Welcome to lesson 1.2.
In this lesson we'll be reviewing
some of the basic skills involving fractions.
We'll first review how to convert a mixed number
to an improper fraction.
A mixed number is a whole number and a proper fraction.
An improper fraction is a single fraction
where the numerator is greater than
or equal to the denominator.
So we first want to convert 3 2/7 to an improper fraction
following the procedure outlined above,
we first multiply the denominator in the whole number,
which would be seven times three,
step two, we add the numerator,
which in this case would be plus two.
We write the result over the original denominator,
which in this case is seven.
And now we'll simplify using the order of operations.
So we multiply first, seven times three is equal to 21.
So we have 21 plus two all over seven,
which gives us 23/7 as our improper fraction.
Let's look at the model below for 3 2/7,
to understand why this procedure works.
Notice these three bars represent the three,
where each part is partitioned into seven equal pieces,
so three is equal to seven plus seven plus seven or 21/7.
And then we have plus these two sevenths,
which gives us 23/7.
Notice how the seven times three gives us
a number of sevenths and three
and the plus two here gives us the remaining 2/7,
giving us a total of 23/7
for the improper fraction for 3 2/7.
For 12 1/3,
again is an improper fraction.
We have the denominator times the whole number,
so three times 12 plus the numerator of one
all over the original denominator of three.
Three times 12 equals 36,
so we have 36 plus one all over three.
So 12 1/3 equals 37/3.
Now let's review how to write an improper fraction
as a mixed number.
We first want to write 42/5 as a mixed number.
So step one we divide the numerator by the denominator.
So we'd have 42 divided by five
and there are eight fives in 42, eight times five equals 40.
We subtract and notice out here we have a remainder of two.
The quotient becomes the whole number part
of the mixed number and the fraction part
is your remainder over the denominator.
So 42/5
is equal to eight and the fraction
is going to be the remainder two over the divisor
or the original denominator which would give us 2/5.
Now let's look at the model below to see why this works.
Here we have a total of 42/5 where every time we have 5/5
we have one whole.
So to tell 42/5 is equal to one, two, three,
four, five, six, seven, eight and 2/5.
Next we have 53/9.
So we'll take 53
and divide by nine
and there are five nines in 53, five times nine equals 45.
And we subtract and we have a remainder of eight,
which means 53/9 is equal to five
and the remainder of eight over the denominator
or divisor of nine.
Now let's talk about equivalent fractions.
Equivalent fractions are different fractions
equal to the same amount or same value.
Here we have three models of three equivalent fractions,
where if this rectangle is one whole,
notice how for this model, we've cut or partitioned
one whole into four equal pieces, one of which is shaded.
So this model is the fraction 1/4.
The next model, notice how we have one whole
cut or partitioned into eight equal pieces,
two of which are shaded, so this is the model for 2/8.
Notice here we have two different fractions,
but they're equal to the same amount or the same value,
because notice how the same amount is shaded.
So 1/4 and 2/8 are equivalent fractions.
Here one whole is cut or partitioned into 12 equal pieces,
three of which are shaded, so this model is 3/12.
Notice how these three fractions are equivalent fractions.
So if we're given a fraction, for example, 1/4,
to come up with another equivalent fraction,
we can simply multiply the numerator and denominator
by the same value other than zero and one.
So for example, if you multiply the numerator
and denominator by two, we'd have one times two
over four times two, notice how this gives us 2/8,
which we can see from our model
is an equivalent fraction to 1/4.
If we multiplied 1/4 by three over three,
or multiplied both the numerator and denominator by three,
notice how we'd have 3/12,
our second equivalent fraction to 1/4.
By multiplying by two over two or three over three,
we're simply multiplying by one
by changing the form of the fraction.
So, for example, three, we're asked to find
two equivalent fractions to 2/7.
So there's an infinite number
of equivalent fractions we can find.
For our first equivalent fraction,
let's multiply the numerator and denominator by five.
So we'd have two times five over seven times five
which is equal to 10/35.
10/35 is equivalent to 2/7.
For our second equivalent fraction, let's multiply it by 11.
So we have 2/7 would be equal to two times 11
over seven times 11 which is equal to 22/77.
These would be two equivalent fractions to 2/7.
Of course, there's many others we could find.
Next we want to simplify the following fractions.
A fraction is considered simplified
if the only common factor
between the numerator and denominator is one.
So for 3/18, notice how both three and 18
share a common factor of three,
and therefore 3/18 simplifies.
So just like before when we were multiplying
the numerator and denominator by the same value,
we can simplify a fraction by dividing
both numerator and denominator by the same value.
So here, we can divide out the common factor
by dividing the numerator by three
and dividing the denominator by three.
Notice how now we'd have one over six or 1/6,
which is a simplified form of 3/18,
because one and six only share a common factor of one.
Next we have 42/54.
Notice how 42 and 54 do share a common factor of two,
but they also share a common factor of three,
and, therefore, the greatest common factor
of 42 and 54 is six.
So to simplify this fraction in one step,
we would divide both the numerator and denominator by six.
42 divided by six is equal to seven,
and 54 divided by six is equal to nine.
The only common factor between seven and nine is one
and therefore, we know 7/9 is a simplified form of 42/5.
But I do want to take a look at this again.
Sometimes you may not recognize the greatest common factor
and it may take more than one step to simplify a fraction.
So, for example, if we start again with 42/54.
And let's say we first recognize as a common factor of two.
So we'd have 42 divided by two over 54 divided by two,
this would give us 21 over 27,
but this fraction still simplifies,
because 21 and 27 share a common factor of three.
So now we would divide the numerator
and denominator by three which would give us
the final simplified fraction of 7/9.
So nothing wrong with taking more than one step
to simplify a fraction.
We can only simplify it in one step
if we recognize the greatest common factor
between the numerator and denominator.
I also want to mention there's also another common way
to simplify a fraction by writing
out the prime factorization
of both the numerator and denominator.
For example, looking again at 42/54, again,
the prime factorization of 42 would be
two times three times seven,
and the prime factorization of 54 would be
two times three times three times three.
The nice thing about doing it this way
is that we can actually see the common factors.
Two divided by two simplifies to one,
three divided by three simplifies to one,
giving us our final simplified fraction of 7/9.
So there is more than one way to simplify fractions.
And now for the last example, we're asked to simplify
the following fractions if possible.
Before we take a look at these, though,
remember a fraction bar means division.
So, for example, if we have 15/3,
that means 15 divided by three which is equal to five.
And every division problem has
a corresponding multiplication problem.
15 divided by three equals five
because five times three equals 15.
Notice how the corresponding multiplication problem
is formed by looking at the division problem
in the opposite direction.
Five time three equals 15.
So our first fraction is 1/4.
And we know from above this does not simplify,
because the only common factor between one and four is one,
so let's just write 1/4 equals 1/4.
If we did want to model this fraction,
we'd have one whole cut or partitioned
into four equal pieces, one of which would be shaded.
Next we have four over one which means four divided by one
which we know equals four.
The model for this, because the denominator is one,
we'd have one whole cut or partitioned into one piece
which we already have and we'd have four of these,
giving us a total of four.
And also notice the corresponding multiplication problem
would be four times one equals four.
Next we have 4/4, which means four divided by four
which equals one.
Notice how the corresponding multiplication problem would be
one times four equals four which we know is true.
The model for this would be we have one whole
cut or partitioned into four equal pieces,
and because the numerator is four,
we have 4/4 which equals one whole.
Next we have zero over four,
which means zero divided by four, which is equal to zero.
The reason we know this quotient equals zero
is because the corresponding multiplication problem
formed in this direction would be
zero times four equals zero which we know is true.
The model for this would be we'd have one whole,
because the denominator is four,
we'd cut or partition this into four equal pieces,
but because the numerator is zero,
we'd have zero of these pieces which would be equal to zero.
And finally we have four over zero
which means four divided by zero
and division by zero is always undefined.
To justify why it's undefined,
notice how the corresponding multiplication problem
formed in this direction would be
some number times zero must equal four.
We know there's no number times zero which equals four,
verifying this quotient or division by zero is undefined.
Using a model for a fraction,
there's no way we can take one whole
and cut it or divide into zero equal pieces,
it's just not possible, again,
verifying our quotient is undefined.
Okay, I hope you found this helpful.