MATH TIPS

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QUICK MATH REVIEW TIPS3
Step into Algebra and Conquer it
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Exponents

• Exponents are used to simplify expressions where
  the same number multiplies itself several times.
• For example the number 8 can be written as the
  product 2x2x2 or 222
• Instead of writing 8222 we can shorten the
  expression by using exponents.
• So we write 823 23 is short for 222)
• 2 is called the base and 3 is the power or
  exponent
• 23 is read as Two to the power three or Two
  raised to the power three
• Other examples
• 27333 33
• 162222 24

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• If you raise any number to the power of 0 you
  will get 1
• 20 1
• 30 1
• 1000 1
• x0 1
• 10000 1

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• If you raise any number to the power
• of 1 you will get the original number
• 21 2
• 31 3
• 51 5
• 1001 100

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• If you raise any number to the power of
• NEGATIVE ONE (-1) you will get the
• reciprocal of the original number
• 2-1 1
• 2
• 3-1 1
• 3

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• If you raise a number to a negative power
• you get the reciprocal of the original number
• raised to the positive of the original power
• 2-3 1 1
• 23 8
• 3-4 1 1
• 34 81

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• When you multiply exponents that have a common or
  similar base, their powers simply add up
• 8 2.2.2 23
• 27 3.3.3 33
• 36 4.9 2.2.3.322.32
• 23.2528 2(35)
• 26.2-422 2(6-4)

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Practice Questions

• Write the product 333777 using exponents.
• Write 108 as a product of its prime factors in
  expanded form and then in exponential form.
• Find the value of n in 813(n-2)
• Find the value of d in the below equation if n7
• d2(n3).3 (n-4)
• If 722(x-2)32, what is the value of x?

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Solving for the Unknown.

• You can use any letter to represent an UNKNOWN in
  a
• any Math problem.
• For example if we are told that there are 36
  students in a class out of which 23 are boys and
  we are required to find the number of girls, we
  can start out by choosing a letter to represent
  the unknown (in this case the number of girls).
  Then write down a simple equation for the total
  number of students in the class with the
  information we know so far.
• If we choose the letter g to represent the number
  of girls, then we can write
• 23 g 36
• where 23 is the number of boys
• g is the letter we have selected to represent the
  number of girls
• 36 is the total number of students in the class
• Using letters to represent unknowns comes very
  handy when dealing
• with very long statement math problems. Just
  follow the statements in the question
• patiently and use letters to represent the
  unknowns as necessary. You will
• then be able to write down a mathematical
  statement in place of the long

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• In Algebra you are mostly looking for some
  UNKNOWN value in a given equation.
• The UNKNOWN is also called a variable and may be
  represented by any letter.
• For example in the equation 2p 24,
• p is the unknown or the variable.
• 2p means 2 times p (that is 2 x p )
• The number 2 in this case is called the
  coefficient of p
• So 2p24 is the same as 2 x p 24
• In this simple situation it is easy to see that 2
  x 12 24 so p12
• How did we get p to be equal to 12 ?
• By looking for the number that will multiply 2 to
  give 24

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• In a Math problem if you see the unknown, such as
  p, standing by itself it is the same as 1p ( that
  is to say p1p)
• So p 2p means 1p 2p which is equal to 3p
• And 3p p 2p (Notice that 3p p ¹ 3)
• If you have 3 apples and you give away 1
  apple you will be left with 2 apples.
• Each item in the equation is called a term, with
  the exception of the operators (, -) and the
  equal to sign ().
• In the equation 2p 6 20, the terms are
  2p, 6 and 20
• If you add or subtract two or more unknown terms,
  simply add or subtract the actual numbers and
  then apply the unknown to the result
• 2p 5p 7p
• 4p -3p p

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• There are a couple of ways to deal with a basic
  algebra equations in order to find the unknown.
• One method is to continue performing the same
  actions to both sides of the equation (to the
  left and the right of the equal to sign) until
  you have the unknown terms standing on one side
  of the sign and all other terms standing on
  the opposite side of the sign.
• Performing the same actions to both sides of the
  equation means that if you add, subtract, divide
  or multiply one side of the equation by a number,
  you must do the same on the opposite side of the
  equation.
• An example will help clarify this.

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• We want to find p in the equation, 2p 7 21
• Step 1 Decide where you would finally want the
  unknown, p, to stand when you
• finish solving the problem. Would you
  want it on the left or the right side of the
• sign
• Step 2 Start adding and subtracting terms you
  would like to disappear from one side of the
  equation and appear on the other side until you
  have the unknown term standing alone.
• Step 3 The last step normally involves dividing
  or multiplying each side of the equation by
• the number associated with the
  unknown (or coefficient)
• Now the solution
• 2p 7 21
• - 7 -7
• 2p 0 21 -7
• 2p 14

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• What is the value of x in the following equation
• 2x -6 15 x
• Solution
• 2x -6 15 x lt--- this is the given
  equation
• 6 6 lt--- add 6 to both sides of the
  equation to zero out the -6 on the left
• 2x -0 21 x lt--- after adding 6 to each
  side
• -x - x lt--- subtract x from both sides
  to zero out the x on the right side
• x 21
• (Remember that 2x x is the same as 2x -1x which
  is
• equal to 1x or simply x)

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• How difficult was that?
• Just keep in mind that any action you take on the
  left side of the sign must be taken at the
  same time on the right side of the . Do this
  before you even blink.
• Now try the following
• 2p -6 32
• 3x 18 45 - x
• 2p -6 15 p
• 13 5x 35 x
• 3(8 x) 5 2x -16

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• Another way to solve an algebra equation is by
  re-arranging the terms in the equation so that
  all LIKE terms are grouped together on either
  side of the sign.
• Before you move each term, note that the operator
  in front of a term makes the term either positive
  or negative.
• Any term that you move from one side of the
  sign to the opposite side will have its sign
  changed.
• You will re-arrange the equation by moving terms
  to join their likes on either side of the
  sign.
• The unknown terms together with their
  coefficients are considered like or similar
  terms. Move them to one side of the equal to
  sign.
• All the other terms (without the variables)
  belong to a different group of LIKE terms. Move
  them to the opposite side of the equal to sign

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• 2p 7 21
• Grouping Like terms together
• 2p 21 7
• You notice that 7 has become -7 as soon as it
• crossed the sign from the left side to the
  right
• side.
• 2p 14
• 2p 14
• 2 2
• p 7

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• You can normally carry out the rearrangement in a
  single step but it is okay to use as many steps
  to group LIKE terms together as you feel
  comfortable
• Let us find the value of p in the equation
• 2p - 6 15 p
• Solution
• 2p - 6 15 p
• Grouping like terms together
• 2p p 15 6
• p 21

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• Try the following
• 2x - 6 32
• 3p 18 p 45
• 2p -6 27 p
• 13 5p 35 p
• 3(8 x) 5 2x -16

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RATIOS PROPORTIONS

• A ratio compares two or more actual quantities
  using smaller equivalent quantities or numbers.
• For example if we are told that a basket contains
  20 apples and 30 oranges, we can represent these
  actual quantities using a ratio by saying that
  the ratio of apples to oranges is 20 to 30.
• We can also write the ratio of apples to oranges
  as 2030
• Using smaller equivalent numbers we can simplify
  2030 and represent the ratio as 23 or 2 to 3
• From the question it is clear that there is a
  combined total of 50 (apples oranges) in the
  basket

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• The previous question could have been asked in a
  different way
• A basket contains apples and oranges in the ratio
  of 2 apples to 3 oranges and there is a total of
  50 apples and oranges together. How many of each
  fruit is in the basket?
• In this case we are given the smaller equivalent
  numbers or apples and oranges and are required to
  find out the actual quantities of each fruit in
  the basket
• We can write the ratio of apples to oranges as
  23 or 2 to 3
• This means that if we decide to group the 50
  fruits in equal quantities using the given ratios
  then for every 5 (i.e. 23) fruits there will be
  2 apples and 3 oranges.
• 23 5 is the total ratio
• In other words 2 out of 5 (two-fifth) Of the
  total number of fruits are apples and 3 out of 5
  (three-fifth) Of the total number of fruits are
  oranges.
• In fractions
• 2 Of 50 are apples
• 5

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• When a given quantity ,Q, is split in the ratio
  abc, we can find the actual quantities of a, b
  and c by writing the fractions for the ratios.
• If we represent the actual quantities for a, b
  and c with the symbols
• A, B and C respectively then we can calculate
  as follows
• A a x Q
• abc
• B b x Q
• abc
• C c x Q
• abc
• The above formulas do not need to be memorized.
  They are only intended to be understood and
  applied in ratio and proportions calculations.

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Another Example

• There are 350 students in a school. The ratio of
  boys to girls in the school is 5 to 2. What are
  the actual numbers of boys and girls in the
  school.
• Solution
• Total number of students 350
• Ratio of Boys to Girls 52
• Number of Boys 5 x 350 5 x 35050
• 5 2 71
• 250
• Number of Girls 2 x 350 2 x 35050
• 5 2 71
• 100

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Now Try These

• There are 18 girls in a class. If there are six
  more boys than girls in the class, find the ratio
  of boys to girls in the class. What is the total
  number of students in the class?
• A box contains 24 pencils and 42 pens. What is
  the ratio of pens to pencils in the box?
• David, Kim and Isaiah want to share an amount of
  120 in the ratio 235. How much will each
  person get?