Vanessa Graulich

Certified Math Teacher

6th-12th

If they ask you "What are the multiples of 24?"

You start with the question

"What is the product of 24 with all the natural numbers?"

That means 24*1=24,

The multiples of 24 are: 24,48,72,96.......

The Greatest Common Factor or GCF and the Least Common Multiple will help you to do factoring and solve equations for the algebra section.

These are two examples of finding the Greatest Common Factor

and the Least Common Multiple for 12 and 24.

A) Find the GCF of 12 and 48

The first step is to find the individual factors of 12 and 36.

12: 1,2,3,4,6,12

48: 1,2,3,4,6,8,12,16,24,48

You can see the COMMON FACTORS ARE (1,2,3,4,6,12)

12: 1,2,3,4,6,12

36: 1,2,3,4,6,8,12,16,24,48

The GREATEST COMMON FACTOR IS 12

Let's practice

Find the GCF of 100 and 50:

The first step is to find the individual factors of 100 and 50.

100: 1,2,4,5,10,20,25,50,100

50: 1,5,25,50

You can see the COMMON FACTORS ARE (1,5,25,50)

100: 1,2,4,5,10,20,25,50,100

50: 1,5,25,50

The GREATEST COMMON FACTOR IS 50

Find the LCM of 12 and 48

The first step is to find multiples of 12 and 48

12: 12,24,36,48,60,72,84,96,108,120,132,144…

48: 48,96,144,192……

You can see the COMMON MULTIPLES ARE SO FAR (48,96,144…)

12: 12,24,36,48,60,72,84,96,108,120,132,144…

48: 48,96,144,192……

You need the LEAST one from the list, that is 48

The LEAST COMMON MULTIPLE 48

Prime Factorization

Prime factorization will help you find the

Greatest Common Factor and least common multiple faster.

Prime factorization is when write the numbers as a product of prime numbers.

Let's do some examples:

Find the prime factorization of 12:

Start by finding the factors of

12=4·3

The number 4 is also 2²

The prime factorization of 12 is 2²·3

Here are some other examples

12=2²·3

49= 7²

50=5²·2

100=5²·2²

120=2³·3·5

1000=5³·2³

Fractions

When you have a fraction you need to remember, you have a numerator and a denominator.

If the numerator is smaller than the denominator

the fraction is proper.

If the numerator is greater than the denominator

the fraction is improper.

3/4 proper vs 4/3 improper

Mixed Numbers are a whole number and a fraction together.

An improper fraction can be converted into a mixed number the following way.

Let's practice

Convert 36/5 into a mixed number

Divide the denominator into the numerator (like: 36 ÷ 5).

Since 5 goes into 36, 7 times (5 × 7 = 35).

The remainder is the difference 36 – 35 = 1. You can now re-write the expression the following way:

How to add and subtract fractions

When you add or subtract fractions, they need the same denominator.

Ratios

If you need to compare two quantities with the same unit, you can use ratios.

Let’s say you have a basket with 2 apples and 5 oranges, then you can say:

The number of apples to the number of oranges can be written the following ways:

1)As a fraction: (2 apples)/(5 oranges)

2) 2 to 5

3) 2:5

BE CAREFUL, IF YOU WANT TO REPRESENT THE RATIO, then you need to find the whole, in this case how many fruits in total you have.

Whole= 2 apples + 5 Oranges= 7 fruits in total

The ratio of apples to the whole (both fruits)

2/(2+ 5) =2/7

The ratio of oranges to the whole (both fruits) 5/(2+5)= 5/7

Let’s do an example:

In a classroom there are 24 students, the ratio of girls to boys is 5:7.

How many girls are in the classroom?

Let's do the fraction of the number of girls and boys at the classroom:

Girls : 5/(5+7) = 5/12

Boys: 7/(5+7)= 7/12

In order to find out how many girls are at the classroom, just multiply the ratio by the total amount of students:

Girls : (5/12)*24 = 10 girls, then they the rest are boys (24-10= 14) , or you find how many boys are in the classroom by using the ratio again :

Boys: (7/12) *24= 14 boys

Rates and Proportions

Rates can relate two different units.

For example: miles per hour, dollars per pound, litters per minute

Common Rates Examples:

A car drives 60 miles per hour

A hose pumps 1 litter of water per minute

Proportions are used to compare two ratios.

For example:

If there are 60 minutes in 1 hour,

how many minutes are in 4 hours?

(60 minutes)/(1 hour) = X/(4 hours)

Solve for X by:

60(4)= X

240= X

X=240 minutes

Percentages

Percentages are represented by the symbol %,

Percentages can be written as a percentage, a decimal or a fraction.

These are the most common percentages

Let’s do some examples to understand how percentages are represented

1. 20% is also 0.2 and 20/100=1/5

2. 34.5% is also 0.345 and 34.5/100=69/200

3. 35% is also 0.35 and 35/100=7/20

4. 234% is also 2.34 and 234/100= 117/50

Percentages Increase and Decrease

Let’s say you want to buy a phone that is $105 dollars on Monday,

and then on Friday the same phone is selling for $84 dollars,

you get really happy and wonder what was the percent decrease from 105 to 84.

It is really easy to do by using this formula:

Let's do another example:

Vanessa's salary was $45 dollars an hour, after a raise her hourly rate is $60 dollars, what is the percent increased ?

Notice the original amount is 45, that is the denominator

Percent Change (increased) = (60-45)/45= 0.333 or 33.33%

Let's say Vanessa's salary was $60 dollars an hour, and now her salary is $45 hourly, what is the percent decreased?

Notice the original amount is 60 now, that is the denominator

Percent Change (decreased) = (60-45)/60= 0.25 or 25%

Notice two important things:

1) The original always goes to the bottom

2) The percent change from 45 to 60 is NOT the same as 60 to 45!

The following formula will help you solve percentages

word problems.

1) What is 30% of 120?

You are looking for the part (is) of the problem, you have the percentage and the whole (of), plug the values in to the formula and “X” is what you are looking for.

X/120=30/100

Cross-multiplying both sides:

X(100)=120(30)

X(100)=3600

X= 3600/100= 36

36 is 30% of 120

2) 12 is what percent of 20?

Apply the formula, you need the percent, make percent “X” and solve for it.

12/20=X/100

Cross-multiplying both sides:

12(100)=20X

1200=20X

X= 1200/20= 60

12 is 60% of 20

3) 25% of what number is 300

Using the formula, you need to find the whole(of)

300/X= 25/100

300(100)=25(X)

30000/25=X

1200

25% of 1200 is 300

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