QUICK MATH REVIEW & TIPS

1

Basic Facts & Rules To Remember

Word of Advise

 For a good and lasting foundation in Math, know your multiplication tables by all means.

 Knowing multiplication translates to being able to figure out division problems in the shortest amount of time.

 Working with fractions, algebra, ratios and percentages also become easy to handle.

 Start solving Math problems using the facts, the rules & information you already know to guide you.

If you multiply two negative numbers the result will be positive:     4 × 9 = 36 3 × 9 = 27 7 × 8 = 56 6 × 9 = 54

If you multiply a negative and a positive number the result will be negative :    4 x 9 = 36 4 x 9 = 36 5 x 8 = 40

If you subtract a larger number from a smaller number the result will be negative:   12 - 25 = -13 18 - 38 = -20

If you add two negative numbers the answer will be a “bigger” negative number:    5 + 7 = 5 + ( 7) = 12 23 + 12 = 23 + ( 12) = 35 51 + 10 = 61 ( If you owe money and you borrow more you will owe more money. More negative )

Any number multiplied by ONE gives the same number:   5 x 1 = 5 100 x 1 = 100

Any number multiplied by ZERO gives ZERO as the result:  3 x 0 =0  12 x 0 = 0  A x 0 = 0  10,000 x 0 = 0

If you divide two negative numbers the answer will be positive:  24 ÷ 8 = 3  -42 = 6 -7  -72 = 9 -8

If you divide a negative number by a positive number the answer will be negative:  24 ÷ 8 = 3  42 = 6 7  72 = 9 8

If you divide a positive number by a negative number the answer will be negative :  24 ÷ 8 = 3  42 = 6 -7  72 = 9 -8

 18 ÷ 6 is the same as 18 6  24 ÷ 3 is the same as 24 3  18 is the same as 18 ÷ 3 3

          

Some tips on Simplification

Minus, Minus is Plus 5

-

8 = 5 + 8 = 13 Minus, Plus is Minus 15

- +

8 = 15

-

8 = 7 Plus, Minus is Minus 15

+ -

8 = 15

-

8 = 7 The easiest way to simplification problems involving order of operations is to use the BEDMAS or PEDMAS or PEMDAS approach. Learn and use the one you can easily remember.

PEMDAS is short for

P

arenthesis

E

xponents

M

ultiplication

D

ivision

A

dditions

S

ubtraction BEDMAS is short for

B

rackets

E

xponents

D

ivision

M

ultiplication

A

dditions

S

ubtraction PEDMAS is short for

P

arenthesis

E

xponents

D

ivision

M

ultiplication

A

dditions

S

ubtraction This means when working on long simplification problems, do Brackets or Parenthesis first, then Exponents, then Division, then Multiplication, followed by Addition and finally Subtraction 13 + 3(14 -5)-5+20 16 - 2 + 39 ÷ 3+(7×2-18) Always follow the order but skip operations that are not mentioned in the simplification question.

Practice Questions

 (4(12 - 4) + 10) ÷ 7=  Calculate 113 – 3(3 + 2) 2 + 12 ÷ 2

How Percentages, Fractions and Decimals relate to each other

     2% is the same as 2 which is the same as 0.02

100 25% is the same as 25 which is the same as 0.25

100 10% is the same as 10 100 which is the same as 0.10 12.5% is the same as 12.5

100 and also be written as 0.125

Notice that when you write the percentage as a fraction each zero in the denominator represent a single move of the decimal point to the left in the numerator when you convert it to decimals.



Some Tips & Tricks in Converting fractions to decimals.

In the absence of a calculator always check to see if the denominator of the fraction can be converted to a ten, a hundred, a thousand and so forth by multiplying by a number.

 If you can multiply the denominator by a number to get 10, 100, 1000 etc., then multiply both the numerator and denominator by this number.

 Now convert the numerator to the decimal number by moving the decimal point to the left as many times as there are zeros in the denominator.

 Note that each zero in the denominator represents a “one decimal place move” to the left.

  If the numerator did not originally contain a decimal point, start by assuming that there is a decimal point right after the last digit.

(14 is the same as 14.0 or 14.)

Examples

 Write 3 5 as a decimal.

First we look at the denominator We can convert this to 10 by multiplying by 2.

We have to also multiply the numerator by 2 so the value of the fraction remains the same 3 = 3 x 5 5 x 2 = 6 = 0.6

2 10

 Write 7 as a decimal.

20 7 = 7 x 5 = 35 = 0.35

20 20 x 5 100

THE

OF

KEYWORD A given percentage equal to the Percentage multiplied by that quantity.

OF

a certain quantity is  10%

of

a certain quantity can be expressed as (10% x the quantity ) 10%

of

200 = 10%

×

200 = 10

×

200 = 20 100 1

A given Fraction

OF

a certain quantity is equal to the Fraction multiplied by that quantity:  ¾ ( ¾

of

a certain quantity can be expressed as x the quantity ) ¾

of

20 = ¾

×

20 = 3 4 1

×

20 5 1 = 15

EXAMPLES

1.) What is 15% of 500 ?

Answer: 15% of 500 15 x 500 = 15 x 5 = 75 100 1 1 1

2.) What is two-fifth of 80 ?

Answer: 2 of 80 5 2 5 1 x 80 16 = 2 x 16 1 1 1 = 32

3.) What percentage of 250 is 40 ?

Answer: Lets represent “what percentage” which we don’t yet know with the letter p ( you can use any letter).

Then we can write the following: p% of 250 is 40 p x 250 100 1 = 40 (We can now solve for p) 2.5p = 40 p = 40 = 16 2.5

So 16% of 250 is equal to 40

4.) What fraction of 120 is 36?

Answer: Let p represent the unknown fraction (i.e “what fraction”) p of 120 is 36 p x 120 = 36 (Notice that we did not divide p by 100 because we want a fraction) p = 36 = 3 120 10

5.) 55% of the students in a school are boys. If there are 330 boys, what is the total number of students in the school?

Answer: We ask ourselves, “What is the unknown here?” The unknown is the “total number of students” Lets represent the total number of students by T.

We can write the following mathematical statement: 55% of T is equal to 330 55 100 x T = 330 0.55T = 330 T = 330 = 600 .55

What is a Reciprocal?

 The reciprocal of a whole number is 1 divided by the whole number.

So the reciprocal of 5 will be 1

5

 As you can see, the reciprocal of a whole number becomes a fraction.

 The reciprocal of a fraction is the fraction you get when the numerator and denominator switch places.

So the reciprocal of 7 9 will be 9 7

LEAST COMMON MULTIPLE (LCM)

 In LCM we are looking at the multiples of two or more numbers to find out which of the multiples appear in all (COMMON) the numbers and at the same time the smallest.

 For example to find the LCM of 8 and 12 lets write out the multiples of each to a point: 8=>8,16, 24 ,32,40, . . .

12=>12, 24 ,36,48, . . .

 Right away you notice that 24 is the first multiple of both 8 and 12. It is also the smallest or least of the multiples.

So the LCM of 8 and 12 is 24  To summarize 24 is both the C ommon M ultiple and the L east.

Finding LCM using the traditional method

 Start out by making a list of the multiples of each given number  Look through the multiples for each given number and find which of the multiples appear in both lists or are

common

to both numbers.

 For example we want to find the LCM of 16 and 24:   16 : 16, 32, 48 , 64, 80 …….

24 : 24, 48 , 72, 96 ……..

 We notice that 48 is the first multiple that is common to both 16 and 24 so 48 is our LCM



LEAST COMMON MULTIPLE (LCM) IN FOUR STEPS:

1.)Write out the prime factors of each given number.

2.) Look for each COMMON factor and write it down only once for each time the common factor appears.

3.)Look for each Non-Common factor and write it down once.

4.)Multiply the factors from steps 2. and 3. above.

For example to find the LCM of 16 and 24, write each number using its prime factors: 16 = 2 .

2 .

2 .2

24 = 2 .

2 .

2 .3

LCM= 2 .

2 .

2 .2.3 =48

Practice Questions

 What is the least common multiple of 4, 6 and 10 ?

 What is the least common multiple of 6,10 and 14?

 Try using both method to arrive at your answers and see which one is faster.

GREATEST COMMON FACTOR (GCF)

 In GCF we are looking at the factors of two or more given numbers to determine which of the multiples appear in all (COMMON) the given numbers. This common factor should also be the greatest or largest or highest.

 GCF is also referred to as HCF (Highest Common Factor). They both mean the same thing.

 For example to find the GCF or HCF of 8 and 12 lets write out the factors of each number: 8=> 1 , 2 , 4 ,8 12=> 1 , 2 , 3, 4 , 6, 12  We notice from the list of factors that 1, 2 and 4 are common to both lists. Of these three common factors 4 is the greatest or highest.

So the GCF of 8 and 12 is 4  To summarize two or more given numbers may have more than one factor which is common to them but we are only interested in the greatest of the common factors.

GCF or HCF using the traditional method

 Start out by making a list of the factors of each given number.

 Look through the factors for each number and find which of the factors appear in both lists.

 The largest or highest of the common factors is the GCF or HCF.

 For example we want to find the GCF or HCF of 16 and 24:   16 : 1, 2 , 4 , 8 ,16 24 : 1, 2 ,3, 4 ,6, 8 ,12,24  We notice that of the common factors 1, 2 , 4 , 8 the greatest or highest one is 8 so 8 is our GCF or HCF.



Greatest Common Factor (GCF) or Highest Common Factor (HCF) in THREE STEPS

 1.) Write out the given number as a product of its prime factors  2.) Look for each prime factor that is COMMON to all the given numbers and write it down only once for each time that the factor appears common.

 3.) Multiply the common prime factors from steps 2 to get the GCF or HCF.

    e.g.

16 = 2 · 2· 2· 2 24 = 2· 2· 2· 3 The GCF is 2· 2· 2 = 8  What is the greatest common factor of 9, 12 and 15?

 Find the GCF of 24, 36 and 54