Fractions

Basics

A fraction is a division problem written as a single number. Writing numbers this way is called fractional notation. It has two parts the top number and the bottom number. The top number is to be divided by the bottom number. We end up writing the division problem as a single fraction.

1÷2 = ½ 2÷3 = ⅔ 2÷5= ⅖

The top number is called the numerator, the bottom number is called the denominator. The top number is also called the dividend, the bottom number is also called the divisor. The result of this division is called a fraction, or a quotient.

The numerator is a product of the denominator and the quotient.

1÷2 = 0.5 = ½ 2 ∙ 0.5 = 2 ∙ ½ = 2/2 = 1 2÷3 = 0.6 = ⅔ 3 ∙ 0.6 = 3 ∙ ⅔ = 6/3 = 2 2÷5 = 0.4 = ⅖ 5 ∙ 0.4 = 5 ∙ ⅖ = 10/5 = 2

The numerator divided by the denominator equals a fraction. The dividend divided by a divisor equals a quotient.

We can write any number as itself divided by 1: 1÷1 = 1 10÷1 = 10 5÷1 = 5 Any number divided by 1 is itself: a÷1 = a 1/1 = 1 10/1 = 10 5/1 = 5

1÷1 = 1/1 = 1 10÷10 = 10/10 = 1 5÷5 = 5/5 = 1

In multiplication two numbers called factors multiply to produce a product. A product is the result of multiplication. We say multiplication it the inverse operation of division. Inverse means to reverse the operation and get the original number we started with. If I divide 1 by 2 I will get “one-half”. If I take one-half and multiply it by the number I used to divide I will get 1 back. 1 was my original number. This is called a reciprocal operation.

Any number divided by itself equals 1, (except 0 because we can not divide by 0). When a number is divided by 0 we say the answer is “undefined”. This means the answer is unknowable. Using variables to represent any number we write these laws for fractions:

1. Any non-zero number a divided by itself is 1: a/a = 1. 2. Any reciprocal operation equals 1: a/1 ∙ 1/a = a∙1/1∙a = a/a = 1. 3. Any number a divided 0 is undefinable: a÷0 = a/0 = undefined or “und.” for short.

We also say the answer “does not exist” or dne for short because you can not divide by 0 and get a real number answer. We prove this later.

• To add or subtract fractions you need to have a common denominator.
• To multiply fractions you write the product of the numerators over the product of the denominators.
• To divide fractions you multiply the first fraction with the reciprocal of the second fraction.
• An identity of a number is an operation that does not change the value of the number like multiplying by 1 or adding 0.

Multiplicative identity of 1: a∙1 = a a÷1 = a Additive identity of 0: a+0=a a – 0 = a

We can perform addition, subtraction, multiplication, and division of two fractions. Multiplication is the easiest:

The parentheses are grouping symbols telling what order to perform operations. This is covered during pemdas. Algebra uses the multiplication dot called center dot or interpunct instead of the multiplication cross arithmetic uses. We combine the numerators on top and denominators on the bottom and multiply. To multiply is to find a product. We say “we take the product of numerators and divide by the product of the denominators”. Since a fraction is a division problem you should know the different ways of writing them and names of their symbols. 1÷2 = 1/2 = ½ . Even though 1/2 and ½ both have forward slash consider ½ to have a fraction bar since the numbers are smaller and the numerator is raised above the slash.

1. The fraction in the illustration use a fraction bar “ – ” to indicate division.
2. On a computer we normally use forward slash “ / ” to indicate division. 1/2 = ½ .
3. The division symbol “ ÷ ” is called obelus. It is a Latin word.

Multiplication of fractions

For any two fraction a/b and c/d, a/b ∙ c/d = ac/bd. The numerator is the product of the two numerators and the denominator is the product of the two denominators. Multiply both numerators to make a new numerator, then multiply both denominators to make a new denominator.

Multiply by 1 gives that same number: ⅘ ∙ 5/5 = 4∙5/5∙5 = 20/25 . Since 5/5 = 1 all we did was multiply ⅘ by 1.

Notice that when multiplying we combine the two fractions using 1 fraction bar. Doing this means writing two multiplication dots instead of 1. This is a big deal. We changed the way our equation looked. Changing the way it looks is called “algebraic manipulation”. Changing the way it looks makes it easier to solve, it gets us started in the right direction.

Two factors produce a product. a÷b = ab, here we have 4∙5 = 20 and 5∙5 = 25. We say the factors of 20 are 4 and 5 and that the factors of 25 are 5 and 5.

Since we multiplied â…˜ by 1 and got 20/25 then â…˜ = 20/25. â…˜ and 20/25 are the same fraction therefore the same number.

We can simply a fraction by reducing it to it's simplest form. It works like this:

20÷5 /25÷5 = 4∙(5÷5)/5∙(5÷5) 4∙(5÷5)/5∙(5÷5) = 4∙(1)/5∙(1) = 4∙1/5∙1= 4/5

We can manipulate fractions this way to cancel factors that appear in the number and the denominator. If a factor is in both the top and bottom then the are a fraction equal to 1 and are said to cancel. Here are some different ways of writing this:

When we cancel factors in a fraction we are reducing the fraction to its simplest form. Simplest form for a fraction removes fractions equal to 1.

½ ∙1 = ½ ½ ∙2 = 2/4 ½ ∙3 = 3/6 ½ ∙10 = 10/20

2/4 , 3/6 , and 10/20 all equal ½ . They are multiples of ½ made by multiplying ½ by a fraction equal to 1. The fractions equal to 1 used were 1/1, 2/2, 3/3, and 10/10.

½ is the simplest form of these multiples of ½. To reduce a fraction to simplest terms we divide the top and bottom by the same factor. This factor is called the greatest common factor or GCF.

10/20 The greatest factor that is common to both 10 and 20 is 10.

10/20 = 1∙10/2∙10 and then we separate into two fractions ½∙ 10/10 and then we write 10/10 as 1. Observe: 10/20 = 1∙10/2∙10 = ½ ∙10/10 = ½∙1 = ½. We are left with ½ as the simplest form of 10/20. Canceling the other fractions using their gcf's of 2 for 2/4 and 3 for 3/6 yields similar results.

The original picture had a factor of 2 in the top and bottom. We can reduce the fraction by canceling 2/2 to 1.

1/2∙ 2/3 = 1∙2/2∙3 = 1∙2/2∙3 = 1/3 ∙ 2/2 = 1/3 ∙1 = 1/3. There are trickier ways to write this where you can swap denominators of both fraction and get same result. This is advanced: ½ ∙ ⅔ = ⅓ ∙2/2.

Because multiplication is commutative we can switch the order of numbers to be multiplied: 2∙3 = 3∙2. ½ ∙⅔ = ⅓∙2/2 the denominators have switched places. We really did this:

1/2∙2/3 = 1∙2/2∙3 = 1∙2/3∙2. Writing this using fraction notation we get: ½ ∙⅔ = 1∙2/2∙3 = 1∙2/3∙2. We changed the order of the factors in the second fractions denominator. We have 1∙2/2∙3 = 1∙2/3∙2. When we separate 1∙2/2∙3 it becomes ½ ∙ ⅔. When we separate 1∙2/3∙2 it becomes ⅓ ∙ 2/2.

1. When multiplying fractions combine all fractions by fraction using a single fraction bar. 2. Write all terms as being multiplied by each other

Canceling

There are two types of canceling: Additive and multiplicative. Additive cancellation is when you start at a number then add or subtract a number and to cancel you add it's additive inverse. Multiplicative cancellation is when you reduce a fraction by removing factors that as a fraction equal 1/1 = 1.

3+7 =10 I start with 3 and add a number “7” which results in a sum of “10”. 3+7 – 7 = 3 I cancel “+7” with “ – 7”. The operation is canceled and I have my original number

3–7 = –4 I start with 3 and subtract a number “7” which results in a difference of “ –4”. 3–7+7 = 3 I cancel “–7” with “ +7”. The operation is canceled and I have my original number 3

An additive inverse of a number is the number with the opposite value sign. 3 and –3 are additive inverses. The definition of an additive inverse is two numbers that sum to 0. By definition two additive inverses cancel each other under addition. “Under addition” means addition the operation we are using.

3 + (–7) = –4 We are adding –7. 3 + (–7) = 3 – 7 Adding a negative is the same as subtracting a positive. 3 + (–7) +7 = 3 We cancel the addition of –7 by adding 7 (or by subtracting –7).

3 – (–7) = 10 We are subtracting –7. 3 – (–7) = 3 + 7 Subtracting a negative is the same as adding a positive. 3 – (–7) –7 = 3 we cancel the subtraction of –7 by subtracting 7 (or by adding –7).

15/40 I have the fraction 15/40. 3∙5/2∙2∙2∙5 I find the prime factors of 15 and 40. This is known as the prime factorization. 3∙5/2∙2∙2∙5 I cross out or “strike through” factors when they matching the top and bottom. 3/2∙2∙2 I write my factors with the canceled factors removed. 3/8 = ⅜ I write my final result in fraction notation.

15/40 A fraction not in simplest form. 3∙5/2∙2∙2∙5 Writing numerator and denominator as the products of their prime factors. 3∙5/2∙2∙2∙5 Removing pairs of matching factors. 3/2∙2∙2 Writing the factors without the canceled factors. ⅜ Simplifying by performing all operations until none are left.

The fraction 3/2∙2∙2 still has operations left. We still have to multiply the factors in the denominator. We need to find the product of these factors so we can write it as a single number. Once we have found the product and written ⅜ there are no more operations and the fraction is as reduced as I can make it. Something in simplest form is when there are no more operations to perform and all fractions are fully reduced.

A prime factorization is the unique set of factors for an individual number. Every number has a set of factors that is unique to that number. Every number has a prime set of factors, no two numbers have the same unique set of factors. Prime factorization is used when you have trouble find the greatest common factor. If you know the gcf then using it to cancel is faster. By the way even numbers or whole numbers are called integers.

15/40 What is the largest factor that 15 and 40 both have? 15/40 What is the largest number that divides 15 and 40 and gives an even number? 15÷5/40÷5 The largest number is 5. 3/8 = ⅜ Dividing by 5 will give even numbers. We call this “ integer solutions”.

Fractions should have integers for the top and bottom. If they don't you have a complex fraction like 2 = 1/½ or 1 = ½/½. To avoid complex fraction find the gcf or use prime factorization. Complex fractions are used for rational expressions or ratios. This is covered after polynomials where rational expressions look like polynomials but are not. If you have a complex fraction clear the fractions or decimals to get an integer for the top and bottom. ½÷½ = 1 because a number divided by itself is 1. This applies to fractions, decimals, or variables.

½÷½ = 2(½÷½) = 2∙½ ÷ 2∙½ = 2/2 ÷ 2/2 = 1÷1 = 1 ½÷½ = 2(½) ÷ 2(½) = 1

0.5÷0.5 = 2(0.5÷0.5) = 2∙0.5 ÷ 2∙0.5 = = 1÷1 = 1 0.5÷0.5 = 2(0.5) ÷ 2(0.5) = 1

1.5/3 The numerator contains a decimal 0.5 which as a fraction is ½. 1.5/3 Multiplying what to ½ will equal an integer? 2∙ ½ = 1. 2/2∙ (1.5/3) Multiply the top and bottom by 2 using 2/2. 2/2 ∙ (1.5/3) 2∙1.5 = 3 and 2∙3 = 6 so we have 3/6 which is 3÷6 which is ½.

2/2 ∙ (1.5/3) 2∙1.5/2∙3 Combining fractions as one fraction. 3/6 Finding products. 3÷3/6÷3 Factoring a 3 out of 3 and 6 because 3 and 6 have a gcf of 3. ½ 3÷3 = 1 and 6÷3 = 2 so we have 1∙2 which is ½.

There are shortcuts we instinctively learn to use later. We have many options for writing equations because some forms are easier than others depending on the circumstances. 1.5/3 had a decimal in the numerator. This makes it a complex fraction where we have a fraction in a fraction. Fractions are not whole numbers so they are not integers. Normally we want integers. Having a fraction implies division and so this is not the simplest form because we have the operation of division to do.

Addition of fractions

Because 1.5 = 1+½, the fraction 1.5/3 can be written messy as (1+½)÷(3) or 1+½/3. We can separate this into two fractions.

1.5/3 We have a complex fraction. 1+½/3 Splitting 1.5 into its whole and fractional parts of 1 and 0.5 written as 1 and ½. 1/3 + ½/3 or 1/3 + 0.5/3 We can write ½ ÷3 as ½/3 or 0.5/3. 1/3 + 0.5/3 For this “ 1/3 + 0.5/3 ” looks better than “ ½/3 ” because it is easier to read. 1/3 + 2/2( 0.5/3) We have to multiply the top and bottom by 2 so we multiply by 2/2. 1/3 + 2∙0.5/2∙3 When multiplying write as one fraction to combine the operations. 1/3 + 1/6 We are adding two fractions. To add we need a common denominator. 1/3+1/6 = 2/2 ∙ 1/3 + 1/6 This is hard to read because we are not using fraction notation. 1/3+1/6 = 2/2 ∙ 1/3 + 1/6 This is easier to read. 2/2 ∙ 1/3 + 1/6 To add we scale ⅓ by 2/2 to get 2/6. One third equals two sixths so value is the same. 2/6 + 1/6 To add fraction with the same denominator find the sum of the numerators. 3/6 2+1 = 3 so we have 3/6. ½ 3/6 reduces to ½ when you factor out the gcf of 3. 3/3 = 1 and 3/6 =2 we have 1/2.

This was a easy example made more complicated by bad notation. The next examples with be from first basics. Fractions are parts of a whole number of something. When we take something and divide it into equal parts each part is an equal fraction of each other. If I have a piece of candy and divide it into two equal parts I have two halves. That is I have 2 parts that are each ½. If I add the parts I should get the original number of whole pieces of candy I had: ½ + ½ = 1 whole piece of candy.

This is the difference between a fraction and a piece. A piece is 1 whole piece and a fraction is a fraction of that whole piece. Pieces are not fractions but rather 100% of something. When we have “a piece of a piece” we call that a fraction.

½ + ½ =1 This says that ½ of 1 piece plus ½ of 1 piece equals 1 whole piece.

½ of a piece plus ½ of a piece = 1 whole piece. We could say ½ a candy + ½ a candy = 1 candy. I have a chocolate candy bar that is 4 squares each the same size. Each square would then be ¼ of a whole bar. I have an orange and it has 8 equal slices. Each slice is ⅛ of a whole orange. If I eat 2 slices I ate 2/8 of a whole orange.

â…›+â…› = 1+1/8 = 2/8 Add the numerators. The denominator stays the same.

If I ate 3 squares of the candy bar then:

¼ + ¼ + ¼ = ¾ I ate ¾ of the whole bar. 1 – ¾ = ¼ I have ¼ of the bar left. 4/4 – ¾ = ¼ I started with 4 out of 4 squares, then ate 3 out of 4, and now have 1 out of 4 left.

The numerator is how many we have and the denominator is how many it takes to have a whole piece.

If I eat 1 square then I ate ¼ of the bar. If I eat 2 squares then I ate 2/4 of the bar. 2/4 reduces to ½ or the same as saying ½ of the bar. If I eat 3 squares then I ate ¾ of the bar. If I eat 4 squares then I ate 4/4 of the bar. I ate the entire bar. 4/4 = 1 so I ate 1 whole bar.

If I eat 2 out of 8 orange slice then I ate 2/8 of the orange. This reduces to ¼. Really I ate ¼ of the entire orange. If I eat 7 orange slice then ⅛+⅛+⅛+⅛+⅛+⅛+⅛=⅞. I ate 7 out of 8 which as a fraction is ⅞. To add these fractions we check to make sure we have a common denominator. A common denominator means that all fractions have the same denominator. All fractions have the same denominator so we can add these fractions by simply adding the top numbers: ⅛+⅛+⅛+⅛+⅛+⅛+⅛= (1+1+1+1+1+1+1)÷8 = (1+1+1+1+1+1+1)/8 = ⅛.

Repeated addition is called multiplication. We can multiply instead of adding here since we are adding the same number over and over. ⅞ = (1+1+1+1+1+1+1)÷8 = 7(1)÷8 = 7÷8 = ⅞.

Writing (1+1+1+1+1+1+1) as 7(1) is called factoring. We factored “1” out seven times so we have 7 groups of “1”. We write this as 7∙1 or 7(1). Parentheses implies we need to multiply if there is no other operation sign outside the ().

Factoring is a form of division. Here are the gory details:

1+1+1+1+1+1+1 (1+1+1+1+1+1+1) (1÷1+1÷1+1÷1+1÷1+1÷1+1÷1+1÷1) or (1/1+1/1+1/1+1/1+1/1+1/1+1/1) or (1/1+1/1+1/1+1/1+1/1+1/1+1/1) = 7(1)

We count how many “1's” are in the () and then write that number outside the () and write 1 inside the () only once.

More Factoring

2+4+6+8+10 All of these numbers are multiples of 2. 2 is the least common multiple or LCM. (2+4+6+8+10) Please group this series of numbers with (). 2(2/2+4/2+6/2+8/2+10/2) Please write the LCM outside () and divide every number by the LCM 2(2/2+4/2+6/2+8/2+10/2) This writes the same thing above but in fraction notation. 2(1+2+3+4+5) Simply fractions. 2(15) Sum terms inside (). 30 Find product of 2∙15 to be 30.

If we sum the series we still get 30:

2+4+6+8+10 6+6+8+10 12+8+10 20+10 30

Sometimes factoring is faster for harder problems or allows us to cancel or do something else. Notice that we divided every number, called a term, by the factor we were using? To factor find a common factor of all the terms in the (). If they have a common factor other than 1 you can factor that out by dividing every term by that factor.

⅛+⅛+⅛+⅛+⅛+⅛+⅛ Write our sum. (⅛+⅛+⅛+⅛+⅛+⅛+⅛) Write it in (). (⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛) Divide every term by ⅛. ⅛(1+1+1+1+1+1+1) This means we factored out ⅛ out of every term. ⅛(7) Whats in the () equals 7 (⅛+⅛+⅛+⅛+⅛+⅛+⅛) Our sum. 7(⅛) This says we have 7 factors of ⅛.

More Adding Fractions

Find a common denominator to add fractions:

⅖ + ¾ These have different denominators. (4/4 ∙ ⅖ ) + (5/5 ∙ ¾) This creates a common denominator by cross multiplying. (8/20) + (15/20) We now have a common denominator of 20. 23/20 We add 8+15 = 23.

Cross multiplying multiples the the top and bottom of the first fraction with the second fractions denominator. Cross multiplying multiples the the top and bottom of the second fraction with the first fractions denominator.

a/b + c/d = ad/bd + bc/bd = ad+bc/bd a/b + c/d = ad/bd + bc/bd = ad+bc/bd

We cross multiply when we struggle to find a lcm. Other fractions are easy to find a lcm so we do not need to cross multiply. If we denominators are multiples of each other then we only multiply the smaller denominator to math the bigger one.

Examples of adding fractions

4/8 + 5/8 = 9/8 Add numerators since denominators are same.

2/5 + 5/10 5 and 10 are multiples of 5 so use 10 as the lcm. Multiply the first fraction by 2/2. 4/10 + 5/10 = 9/10 2/2 ∙ 2/5 = 2∙2/2∙5 = 4/10. 2/5 + 5/8 The LCM of 45 and 8 is 40. 16/40 + 25/40 Cross multiply to find common denominator since 5 and 8 are not multiples. 31/40 31/40 is simplest form because 31 and 40 do not share any factors to factor out.

24/100 + 3/50 Find lcm. 12/50 + 3/50 Use 50 as lcm by factoring 2 out of 24 and 100. 24÷2/100÷2 = 12/50. 15/50 Check for gcf of 15 and 50. The gcf is 5. Factor 5 out of 15 and and 50. 3/10 This is the simplest form.

Subtracting fractions

Same as adding except subtract numerators instead of adding them. Use same methods to combine fractions.

1. Check to see if denominators are the same.
2. If they are same subtract numerators and write difference as the new numerator over given denominator.
3. LCM stands for least common multiple and GCF stands for greatest common factor.
4. If denominators are not the same find LCM and write one of the fractions so that you can subtract.
5. If denominators are not multiples of each other then the LCM will be found by cross multiplying.
6. Once denominators are the same and you have subtracted reduce the fraction by factoring out the GFC.

7/8 – 3/8 = 4/8 = ½ 4/8 reduces to ½. 4 and 8 share a factor of 2 that cancels.

4/8 We have 4/8. 2∙2/2∙4 Writing as its factors shows that the greatest common factor is 2. 2/2 ∙2/4 We separate the fraction into two fractions to treat 2/2 separately. 1∙2/4 2/2 is said to “ cancel “ to 1. We rewrite 2/2 as 1. 2/4 Multiplying by 1 cancels since multiplying by 1 does not change the number. 2∙1/2∙2 Writing as its factors shows that 2 and 4 have a gcf of 2. 2/2 ∙1/2 Separate the two fractions. Rewrite the cancellation as the gcf over itself. 1∙ ½ Write the cancellation as 1. ½ Multiplying by 1 cancels the factor 1. ½ is the simplest form of 4/8.

Obviously there are steps that can be combined or skipped. This shows every pro45cess to completion so there are no questions about a particular step. For simple problems basic steps are combined. For harder problems it helps to be more cautious and write steps out more fully as needed. We could have simply said 4/8 = ½ and been done.

3/8 – 7/8 = –4/8 = –4/8 = –½ 3 – 7 = – 4.

When you have a negative fraction write the negative sign in front of the fraction and not in the numerator nor denominator. Some formulas have subtraction performed in either the numerator or denominator. When a formula does many things to create a fraction we tend to call it a quotient. Famously the formula for slope of a linear equation can have a negative denominator if written backwards. The numerator would be positive but the denominator is negative. We move the negative sign to the front of the fraction to quote the entire fraction as negative not just the top or bottom. The word quotient means divide and comes from the word “ quote”.

If we have a quotient of –3/7 then: –3/7 = 3/–7 = – 3/7 Move the sign to the front. Here is what is going on: Dividing a positive by a negative or a negative by a positive gives a negative result.

–10/2 = –5 because: –10 ∙ 2 = –5 10/–2 = –5 because: 10 ÷ –2 = –5

Either way we get –5 or – 5/1 .

Remember that any whole number (or fraction) can be written as itself over 1 because any number divided by 1 is itself. This is the called the division identity fo 1 where n ÷ 1 = n.

2 = 2/1 3 = 3/1 2+1 = 2+1/1 5∙7 = 5∙7/1 x = x/1 2x+y –7 = 2x+y –7/1

Indeed even equations can be written as themselves divided by 1. For any two fractions a/d and b/d: a/d + b/d = a+b/d and a/d – b/d = a – b/d. If denominators are unlike multiply one fraction by the lcm/lcm so that you have two common denominators. If denominators are not multiples of each other the lcm can be found by cross multiplying: a/b + c/d = ad+bc/bd.

a/b+c/d = d/d∙a/b + b/b∙c/d = d∙a/d∙b +b∙c/b∙d = da/db +bc/bd = ad/bd +bc/bd = ad+bc/bd

a/b+c/d We have two fractions to be added with different denominators. d/d∙a/b + b/b∙c/d We cross multiply to find the LCM to use as a denominator for both fractions. d∙a/d∙b +b∙c/b∙d We combine fractions like this if we want to show our steps more fully. da/db +bc/bd Writing variables without operation signs between them means they are being multiplied. ad/bd +bc/bd Reorder the variables in alphabetical order. ad+bc/bd The final answer. If we had numbers instead of variables we would combine numerators.

Division of fractions

To divide two fractions multiply by the reciprocal of the divisor: a/b ÷ c/d = a/b ∙ d/c = ad/bc.

½ ÷⅔ = ½ ∙ 3/2 = ¾ ⅔ ÷ ⅗ = ⅔ ∙ 5/3 = 10/9 ⅔ ÷ ¼ = ⅔ ∙4/1 = ⅔ ∙4 = 8/3 10÷⅕ = 10∙5 = 50

When dividing by a fraction the rule used says to “flip and multiply”. We say “flip” to mean flip the numerator and denominator of the dividing fraction. Flipping means to swap the top number with the bottom number. Flipping a fraction gives you the reciprocal of that fraction. The reciprocal of a whole number is a fraction with that number on the bottom and 1 as the numerator as this is the opposite of a whole number which can be written as that number over 1 as the denominator.

6÷7/3 = 6∙3/7 = 6/1 ∙ 3/7 = 6∙3/1∙7 = 18/7 This show some steps which can be eliminated once you know the rules. 6÷7/3 = (6∙3)÷7 = 18÷7 = 18/7 This shows a sloppy way of doing it but is valid. 6÷7/3 = 6∙3/7 = 18/3 This is the fastest and preferred method. Too many steps confuse beginners.

10÷2 = 10∙½ = 5 Dividing by a whole number is the same as multiplying by its reciprocal. 10÷½ = 10∙2 = 20 Dividing by a fraction is the same as multiplying by its reciprocal.

⅔÷¼ = ⅔∙4/1 = 8/3 Dividing by a fraction is the same as multiplying by its reciprocal. ⅔÷4 = ⅔∙¼ = 2/12 = 1/6 Dividing by a whole number is the same as multiplying by its reciprocal. ⅔÷4 = ⅔∙¼ = 2/3∙4 = 1/6 Note that dividing by 4 is the same as multiplying the denominator by 4.

Dividing a fraction by a number is the same as multiplying the fraction's denominator by that number.

⅔÷ 4 = 2÷(3∙4) = 2÷12 = 1/6 = 1/6 ⅔÷4 We are dividing by 4... ⅔÷4 ... ⅔∙¼ ... 2/3∙4 … so we are multiplying the denominator of ⅔ by 4. 2/12 1/6

⅔÷¼ = 2÷(3∙¼) = 2÷¾ = 8/3 ⅔÷¼ We are diving by ¼. ⅔∙4/1 Flip and multiply... 8/3 Also we could...

⅔÷¼ We are diving by ¼... 2/3∙¼ ...so we are multiplying the denominator by ¼. 2/¾ The denominator is a fraction so the entire term is a complex fraction. 2÷¾ We are dividing by a fraction so we need to flip and multiply. 2∙4/3 Multiply 2∙4 to get a numerator of 8. 8/3

We do not want complex fractions. Flip and multiply avoids them. If you have a complex fraction you need to know how to solve it. These complicated forms are to be avoided if possible. If you have a complex fraction use these methods as shown to eliminate them.

Fractions plus, minus, times, or divided with a whole number

7+½ = 7.5 We can convert ½ to 0.5 and just add as decimals or... 7+½ 14/2+ ½ ...we can convert 7 to a fraction with a like denominator as ½. 15/2 15÷2 = 7.5

Since ½ has a denominator of 2 we multiply 7/1 ∙2/2 = 14/2 which equals 7 because 14∙2 = 7. Multiplying something by a fraction that is equal to 1 does not change the value of the number but changes the form. When we have a whole number and need to perform operations on it with a fraction we convert the number into a fraction by writing the number over 1. All whole numbers are fractions that have a denominator of 1.

4+5/23 = 4∙23/1∙23 + 5/23 = 96/23 + 5/23 = 101/23

101/23 is in simplest form but is an improper fraction. We can convert it to a mixed fraction if we like. 101/23 = 23+23+23+23+9/23 = 4+9/23 we can pull 23 out of 101 4 times and be left with a remainder of 9.

When multiplying a whole number to a fraction we can usually do this mentally. The denominator stays the same and the whole number and the fractions numerator multiply to produce a new number. If the fraction has 1 for a numerator then the whole number becomes the numerator. Just move whole number up top.

4∙½ = 4/2 = 2 4∙⅓ = 4/3 4∙12/5 = 48/5 x∙3/14 = 3x/14 3∙x/4 = 3x/4 7∙x/y = 7x/y

Let x be any whole number and p/q be any fraction. If you do something with a fraction then x is written like x/1.

Addition: x + p/q = x/1 + p/q = q/q ∙ x/1 + p/q = qx/q + p/q = qx+p/q Subtraction: x – p/q = x/1 – p/q = q/q ∙ x/1 – p/q = qx/q – p/q = qx – p/q Multiplication: x ∙ p/q = x/1 ∙ p/q = px/q Division: x ÷ p/q = x/1 ÷ p/q = x/1 ∙ q/p = qx/q

Make x, p, q and numbers and follow the formulas listed to solve any arithmetic operation. Multiply by the LCM to clear decimals to change complex fractions to regular fractions. Any number divided by itself is 1: x÷x = x/x = 1.

Multiplying by 1 as a fraction of lcm/lcm finds equivalent fractions that are scaled multiples of the original. Scaled multiples of a fraction equal the original and thus do not change the value, we say “ multiplying by 1 does not change the value”. Use lcm/lcm to scale a fraction to find a common denominator so that fractions can be added. If x/y + x/5y then to add multiply x/y with 5/5 to get 5x/5y + x/5y so that denominators are the same. lcm/lcm can be used here because y and 2y are multiples of y. If x/y + x/z then lcm/lcm will not work because y and z are not multiples of each other. To find the lowest common multiple you need to cross-multiply. This will find a common denominator as yz. x/y + x/z = xy/yz + xz/yz = 2xyz/yz .

Canceling only works by reducing fractions: 48x/120xy = 2∙2∙3∙4∙x/2∙3∙4∙5∙x∙y = 2/5y Terms do not cancel under addition: x+2/y+2 ≠ x/y. Because x+2 is (x+2) “the entire sum is the numerator”... you can not cancel addition with division since they are not reciprocal operations. x+3/3 ≠ x. The constants do not cancel because they are not factors. For division only factors cancel not added or subtracted parts.

Fraction notation Division notation using obelus Division notation using slash Result ½/½ ½ ÷ ½ ½ / ½ 1 0.5/0.5 0.5÷0.5 0.5/0.5 1 x/x x÷x x/x 1 ½ 1÷2 1/2 0.5 2/7 2÷7 2/7 0.285714 2x/4y 2x÷4y 2x/4y x/2y

Standard math does not use this symbols but are included for reference. The x and y are inverted: Fraction notation Division notation using backslash Division notation using vertical bar Result ½/½ ½ \ ½ ½ | ½ 1 0.5/0.5 0.5\0.5 0.5|0.5 1 x/x x\x x|x 1 ½ 2\1 2|1 0.5 2/7 7\2 7|2 0.285714 2x/4y 4y\2x 4y|2x x/2y