Math Reviewer

Fundamentals of Mathematics

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Arithmetic • • • •

Sum = addend + addend (or terms) Difference = minuend – subtrahend Product = multiplicand  multiplier (or factors) Quotient = dividend  divisor

a • Take note! is undefined 0 2 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Arithmetic • Addition – Same sign: add the numbers, copy the sign – Opposite signs: subtract the numbers, copy sign of larger number

• Subtraction – Change the operation into addition, then change the sign of the subtrahend

• Multiplication/division – Count the number of negative signs; when even = 3 positive; when odd = negative Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Properties of arithmetic • • • • •

Commutative (,) a + b = b + a Associative (,) Distributive ( over  and  over +) Identity element (0 for  and 1 for ) Inverse elements – Additive inverses have the sum 0 – Multiplicative inverses have the product 1

• Multiplicative property of zero 4 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Properties of equality • Reflexive • Symmetric

aa

a  b; then b  a

• Transitive

a  b and b  c, then a  c

• Substitution

If a  b, then any expression involving a retains the same value if a is replaced by b. 5

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Order of operations: PEMDAS • Parenthesis or grouping symbols • Exponents • Multiplication and division (whichever comes first) • Addition and subtraction (whichever comes first)

http://www.mahalo.com/pemdas/

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Laws of exponents • A power is an expression involving repeated multiplication. • base: b; power or exponent: m or n • Take note! 0

a 1 1 a a

m n

m n

b b b m n mn (b )  b m b mn 0  b ; hence b  1 n b m m m (ab)  a b m m a a    m b b 0 1 a 1 n a  n ; since n  n a a a

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Scientific notation • Write numbers as the product of a number less than 10 and a power of 10. • The sign of the exponent is positive when the original figure is less than 1 and negative when more than 1.

2

245  2.45  10 3 0.0056  5.6  10 8 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Prime, Composite or Denominate? • Prime – Numbers whose only factors are 1 and itself

• Composite – Numbers with factors other than itself and 1

• Denominate – Numbers with a unit of measurement

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GCF and LCM • Greatest common factor – Useful in reducing fractions to lowest or simplest terms

• Least common multiple – Useful in operations involving dissimilar fractions

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Fractions • Improper fractions to mixed numbers

a remainder  quotient (a  b) b b

• Mixed numbers to improper fractions

a (b * c)  a c  b b 11 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Fractions • Equivalent fractions – Have the same value – Result from the multiplication and division of the numerators and denominators of a fraction by the same number

• Reducing to simple fractions – Determine the GCF of the numerator and denominator; divide them by the GCF 12 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Ratio and proportion • Ratios may be written as fractions or by using colons

2 is the same as 2 : 3 3 • Proportions are the equality between two ratios

2 4  is the same as 2 : 3 :: 4 : 6 3 6 13 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Sharing/dividing in a ratio When sharing/dividing any amount/quantity (e.g., n) in a ratio (e.g., a : b : c): • Add

a  b  c  number of parts

n  share per part • Divide number of parts • Multiply share per part * a  share of a

or b or c

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or b or c

Philippines.

Rules for ratio and proportion a c If  , then ad  bc. b d a c a b If  , then  b d c d. a c b d If  , then  . b d a c Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Variation • Direct:

y  kx

– Variable y varies at the same rate as variable x

• Inverse:

k y x

– As variable x increases, y decreases and vice-versa

• Joint – More than one variable is involved in a direct variation Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Algebra

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Mathematical sentences • is – was – will be – has:  • plus – sum – more than – greater than – farther than – years from now – increased by – exceeds by:  • decreased by – minus – less than – diminished by – younger than – years ago – subtracted from:  • divided by – quotient – per:  18 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Factoring polynomials • Factor out the common monomial factor • Difference of two squares 2

2

a  b  (a  b)(a  b) • Perfect square trinomial 2

2

a  2ab  b  (a  b)

2

• FOIL method and quadratic polynomials

(a  b)(c  d )  ac  (ad  bc)  bd 2 standard form : ax  bx  c 19 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Notes in solving word problems • Consecutive numbers: x , x  1, x  2 • Consecutive even/odd numbers: x , x  2, x  4 • Mixture (e.g., of coins, vehicles, chemicals): total amt = number of coins * denomination number of feet/wheels = animals/vehicle * # of feet/wheels of each 20 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Notes in solving word problems • Motion:

• Work:

distance speed  time part of a job done by one person (in 1 day/hour)  1 number of days/hours it takes the person to complete

• Age: add when it says “# of years from now” and subtract when it says “# of years ago” 21 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Notes in solving word problems • Interest

I  Prt

• Compound interest

r nt A  P(1  ) n

or future value FV  PV (1  r ) t • Continuous compound interest

A  Pe

rt

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Notes in solving word problems • Percent change

new  original % change  original • Discount

Discount  rate  original price Discounted price  original price  discount • Commission

Commission  rate  total earnings 23 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Functions • A function is a set of ordered pairs such that for each first component, there is at most one value of the second component • Three things that specify a function: – Domain – Range – A rule which assigns to each member of the domain exactly one member of the range 24 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Functions • Domain – set of all first components in each element of the set

• Range – set of all the second components in each element of the set

• When y is a function of x, the set of values of x is the domain of the function and the set of values of y is the range of the function. 25 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Geometry

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Lines • Parallel lines do not meet. • Intersecting lines meet at one point. The sum of any 2 adjacent angles formed is 180°. • Perpendicular lines meet at one point, at 90°.

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Kinds of Angles • • • • •

Acute: less than 90° Right: 90° Obtuse: more than 90°, but less than 180° Straight: 180° Reflex: more than 180 degrees, less than 360°

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Angle relationships • Supplementary angles – Sum of measures is 180°

• Complementary angles – Sum of measures is 90°

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Pythagorean theorem • Applies to right triangles • Legs: a and b • Hypotenuse: c

2

2

c  a b

2

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Polygons • Polygons are flat, closed figures consisting of line segments. • Exterior angles of a polygon add up to 360°. • The sum of the interior angles of a polygon, with the number of sides n: 180° (n  2) • The sum of the exterior angles of a simple, closed polygon is 360°. 31 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Measurements • Perimeter: sum of the length of all sides

Prectangle  2l  2w

Circumference  2r

Ptriangle  a  b  c Psquare  4s

• Area: size covered by a surface

Asquare  s

2

Acircle  r

2

Arectangle  lw 1 Atriangle  bh 2

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Measurements • Volume: the space occupied or contained by a substance or shape

Vcube  s V pyramid

3

1  bh 3

4 3 Vsphere  r 3 1 2 Vcone  r h 3

Vrectangular prism  abc or lwh 33 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Statistics, Probability and Number Series

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Measures of central tendency • Mean – Add the numbers and divide the sum by the number of values (which were added).

• Median – The middle number when numbers are written in order. (When the number of values is even, the median is the mean of the two middle values.)

• Mode – The number which occurs most often 35 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of

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Counting techniques • If two or more actions are performed in a particular order, such that the first action produces a results, the second action produces b results, the third action produces c results, and so on, then:

a * b * c * ...  total number of possible results

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Counting techniques: Permutation • Objects are arranged in a definite order: – n different things taken r at a time – n things taken n at a time in which q are alike, r are alike and so on – n distinct objects are arranges along a fixed circle

n! n Pr  (n  r )! n! n Pr  (q! r!...) Pn  (n  1)!

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Counting techniques: Combination • Objects are not arranged in a particular order – n different things taken r at a time

n! n Cr  (n  r )! r!

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Probability Probability of an outcome  number of ways an outcome can happen the total number of possible outcomes

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Arithmetic sequences and series • Arithmetic sequence/progression: a sequence of numbers where successive members of the sequence have a constant difference (d). • Formula for n-th term an  a1  (n  1)d • Formula for sum of first n values of a sequence n(n  1)

Sn 

2 40

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Geometric sequences and series • Geometric sequence/progression: a sequence of numbers where successive members of the sequence are produced by multiplying them by a constant multiplier (r, or common ratio)

a n  a1 * r • Formula for n-th term • Formula for sum of first n values of a n sequence a1 (1  r )

Sn 

n 1

1 r

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References and further reading British Broadcasting Corporation. (2012). GCSE Bitesize: Maths. Retrieved 27 December 2012, from http://www.bbc.co.uk/schools/ gcsebitesize/ maths/ Fisico, M., L. Sia, J. Ho, Z. Maralit, J. Tan, . . . J. Rosello, R. Franco. (1998). 21st Century Mathematics Series. Quezon City: Phoenix Publishing House, Inc. Pierce, Rod. (2012). Math Is Fun. Retrieved 28 December 2012, from http://www.mathsisfun.com/index.htm Stapel, E. (2012). The Purplemath Forums. Retrieved 27 December 2012, from http://www.purplemath.com/ index.htm

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