Fractions Basic Introduction - Adding, Subtracting, Multiplying & Dividing Fractions
Summary
TLDRThis lesson focuses on adding, subtracting, multiplying, and dividing fractions. It explains step-by-step methods to find common denominators, add or subtract fractions, and simplify the results. The lesson also covers how to handle multiple fractions, find the least common denominator, and provides useful tricks to break down large numbers for easier calculations. Additionally, the 'keep-change-flip' technique is introduced for dividing fractions. Through various examples, the lesson demonstrates efficient methods to solve fraction problems, both with simple and complex numbers, without needing a calculator.
Takeaways
- 🔢 The process of adding fractions involves finding a common denominator by multiplying the denominators.
- ✖️ To add fractions like 3/5 and 4/7, multiply the denominators (5 and 7) to get a common denominator of 35, then adjust the numerators accordingly.
- ➖ When subtracting fractions like 7/8 - 2/9, the same principle applies: find a common denominator, subtract the adjusted numerators, and simplify if possible.
- 🔄 The least common denominator (LCD) is essential when adding or subtracting multiple fractions, and it can be found by listing multiples or multiplying denominators together.
- 📉 When combining three fractions, first find the least common denominator by listing multiples or multiplying the denominators (e.g., for 3/4 + 5/3 - 7/2, the LCD is 12).
- ➕ Once a common denominator is found, combine the numerators and adjust for any negative signs if subtracting fractions.
- 📝 In more complex fraction operations, like adding three fractions, you can use a larger common denominator if needed and simplify at the end.
- 💡 Multiplying fractions involves multiplying the numerators and denominators directly, but you can simplify by breaking down larger numbers to avoid large products.
- 🧮 Simplifying before multiplying helps reduce the numbers, as shown when breaking 24/45 into smaller factors and canceling common terms.
- 🔄 Dividing fractions involves using the 'keep, change, flip' method: keep the first fraction, change the division to multiplication, and flip the second fraction before multiplying.
Q & A
What is the first step in adding or subtracting fractions?
-The first step is to find a common denominator by multiplying the denominators of the fractions.
How do you add fractions like 3/5 + 4/7?
-First, multiply the denominators (5 and 7) to get 35. Then, multiply 3 by 7 (21) and 5 by 4 (20). Finally, add the numerators to get 41, resulting in the fraction 41/35.
What should you do when subtracting fractions like 7/8 - 2/9?
-Multiply the denominators (8 and 9) to get 72. Multiply 7 by 9 (63) and 8 by 2 (16), then subtract the numerators to get 47/72.
How can you find the least common denominator (LCD) when adding or subtracting fractions?
-List the multiples of the denominators and find the smallest multiple they have in common. For example, the LCD of 2, 3, and 4 is 12.
What happens if you use a common denominator that is not the least common denominator?
-You can still get the correct answer, but you will need to simplify the final result at the end.
In the example 3/4 + 5/3 - 7/2, why is 12 chosen as the common denominator?
-12 is the least common denominator (LCD) because it is the smallest multiple common to 2, 3, and 4.
How do you add or subtract fractions with the same denominator?
-Once the denominators are the same, combine the numerators. For example, in 9/12 + 20/12 - 42/12, add 9 and 20, then subtract 42 to get -13/12.
What is the process for multiplying two fractions?
-Multiply the numerators and the denominators across the fractions. For example, 3/5 * 7/2 = 21/10.
Why is it helpful to break down large numbers when multiplying fractions?
-Breaking down large numbers allows you to simplify the multiplication by canceling common factors, making the process easier and reducing the need for a calculator.
How do you divide fractions using the 'keep, change, flip' method?
-Keep the first fraction, change the division to multiplication, and flip the second fraction. Then, multiply the fractions across.
Outlines
➕ Adding and Subtracting Fractions
This section introduces the concept of adding and subtracting fractions. It explains a method where you first multiply the denominators of the two fractions to get a common denominator. For example, in adding 3/5 and 4/7, you multiply the denominators 5 and 7 to get 35. Then, you multiply the numerators accordingly (3 x 7 = 21, 5 x 4 = 20), add the results (21 + 20), and place it over the common denominator to get 41/35. The same technique is applied to subtract fractions, such as 7/8 - 2/9. After getting a common denominator of 72, you subtract the numerators and arrive at the final simplified result.
➗ Adding or Subtracting Three Fractions
The video covers how to handle the addition and subtraction of three fractions. It introduces finding the least common denominator (LCD), and demonstrates how to do this by listing multiples of the denominators. For the example 3/4 + 5/3 - 7/2, the least common multiple of 4, 3, and 2 is 12. After converting all fractions to have this denominator, the numerators are manipulated (3/4 becomes 9/12, 5/3 becomes 20/12, 7/2 becomes 42/12). By adding and subtracting the numerators (9 + 20 - 42), you get the final result of -13/12.
💯 Adding Three Fractions with a Common Denominator
This section focuses on a more complex example of adding three fractions: 8/5 - 2/3 + 9/4. The common denominator is found by multiplying the denominators (5, 3, and 4), resulting in 60. Each fraction is then adjusted accordingly (8/5 becomes 96/60, 2/3 becomes 40/60, 9/4 becomes 135/60). The video explains how to subtract and add the numerators (96 - 40 + 135), arriving at a final answer of 191/60.
✖️ Multiplying Fractions
In this part, the process of multiplying fractions is demonstrated. The method is to simply multiply the numerators and the denominators across. For instance, multiplying 3/5 by 7/2 gives 21/10. The section also covers larger fractions like 24/45, and encourages breaking down larger numbers into smaller factors to simplify the multiplication, showing how factors can be canceled out before multiplying. For example, breaking down 24 as 6x4 and 45 as 9x5 allows for easier cancellations, leading to a simplified result of 4/3.
🧮 Multiplying Large Fractions by Simplification
This section explains how to multiply larger fractions using simplification techniques. The video gives an example of multiplying 56/77 by 35/40. By breaking down these numbers into their prime factors (e.g., 56 as 8x7, 77 as 11x7), the video shows how to cancel out common factors before multiplying the remaining numbers. This results in a simplified answer of 7/11.
➗ Dividing Fractions Using Keep-Change-Flip
This section introduces the 'Keep-Change-Flip' method for dividing fractions. It begins with an example of dividing 8/5 by 12/7. The first step is to keep the first fraction as it is, change the division sign to multiplication, and flip the second fraction (12/7 becomes 7/12). After simplifying common factors (such as canceling out a factor of 4), the final answer is 14/15. Another example is provided, 4/3 ÷ 9/5, demonstrating the same method, leading to a result of 20/27.
➕ Simplifying Complex Fraction Divisions
In this final section, a more complex fraction division is tackled: 36/54 ÷ 64/48. The video breaks down each number into its prime factors (e.g., 36 as 9x4, 54 as 9x6), then applies the keep-change-flip method. After simplifying by canceling common factors, the result is reduced to 1/2, demonstrating how larger fractions can be simplified step-by-step for an easier calculation.
Mindmap
Keywords
💡Fractions
💡Common Denominator
💡Least Common Multiple (LCM)
💡Numerator
💡Denominator
💡Multiplying Fractions
💡Simplifying Fractions
💡Keep Change Flip
💡Reciprocal
💡Canceling Terms
Highlights
Introduction to adding and subtracting fractions using common denominators.
Technique to find a common denominator by multiplying the denominators of two fractions.
Multiplying both numerators and denominators to add fractions with different denominators.
Example of subtracting two fractions by first finding the common denominator.
Steps to add or subtract three fractions by finding the least common denominator.
Using multiples of numbers to find the least common denominator for three fractions.
Simplification process: multiply fractions by necessary values to make denominators the same.
Adding numerators after converting fractions to the same denominator.
Solving an example with three fractions, finding a common denominator, and simplifying the result.
Introduction to multiplying fractions by multiplying across numerators and denominators.
Simplifying complex fractions by breaking larger numbers into factors before multiplying.
Cancellation of common factors before multiplying for easier fraction multiplication.
Using the 'keep, change, flip' method when dividing fractions.
Simplifying division problems by flipping the second fraction and converting the operation to multiplication.
Simplifying complex fraction division using factor cancellation and reduction to the simplest form.
Transcripts
in this lesson we're going to focus on
adding and subtracting fractions
let's say if we have 3 divided by 5
plus
4 divided by 7. how can we add these two
fractions
well here's a simple technique
first multiply five
and seven this will give you a common
denominator of thirty five
next
multiply
three and seven three times seven is
twenty-one
and also there's a plus in between
multiply
five times four which is twenty
twenty-one plus twenty
is forty-one so the answer is 41 divided
by 35.
let's try another example
let's say if we want to subtract 7 over
8
minus
2 over 9.
let's use the same technique
let's multiply the two denominators
eight and nine
which is uh 72
and then the next one
is going to be uh 7 times nine which is
sixty-three
minus
eight times two which is sixteen
now let's subtract what is sixty-three
minus sixteen
this is going to be 47.
now 47 is not divisible by 2
nor is it divisible by 3.
so this is it that's the final answer so
now you know how to add or subtract two
fractions
now what if we wanted to add or subtract
let's say three fractions instead of two
what should we do in this case
so let's say we wish to combine three
over four
plus
five over three
minus seven over two
whenever you wish to add or subtract
fractions
the denominator has to be the same the
denominator is the bottom part of the
fraction
and right now they're all different
so how can we make them the same how can
we get the common denominator
if you want to find the least common
denominator
make a list
all of the multiples of 2 are 2
4
6 8 10
12 14 and so forth
multiples of 3 are 3
6
9
12 15 18
and so forth and multiples of 4 are 4
8 12 16 20.
what is the least common multiple
we're looking for a multiple that is
common to all three numbers but is the
lowest
the least common multiple is 12.
12 is common to 2 3 and 4
and it's the lowest of such numbers
now granted
24 is also a common multiple
and if you use 24 you can get the right
answer you just got to simplify
at the end
so if you're ever like unsure about how
to find the least common denominator you
can find any common denominator one
simple technique is simply to multiply
these three
four times three times two is 24
and you could use 24 and still get the
right answer
so now that we know the least common
denominator is twelve
let's multiply each fraction in such a
way to get twelve
the first fraction
let's multiply the top and the bottom by
three because three times four is twelve
the second one let's use four
and for the last one let's use six
so looking at the first one three times
three is nine three times four is twelve
four times five is twenty four times
three is twelve
seven times six is forty two
two times six is twelve
now that we have the same denominator we
can combine the numerators
9 plus 20
is 29
and 29 minus 42
is negative 13. so this is the final
answer it's negative 13 divided by 12.
so now it's your turn go ahead and try
this example
8 over 5 minus 2 over 3
plus
9 divided by four
so go ahead and add these three
fractions
so this time
to find the least common multiple
we're just going to multiply five three
and four
it may not be the least common multiple
but it is a common denominator
just so you know
if we multiply 5 times 4 times 3
this will give us 5 times 4 is 20 20
times 3 is 16.
so 60 is going to be the common
denominator that we're going to try to
get
so we're going to multiply the first
fraction by 12
because 12 times 5 is 60
and the second one by 20
because 20 times 3 is 60. by the way if
you want to find out the number
divide it
60 divided by 5 will give you the 12.
60 divided by 3 will give you the 20
and 60 divided by 4
will give us the number that we need to
multiply this fraction by which is 15.
now 12 times 8 that's 96 5 times 12 we
know it's 60.
2 times 20 is 40 3 times 20 is 60
and 9 times 15
15 times 10 is 150
so if you take away 15 from that you'll
get 15 times 9 so that's 135
and 4 times 15 is 60.
now
96 minus 40
that's positive 56.
and 56 plus 135. let's go ahead
and add those two numbers the
old-fashioned way
five plus six is eleven carry over the
one one plus three plus five is nine
plus one
so the final answer is 191 divided by
sixty
in this lesson we're going to focus on
multiplying two fractions
whenever we need to multiply
multiply the numbers across the
fractions
3 times seven
is equal to twenty one
and five times two
is equal to ten
and so this is it
the answer is twenty 21 over 10. that's
all you need to do when multiplying
fractions
but sometimes the numbers may not be
that small
let's say if we have larger numbers
what should we do in this case
now we can multiply across we can
multiply 24 and 45
which will give us a big number
but do we really want to do that
when multiplying fractions with large
numbers
it's in your best interest
to break down the large numbers into
small numbers for instance
24
is basically six times four
27 is nine times three
forty-five is nine times five
and 30 is 6 times 5.
you want to break it in such a way that
you can cancel
some numbers here we can cancel a 5
because we have 1 on top and the other
on the bottom the same is true for the 9
and we can cancel a six
so therefore the final answer
is four over three
so we were able to get the final answer
without multiplying twenty four by forty
five
that step was necessary plus is going to
take some time and you need a calculator
doing it this way requires no use of a
calculator
try this one
multiply 56
divided by 77
by 35 over 40.
now 56 is 8 times 7
77 is 11 times 7
35 is 7 times 5
and 40 is 8 times 5.
so we can cancel an eight
we can cancel a seven
and we can cancel a five
leaving the final answer of seven over
eleven so now you know how to multiply
two fractions
in this lesson we're going to focus on
dividing two fractions
let's use eight over five as an example
and let's divide it by
twelve over seven
now perhaps you heard of the expression
keep change flip
it's useful when dividing fractions
keep the first fraction the same way
change division to multiplication and
flip the second fraction and now you can
do it
so 8 times 7
is 56 but we can simplify it before we
multiply
8 is basically 4 times 2
and
12 is four times three
so we could cancel a four
and now we can multiply
two times seven
is fourteen
and five times three is fifteen
so the final answer is fourteen over
fifteen
try this one what's four divided by
three
divided by
nine over five
so using the expression keep change flip
let's keep the first fraction the same
let's change division to multiplication
and let's flip the second fraction
now there's nothing to cancel so let's
multiply across 4 times 5 is 20
3 times 9 is 27
and so we can't reduce this fraction
that's the answer
now what if you see
a problem that looks like this
36 over 54
divided by
64 over 48
if you have a fraction written this way
what should you do
this expression is equivalent to saying
36 over 54
divided by 64 over 48
and then we can use the
keep
change flip principle let's keep the
first fraction the same
let's change division to multiplication
and then
let's flip the second fraction
and now let's simplify
so 36
is basically
nine times
four
fifty four
is nine times six
forty eight is sixteen times three
and sixty four
is sixteen times four
so right now we can cancel a nine
we can cancel a 16
and we can cancel a four
so what we have left over is three over
six
now three over six can be reduced
we can divide both numbers
by three
three divided by three
is equal to one six divided by three is
two
so the final answer is one over two
you
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