QUICK MATH REVIEW & TIPS 1 Basic Facts & Rules To Remember.

QUICK MATH REVIEW & TIPSQUICK MATH REVIEW & TIPS

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Basic Facts & Rules To Basic Facts & Rules To Remember Remember

Word of AdviseWord of Advise For a good and lasting foundation in Math, know For a good and lasting foundation in Math, know

your multiplication tables by all means.your multiplication tables by all means.

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

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

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

If you multiply two negative numbers If you multiply two negative numbers

the result will be positive: the result will be positive:

--4 4 ××--9 = 369 = 36 --3 3 ××--9 = 279 = 27 --7 7 ××--8 = 568 = 56 --6 6 ××--9 = 549 = 54

If you multiply a If you multiply a negativenegative and a and a

positivepositive number the result will be number the result will be

negativenegative : :

--4 x 9 = 4 x 9 = --3636 4 x 4 x --9 = 9 = --3636 5 x 5 x --8 = 8 = --4040

If you subtract a larger number from aIf you subtract a larger number from a

smaller number the result will besmaller number the result will be

negative:negative:

12 - 25 = 12 - 25 = --1313 18 - 38 = 18 - 38 = --2020

If you If you add two negative numbersadd two negative numbers the the

answer will be a answer will be a “bigger”“bigger” negative number: negative number:

--5 + 5 + --7 = 7 = --5 + (5 + (--7) = 7) = --1212 --23 + 23 + --12 = 12 = --23 + (23 + (--12) = 12) = --3535 --51 + 51 + --10 = 10 = --61 61

( If you owe money and you borrow more you will owe ( If you owe money and you borrow more you will owe more money. more money. More negativeMore negative ) )

Any number multiplied by ONE givesAny number multiplied by ONE gives

the same number:the same number:

55 x 1 = x 1 = 55 100100 x 1 = x 1 = 100100

Any number multiplied by Any number multiplied by ZEROZERO gives gives

ZEROZERO as the result: as the result:

3 x 0 =03 x 0 =0 12 x 0 = 012 x 0 = 0 A x 0 = 0A x 0 = 0 10,000 x 0 = 010,000 x 0 = 0

If you divide two negative numbers the If you divide two negative numbers the

answer will be positive:answer will be positive:

--24 ÷ 24 ÷ --8 = 3 8 = 3

-42-42 = 6 = 6

-7-7

-72-72 = 9 = 9

-8-8

If you divide a If you divide a negativenegative number by a number by apositivepositive number the answer will be number the answer will be negativenegative::

--24 ÷ 8 = 24 ÷ 8 = --3 3

--4242 = = --66 77

--7272 = = --99 88

If you divide a positive number by a negativeIf you divide a positive number by a negativenumber the answer will be negative : number the answer will be negative :

24 ÷ 24 ÷ --8 = 8 = --3 3

4242 = = --66

-7-7

7272 = = --99

-8-8

18 ÷ 6 is the same as 18 ÷ 6 is the same as 1818

66

24 ÷ 3 is the same as 24 ÷ 3 is the same as 2424

33

1818 is the same as 18 ÷ 3 is the same as 18 ÷ 3

33

Some tips on SimplificationSome tips on Simplification MinusMinus, , MinusMinus is is PlusPlus 5 5 - - --8 = 5 8 = 5 ++ 8 = 13 8 = 13

MinusMinus, , PlusPlus is is MinusMinus 1515 - +- +8 = 15 8 = 15 - - 8 = 78 = 7

Plus, MinusPlus, Minus is is MinusMinus 1515 + + - - 8 = 15 8 = 15 - - 8 = 78 = 7

The easiest way to simplification problems involving order of operations is to use the The easiest way to simplification problems involving order of operations is to use the BEDMASBEDMAS or or PEDMASPEDMAS or or PEMDAS PEMDAS approach. Learn and use the one you can easily remember.approach. Learn and use the one you can easily remember.

PEMDASPEMDAS is short for is short for PParenthesis arenthesis EExponents xponents MMultiplication ultiplication DDivision ivision AAdditions dditions SSubtractionubtraction BEDMAS BEDMAS is short for is short for BBrackets rackets EExponents xponents DDivision ivision MMultiplication ultiplication AAdditions dditions SSubtractionubtraction

PEDMASPEDMAS is short for is short for PParenthesis arenthesis EExponents xponents DDivision ivision MMultiplication ultiplication AAdditions dditions SSubtractionubtraction

This means when working on long simplification problems, do This means when working on long simplification problems, do BBrackets or rackets or PParenthesis first, then arenthesis first, then EExponents, then xponents, then DDivision, then ivision, then MMultiplication, followed by ultiplication, followed by AAddition and finally ddition and finally SSubtractionubtraction

13 + 3(14 -5)-5+2013 + 3(14 -5)-5+20

16 - 2 + 39 ÷ 3+(7×2-18)16 - 2 + 39 ÷ 3+(7×2-18)

Always follow the order but skip operations that are not mentioned in the simplification question.Always follow the order but skip operations that are not mentioned in the simplification question.

Practice QuestionsPractice Questions

(4(12 - 4) + 10) ÷ 7=

Calculate 113 – 3(3 + 2)2 + 12 ÷ 2

How Percentages, Fractions and How Percentages, Fractions and Decimals relate to each otherDecimals relate to each other

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

100100

25% is the same as 25% is the same as 25 25 which is the same as 0.25 which is the same as 0.25 100100

10% is the same as 10% is the same as 1010 which is the same as 0.10 which is the same as 0.10 100100

12.5%12.5% is the same as is the same as 12.512.5 and also be written as 0.125 and also be written as 0.125

100100

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

Some Tips & Tricks in Converting Some Tips & Tricks in Converting fractions to decimals.fractions to decimals.

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

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

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

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

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

(14 is the same as 14.0 or 14.)(14 is the same as 14.0 or 14.)

ExamplesExamples Write Write 33 as a decimal. as a decimal. 55 First we look at the denominatorFirst we look at the denominatorWe can convert this to 10 by multiplying by 2.We can convert this to 10 by multiplying by 2.

We have to also multiply the numeratorWe have to also multiply the numeratorby 2 so the value of the fraction remains theby 2 so the value of the fraction remains thesamesame

3 3 = = 3 3 xx 2 2 = = 6 6 = 0.6 = 0.65 5 5 5 xx 2 10 2 10

Write Write 7 7 as a decimal. as a decimal.

2020

77 = = 7 7 xx 5 5 = = 35 35 = 0.35= 0.35

20 20 x 520 20 x 5 100 100

A given percentage A given percentage OFOF a certain quantity is a certain quantity isequal to the Percentage multiplied by thatequal to the Percentage multiplied by thatquantity.quantity.

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

10% 10% ofof 200 200

= 10% = 10% ×× 200 200

= = 10 10 ×× 200200 = 20 = 20 100 1100 1

THE OF KEYWORD

A given Fraction A given Fraction OFOF a certain quantity is equal to a certain quantity is equal tothe Fraction multiplied by that quantity:the Fraction multiplied by that quantity:

¾¾ ofof a certain quantitya certain quantity can be expressed as can be expressed as ((¾¾ x x the quantity the quantity ))

¾¾ ofof 20 20

= = ¾¾ ×× 20 20

= = 33 ×× 202055 = 15 = 15

4411 1 1

EXAMPLESEXAMPLES1.) What is 15% of 500 ?1.) What is 15% of 500 ?

Answer:Answer: 15% 15% ofof 500 500

15 15 xx 500500 = = 1515 x x 55 = 75 = 75 100 1 1 1100 1 1 1

2.) What is two-fifth of 80 ?2.) What is two-fifth of 80 ?

Answer:Answer: 2 2 of 80 of 80 5 5 2 2 xx 80801616 = = 22 x x 1616 ==3232 5511 1 1 1 1 1 1

3.) What percentage of 250 is 40 ?3.) What percentage of 250 is 40 ?

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

Then we can write the following:Then we can write the following:

p% p% ofof 250 250 isis 40 40

p p xx 250250 == 40 (We can now solve for p) 40 (We can now solve for p) 100 1100 1

2.5p = 402.5p = 40

p = p = 40 40 = 16 = 16 2.52.5

So 16% of 250 is equal to 40So 16% of 250 is equal to 40

4.) What fraction of 120 is 36?4.) What fraction of 120 is 36?

Answer:Answer:

Let Let pp represent the unknown fraction (i.e “what fraction”) represent the unknown fraction (i.e “what fraction”)

pp of 120 is 36 of 120 is 36

p x 120 = 36 (Notice that we did not divide p by 100 because we want a fraction)p x 120 = 36 (Notice that we did not divide p by 100 because we want a fraction)

p = p = 36 36 = = 3 3120 10120 10

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

Answer:Answer:

We ask ourselves, “What is the unknown here?”We ask ourselves, “What is the unknown here?”The unknown is the “total number of students”The unknown is the “total number of students”

Lets represent the Lets represent the total number of studentstotal number of students by T. by T.

We can write the following mathematical statement:We can write the following mathematical statement:

55% of T is equal to 33055% of T is equal to 330

55 55 xx T = 330 T = 330100100

0.55T = 3300.55T = 330

T = T = 330330 = 600 = 600 .55.55

What is a Reciprocal?What is a Reciprocal? The reciprocal of a whole number is 1 divided by the whole The reciprocal of a whole number is 1 divided by the whole

number.number.

So the reciprocal of 5 will be So the reciprocal of 5 will be 11 55

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

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

So the reciprocal of So the reciprocal of 77 will be will be 99 99 77

LEAST COMMON MULTIPLE (LCM)LEAST COMMON MULTIPLE (LCM)

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

For example to find the LCM of 8 and 12 lets write out the For example to find the LCM of 8 and 12 lets write out the multiples of each to a point:multiples of each to a point:

8=>8,16,8=>8,16,2424,32,40, . . .,32,40, . . . 12=>12,12=>12,2424,36,48, . . .,36,48, . . .

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

So the LCM of 8 and 12 is So the LCM of 8 and 12 is 2424

To summarize 24 is both the To summarize 24 is both the CCommon ommon MMultiple and the ultiple and the LLeast.east.

Finding LCM using the traditional methodFinding LCM using the traditional method Start out by making a list of the multiples of each Start out by making a list of the multiples of each

given numbergiven number

Look through the multiples for each given number Look through the multiples for each given number and find which of the multiples appear in both and find which of the multiples appear in both lists or are lists or are commoncommon to both numbers. to both numbers.

For example we want to find the LCM of 16 and For example we want to find the LCM of 16 and 24:24:

16 : 16, 32, 16 : 16, 32, 4848, 64, 80 ……., 64, 80 ……. 24 : 24, 24 : 24, 4848, 72, 96 …….., 72, 96 ……..

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

LEAST COMMON MULTIPLE (LCM) IN FOUR STEPS:LEAST COMMON MULTIPLE (LCM) IN FOUR STEPS:

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

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

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

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

For example to find the LCM of 16 and 24, write each numberFor example to find the LCM of 16 and 24, write each numberusing its prime factors:using its prime factors:

16 = 16 = 22..22..22.2.224 = 24 = 22..22..22.3.3LCM= LCM= 22..22..22.2.3 =48.2.3 =48

Practice QuestionsPractice Questions

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

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

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

GREATEST COMMON FACTOR (GCF)GREATEST COMMON FACTOR (GCF)

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

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

For example to find the GCF or HCF of 8 and 12 lets write out the factors of each For example to find the GCF or HCF of 8 and 12 lets write out the factors of each number:number:

8=>8=>11, , 22, , 44,8,8 12=>12=>11, , 22, 3, , 3, 44, 6, 12, 6, 12

We notice from the list of factors that 1, 2 and 4 are common to both lists. Of We notice from the list of factors that 1, 2 and 4 are common to both lists. Of these three common factors these three common factors 44 is the greatest or highest. is the greatest or highest.

So the GCF of 8 and 12 is So the GCF of 8 and 12 is 44

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

GCF or HCF using the traditional methodGCF or HCF using the traditional method

Start out by making a list of the factors of each given number.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 Look through the factors for each number and find which of the factors appear in both lists.factors appear in both lists.

The largest or highest of the common factors is the GCF or HCF.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:For example we want to find the GCF or HCF of 16 and 24:

16 : 1,16 : 1,22,,44,,88,16,16 24 : 1,24 : 1,22,3,,3,44,6,,6,88,12,24,12,24

We notice that of the common factors 1,We notice that of the common factors 1,22,,44,,8 8 the greatest or the greatest or highest one is highest one is 88 so so 88 is our GCF or HCF. is our GCF or HCF.

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

1.) Write out the given number as a product of its prime factors1.) 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 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 numbers and write it down only once for each time that the factor appears common.factor appears common.

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

e.g.e.g. 16 = 16 = 22··2·2·2·2·22 24 = 24 = 2·2·2·2·2·2·33 The GCF is The GCF is 2·2·2·2·2 2 = 8= 8

What is the greatest common factor of 9, 12 and 15?What is the greatest common factor of 9, 12 and 15?

Find the GCF of 24, 36 and 54Find the GCF of 24, 36 and 54