 2 - 4(3)(1) = 4 - 12 = - 8

126. The side of a square is 10 2 inches. Find the length of its diagonal. ________ inches. Note : The length of the diagonal of a square is equal to the length of the side multiplied by 2 . Solution : Length of diagonal = (10 2 )( 2 ) = 10 4 = 10(2) = 20 Example A : The side of a square is 5 2 inches. Find the length of its diagonal. ________ inches. Solution : Length of diagonal = (5 2 )( 2 ) = 5 4 = 5(2) = 10 The following is a variation of this problem. 128. A triangle has integral sides of 7, 10, and x. The largest value of x is ________. Note : A triangle has integral sides of a, b, and x. a ! b < x < a + b Rule : The largest integral value of x is a + b – 1. Solution : 7 + 10 - 1 = 16 Example A : A triangle has integral sides of 5, 13, and x. The largest value of x is ________.

Solution : 5 + 13 - 1 = 17 130. How many positive integers that are less than 15 are relatively prime to 15? Note : Relatively prime numbers : Two numbers are relatively prime if their GCD is 1. Step #1 : Find the distinct prime factors of the number. 15 = 3(5) Step #2 : Find the product of the differences of the reciprocal of the distinct prime factors and 1.

1!1

3

"#$

%&'1!

1

5

"#$

%&'

= 2

3

!"#

$%&4

5

!"#

$%&

Step #3 : Multiply the given number by the result from Step #2.

2

3

!"#

$%&4

5

!"#

$%&(15) =

8

15

!"#

$%&(15) = 8 ; The following numbers are less than 15 and

relatively prime to 15 : 1, 2, 4, 7, 8, 11, 13, and 14. 131. How many positive integral divisors does 24 have? __________. Step #1 : Prime factor the number. 24 = 2 3 x 3 Step #2 : Increase each exponent by 1 and multiply. (3 + 1)(1 + 1) = 4(2) = 8 Example A : How many positive integral divisors does 80 have? __________. Step #1 : Prime factor the number. 80 = 2 4 x 5 Step #2 : Increase each exponent by 1 and multiply. (4 + 1)(1 + 1) = 5(2) = 10 Example B : How many positive integral divisors does 96 have ? __________. Step #1 : Prime factor the number. 96 = 2 5 x 3 Step #2 : Increase each exponent by 1 and multiply. (5 + 1)(1 + 1) = 6(2) = 12 132. A hexagon has __________ distinct diagonals.

An n-gon has x distinct diagonals. x = n(n ! 3)2

Solution : 6(6 ! 3)2

= 6(3)2

= 182

= 9

Note : pentagon has 5 sides ; hexagon has 6 sides ; septagon/heptagon has 7 sides octagon has 8 sides ; nonagon has 9 sides ; decagon has 10 sides undecagon has 11 sides ; dodecagon has 12 sides Example A : An octagon has __________ distinct diagonals.

Solution : 8(8 ! 3)2

= 402

= 20

134. The area of a rhombus is equal to one half the product of the diagonals. Example A : The product of the diagonals of a rhombus whose diagonals are 20 and 12 is _____.

Solution : 20(12)2

= 10(12) = 120

139. A pair of dice is thrown. The probability that the sum is 3 is __________. Sum Number of Outcomes Sum Number of Outcomes 2 1 12 1 3 2 11 2 4 3 10 3 5 4 9 4 6 5 8 5 Sum Number of Outcomes 7 6 Note : If the sum is 7 or less, the number of possible outcomes is 1 less than the sum. If the sum is 7 or more, the number of possible outcomes is 13 less than the sum. Solution : Since the sum is 3, the number of possible outcomes is equal to 3 - 1 = 2.

Probability = Favorable

TotalOutcomes = 2

36 = 1

18

Example A : A pair of dice is thrown. The probability that the sum is 9 is __________.

Solution : Probability = 13! 936

= 436

= 19

144. 1101011

2 = _________

8.

Note : Each of the following 3 digit numbers are being converted from base 2 to base 10. 001 = 1 ; 010 = 2 ; 011 = 3 ; 100 = 4 101 = 5 ; 110 = 6 ; 111 = 7 Step #1 : Beginning from the right, do group of 3 digits. 1 101 011 Step #2 : Convert each group to base 10. 1 = 1 ; 101 = 5 ; 011 = 3 Answer : 153 Example A : 111001101

2 = __________

8.

Step #1 : 111 001 101 Step #2 : 111 = 7 ; 001 = 1 ; 101 = 5 BASIC MATHEMATICS 2. 70 miles per hour is equivalent to _________ inches per second. (A) 840 (B) 1056 (C) 1232 (D) 1680 (E) 6160

70 MilesHour

x 1Hour

3600Seconds x 5280Feet

1Mile x 12Inches

1Foot = 1232 inches per second

3. The average of five tests is 85. If two test scores have 5 points removed from each, 1 test score has 20 points add, and the remaining two remain the same ,the new average is : (A) 84 (B) 85 (C) 86 (D) 87 (E) 88

85(5) ! 2(5) + 20

5 = 87

5. Doug Deap was hired to dig fence post holes. It takes him 1 12

hours to dig 5 holes. His hourly pay

was $12.50 per hour. How many holes did Doug dig if he received $150.00 for the job? (A) 55 (B) 50 (C) 45 (D) 40 (E) 35 Determine how many hours Doug worked.

150

12.50 = 12 hours

Determine how many 1 12

hour intervals there are in 12 hours

12

11

2

= 8

Determine the number of wholes he dug. 8(5) = 40 7. The numerator of a fraction is 6 less than the denominator. If the numerator is doubled and the

denominator is increased by 4, the resulting fraction is 23

. Find the original fraction.

(A) 1319

(B) 1117

(C) 712

(D) 511

(E) 17

Let D = denominator of the fraction

2(D ! 6)

D + 4 = 2

3

3(2)(D - 6) = 2(D + 4) 6D - 36 = 2D + 8 4D = 44 ; D = 11

The original fraction = D ! 6

D = 11! 6

11 = 5

11

8. A legend on a map shows 2.5 cm representing 200 miles. The distance on the map from El Paso to Texarkana is 9.8 cm. According to the map, how far is it from El Paso to Texarkana? (A) 735 miles (B) 750 miles (C) 763 miles (D) 784 miles (E) 800 miles

Solution : 2.5

200 = 9.8

x

2.5x = 200(9.8) ; x = 200(9.8)2.5

= 784 miles

10. Phip Upp’s ruck gets 17 miles per gallon of gas. He has $20.00 to spend on gas. If the cost of a gallon of gas is $3.50, how far can Phil drive? (nearest whole mile) (A) 70 miles (B) 76 miles (C) 97 miles (D) 100 miles (E) 102 miles

Solution : 203.5(17) = 97 miles

11. Find the ratio of the median to the mean of the following list of numbers.

2, 3, 5, 2, 4, 3, 2, 0, 5, 3, 5, 2 (A) 1 : 1 (B) 3 : 2 (C) 3 : 5 (D) 2 : 5 (E) 1 : 2 Solution : Arrange the numbers to find the median. 0, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5 Since there is an even number of terms, the median is the average of the two numbers in the middle. The two numbers in the middle are both 3 and the median is 3. Mean = (sum of 12 terms)/12 = 36/1`2 = 3

Ratio of the median to the mean = 33

= 11

= 1 : 1

12. The sum of three numbers is 98. The ratio of the smallest number to the middle number is 23

.

The ratio of the middle number to the biggest number is 58

. The range of the three numbers is :

(A) 48 (B) 30 (C) 28 (D) 20 (E) 18 Solution : x + y + z = 98

x

y = 2

3 ; 2y = 3x ; x = 2

3y

y

z = 5

8 ; 8y = 5z ; z = 8

5

2

3y + y + 8

5y = 98

152

3y + y +

8

5y

!

"#$

%& = 15 98[ ]

10y + 15y + 24y = 1470 49y = 1470 ; y = 30

x = 23(30) = 20

z = 85(30) = 48

The three numbers are 20, 30, and 48, The range of these numbers is 48 - 20 = 28.

17. Sav-A-Lot has a bookshelf on sale for $84.49. The sales price is 30% off of the regular price. How much money did Sav-A-Lot save you by having the book-

shelf on sale? (tax not included) (A) $25.35 (B) $36.21 (C) $42.07 (D) $54.49 (E) $59.14 Let x = the regular price If you have a 30% discount of the regular price, this can be represented by 70%.

70%x = 84.49 ; .70x = 84.49 ; x = 84.4970

= 120.70

Amount of savings = Regular price - sales price = $120.70 - $84.49 = $36.21 20. Ima Yung's age is one-third of Ur Old's age. Five years ago Ima's age was one-fourth of Ur's age then. How old is Ima now? (A) 12 (B) 15 (C) 18 (D) 21 (E) 24

Let x = Ur Old's age amd 13

x = Ima's age

1

3x - 5 = 1

4(x - 5)

121

3x ! 5

"

#$%

&' = 12

1

4(x ! 5)

"

#$%

&'

4x - 60 = 3(x - 5) 4x - 60 = 3xdd - 15

x = 45 ; Ima's age now = 13

x = 13

(15) = 5

21. If 1 25

of A is 60% of B, then 1 25

of B is what percent of A?

(A) 326 23

% (B) 166 23

% (C) 84% (D) 60% (E) 40%

1 25

A = 60%B ; 75

A = 35

B

B = 53

7

5A

!"#

$%&

= 73

A

7

5B = 7

5

7

3A

!"#

$%&

= 4915

A ; Since 75

B = 4915

A, you must convert 4915

to a percent. 49

15

!"#

$%&

100 = 326 23

%

ALGEBRA 1 1. Tryce Ikle can get to school in 12 minutes riding his bike at an average of 15 miles per hour (mph). How many minutes would it take him to walk to school if he walks at 4 mph? (A) 31 (B) 32 (C) 45 (D) 48 (E) 72 Distance = Rate x Time

D = 1512

60

!"#

$%&

= 3 miles

Time = Dis tanceRate

= 34

Hour = 3

4

!"#

$%&(60) = 45 minutes

3. Kandy Heart had a box of valentines. She gave 23

of them to her classmates. She gave 5 of

the remaining valentines to her brothers and sisters. She had 3 left over for her father, her mother, and herself. How many valentines were in the original box? (A) 12 (B) 18 (C) 24 (D) 30 (E) 36

Let x = original number of valentines ; 23

x = Number of valentines she gave her classmates

1

3x - 5 = 3

1

3x = 8 ; x = 24

5. Six boys and twelve girls are in the senior class. Half of the boys and 25% of the girls wear glasses. What is the probability that a student chosen randomly is a boy, wears glasses, or both?

(A) 16 23

% (B) 25% (C) 33 13

% (D) 50% (E) 66 23

%

Make a Venn diagram. You will find that 3 of the boys and 3 of the girls were glasses. You will also find that 3 of the boys and 9 of the girls do not wear glasses.

Probability = Favorable

TotalOutcomes = 6 + 3

18 = 9

18 = 1

2 = 50%

8. Which of the following is true about the relation h(x) = 5 - x 2 ? (A) odd function (B) even function (C) neither even nor odd function

(D) not a function (E) none of these are true Note : If h(- x) = h(x), then h(x) is and even function. If h(- x) = - h(x), then h(x) is an odd function. Solution : h(x) = 5 - x 2 h(- x) = 5 - (- x) 2 = 5 - x 2 Since h(- x) = h(x), then h(x) is an even function. 9. If two dice are rolled at one time, what is the probability that both dice show a prime number?

(A) 8 13

% (B) 9% (C) 16 23

% (D) 25% (E) 66 23

%

When a die is tossed, the prime numbers that can appear on one of its faces are 2, 3, and 5. Solution : Probability that both dice show a prime number is equal to

1

2

!"#

$%&1

2

!"#

$%&

= 14

= 25%

10. Line L going through points ( - 1, 3) and (k, - 5) is perpendicular to x + 4y = 5. Find k. (A) - 5 (B) - 3 (C) - 1 (D) 2 (E) 5 Solution : x + 4y = 5 4y = - x + 5

y = - 14

x + 54

; The slope of y is - 14

.

Slope of the line perpendicular to y is 4.

m = y2 ! y1x2! x

1

; 4 = !5 ! 3

k ! (!1)

!8

k +1 = 4

4k + 4 = - 8 4k = - 12 ; k = - 3 13. Missy Klass was absent the day of the algebra exam. She took the test the next day and made a 96. Her score raised the class average from 71 to 72. How many students, including Missy, took the test? (A) 22 (B) 24 (C) 25 (D) 26 (E) 28 Solution : Let x = number present when exam was first given

Note : 71x = total number of points earned by the students originally present.

71x + 96

x +1 = 72

71x + 96 = 72x + 72 24 = x Total number of students taking the exam (including Missy) was 24 + 1 = 25 14. The set {…, - 6, - 4, - 2, 0, 2, 4, 6, …} is closed under which of the following operations : I. addition II. subtraction III. multiplication IV. division (A) all of these (B) I & III only (C) I, II, & III (D) II & IV (E) None of these Note : Closure :When you combine any two elements of the set, the result is also included in the set. Solution : If you add, subtract or multiply elements in the given set, the result will be an element of The set. Thus, the answer is C. 15. Let R = {1, 3, 5}, S = {0, 2, 4}, and T = {1, 2, 3}. How many elements are in (R ! T) ! (S ! T)? (A) 6 (B) 5 (C) 4 (D) 3 (E) 2 Solution : R ! T = {1, 2, 3, 5} S ! T = {0, 1, 2, 3, 4} (R ! T) ! (S ! T) = {1, 2, 3} Answer is 3. 35. 2x - 3 and 4x + 5 are factors of which of the following trinomials? (A) 8x 2 - 22x - 15 (B) 8x 2 + 2x + 15 (C) 8x 2 + 2x - 15 (D) 8x 2 - 2x + 15 (E) 8x 2 - 2x - 15 (2x - 3)(4x + 5) (2x)(4x) + (- 3)(4x) + (5)(2x) + (- 3)(5) 8x 2 - 12x + 10x - 15

8x 2 - 2x - 15 54. Five consulting firms agree to contribute equally to the cost of a joint technical library. If three more firms join the plan, the cost of each member of the group would be reduced by $900. Find the cost of the library? (A) $9000 (B) $12000 (C) $6000 (D) $18000 (E) $4500 Let x = cost per firm when there are 5 firms paying for the library

5x

8 = x - 900

5x = 8(x - 900) 5x = 8x - 7200 - 3x = - 7200 x = 2400 ; Cost of the library = 5(2400) = 12000 GEOMETRY 1. The area of a rectangle is 300 cm 2 . The ratio of its length to its width is 4:3. The perimeter of the rectangle is : (A) 125 cm (B) 112 cm (C) 100 cm (D) 70 cm (E) 35 cm Let 3x = the width of the rectangle and 4x = the length of the rectangle. Area = length x width 300 = (4x)(3x) 12x 2 = 300

x 2 = 30012

x = 300

12

Perimeter = 2(length x width)

P = 2(4x + 3x) = 14x = 14 300

12 = 70

4. A tangent and a secant intersect at point A in the exterior of a circle. The measures of the two intercepted arcs are 75° and 50°. What is the measure of angle A formed by the tangent and the secant? (A) 125° (B) 62.5° (C) 37.5° (D) 25° (E) 12.5°

The measure of the angle formed by a secant and a tangent drawn from a point outside a circle is equal to one-half the difference of the two intercepted arcs.

1

2(75° - 50°) = 1

2(25°) = 12.5°

17. It is precisely 2:45 pm on a circular clock. What is the measure of the smaller angle formed by the minute hand and the hour hand of the clock? (A) 192° (B) 187.5° (C) 150° (D) 168° (E) 172.5° Make sketch depicting the information given. The smaller angle is 180° less than Measure of the angle traced by the clock in 15 minutes. Solution : Every hour is equal to 360°/12 = 30°. 15 minutes is equal to ¼(30°) = 7.5°. 180° - 7.5° = 172.5° 29. ! ABC is similar to ! DEF. !A " !D,!B " !E,and!C " !F. AB = 36, DE = 27, BC = 40, and DF = 18. Find AC + EF. Sketch drawing of the two similar triangles. Label the information given.

36

27 = AC

18 ; AC = 36(18)

27

36

27 = 40

EF ; EF = 27(40)

36

AC + EF = 36(18)27

+ 27(40)36

= 54

32. The areas of two circles are in the ratio of 4:9. If the radius of the smaller circle is 4", then the diameter of the larger circle in inches is : (A) 8 (B) 9 (C) 12 (D) 16 (E) None of these Let r = radius of the smaller circle and R = radius of the larger circle

!r2

!R2

= 49

; ! (4)2

!R2

= 49

; 16R2

= 49

; 4R 2 = 16(9) ;

R 2 = 16(9)4

; R 2 = 36 ; R = 6

The diameter of the larger circle is 2(6) = 12. 35. The perimeter of an equilateral triangle is equal numerically to 4 times its area. The length of a side is : (A) 4 3 (B) 3 3 (C) 2 3 (D) 3 (E) None of these

If s = the length of a side of an equilateral triangle, its area is equal to

s23

4.

3s = 4 s23

4

!

"#$

%& ; 3s = s 2 3 ; s 2 3 - 3s = 0 ;

s(s 3 - 3) = 0 ; s = 0 or s = 33

= 3

37. Three tennis balls jus fit in a cylindrical tennis ball can. If teach ball is .5 inches in diameter, what is the volume of air (in cubic inches) left between the balls and the can? (A) 98.2 (B) 122.7 ( C) 8.2 (D) 12.3 (E) 65.4 Volume of cylinder = π(radius) 2 (height) 3 = 98.2 V

can = π(r) 2 (6r) = 6πr

Volume of sphere = 43πr 3

Vballs

= 34

3!r3

"

#$%

&'

Volume between balls and can = 6πr 3 - 4πr 3 = 2πr 3 = 2π(.5) = 98.2

MEMORIZATION

1 hour = 60 minutes 1 minute = 60 seconds 1 foot = 12 inches 1 yard = 3 feet = 36 inches 1 pound = 16 ounces 1 gallon = 4 quarts = 128 ounces 1 quart = 2 pints = 32 ounces 1 pint = 2 cups = 16 ounces 1 cup = 8 ounces 1 gallon = 231 cubic inches 1 square mile = 640 acres 1 inch = 2.54 centimeters 1 foot = 30.48 centimeters Normal body temperature = 98.6°F = 37°C Boiling point of water = 212°F = 100°C Freezing point of water = 32°F = 0°C 1 mile = 5280 feet = 1760 yards

1 cubic foot = 1728 cubic inches 1 cubic yard = 27 cubic feet 1 square foot = 144 square inches 1 square yard = 9 square feet 1 mile = 1760 yards = 5280 feet 10 millimeters = 1 centimeter 100 centimeters = 1000 millimeters = 1 meter 1 hectometer = 100 meters 1000 meters = 1 kilometer 1 dekameter = 10 meters 10 decimeters = 1 meter 1 year = 12 months = 365 days Leap year = 366 days Days in a month January (31) February (28 or 29) March (31) April (30) May (31) June (30) July (31) August (31) September (30) October (31) November (30) December (31)