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Welcome to the 50 there in the
binary number series. In this

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video I'm going to show you how
we can deal with fractions in

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the binary system to be able to
do that, first we're going to

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talk a little bit more about the
power definitions, and then

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we're going to extend the binary
place value system. After that,

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I'm going to show you a couple
of examples how you can convert

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binary fractions into decimal
fractions. So let's extend or

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recap our knowledge about the

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powers. So we all remember that
any number to the zero power, by

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definition, is always one, and
if we talk about the negative

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powers, that is the same as one
over the positive equivalent

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over the same power. So what
does it mean for us? Let's have

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a look at some examples. So if
we talk about 3:00 to the minus

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four, that is just simply one

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over. Three to four, and if we
talk about two to the minus

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three, that is just simply 1 /
2 to three last example.

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Done to the minus two, that's
the same as 1 / 10 to the power

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of 2. Now Lisa, you need to
know the negative powers, but

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remember when we were talking
about the decimal place value

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system, we talked about the
smallest place where you being

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tend to zero and then then to

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the one. Dented it too tentative
310 to the four etc. Now when

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you look at the powers the
powers grow by one as we go from

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right to left. But what happens
when we go from left to right?

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The powers decreased by one and
then if you introduce the

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decimal point you can further
extend this place value table so

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we can talk about 10 to the
minus one because zero last one

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is minus one then.

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Into the minus 210 to the
minus three etc now.

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If we look at it with the number
equivalents, this would be one.

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This would be 1000.

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1000 and 10,000. This way this
would be 1 / 10 this would be

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1 / 100 because 10 to the power
of three is 100 and it will be

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1 / 1000 now.

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It might be a bit more familiar
if we look at this way. 1 / 10

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is 0.11, / 100 is 0.01 and 1 /
1000 is 0.001, so you might

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depending on how you would talk
about it, you either and this

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format or this format now. Why
is it good for us? But basically

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when we start to talk about
decimal numbers and we can talk

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about 3.4. Two, but that's
basically means that I have.

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Three times. 10 to the
0 + 4 * 10 to the

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minus 1 + 2 times.

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10 To the minus two, so
the place values are

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built into the system.

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Let's look at how it differs
when we talk about binary

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numbers. So the binary place
value system we started with two

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to the power of 0, two to the
power of 1 two three power of 2

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to the power of three, and so
on. Now we can also introduce

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something new called a Radix
point and then we can extend our

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knowledge of place values to the
binary system because again, if

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you look at the powers grow from
right to left and then buy one

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and then they decrease by one
from left to right.

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So we can extend this to two to
the minus one.

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Due to the minus 2, two to the
minus three, and so on depending

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on how many digits you want. So
what are these numbers here?

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This is 1.

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This is to this is for this is
it this one? You are quite

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familiar with by now. Now the
radix point comes here. What is

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2 to the minus one? Well that's
1 / 2 one is 2 to the minus two

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that's 1 / 4 what is 2 to the
minus three? That's 1 / 8

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remember two to two is 4 and two
to three is 8. Now again what we

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learned about the binary place
value system is that if I'm go

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from. Right to left. The place.
Why use double?

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So you can see you know double
at 1 S, 2 W, 3 S, 4 W 4 is 8,

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etc. Now if we go from left to
right, the place values gets

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halved, so half of eight is 4.
Half of four is 2, half of two

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is 1. Now that's the true on the
right hand side from the radix

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point because half of 1 is half
half of half is 1/4 and half of

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quarter is an 8. If you can't
really remember this or you have

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forgotten about it, try to draw
them as functions as.

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Slices of pizza to convince
yourself that this is how it

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works, and again, the double
oven 8 is 1/4 and a double of

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1/4 is 1/2. Because remember two
quarters Constanta your half and

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two 8 sconces down to 1/4.

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Now again. In
some cases you will need the

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fraction numbers, but it's
always a good thing to know what

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they look like as decimals. So
half would always be a .5

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quarter would be 0.25 and eight
would be 0.125. Again, if you

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half in .5 you get .25 and if
you half in .25 you got.

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.125 If you're not sure, double
check it with the Calculator.

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So why is it good for us?
Let's look at binary fraction,

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so let's say I've got 11 radix
.1 in binary. Now let's see what

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kind of decimal number is here

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so. Anything that's on the left
hand side from the Radix point

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works like just ordinary binary
numbers, so the place values

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here. One and two, and on the
right hand side from the radix

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point is the new system that we
just introduced. So this place

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value is equivalent to 1/2. So
what I've got here.

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Is I bought 1 * 2
+ 1 * 1.

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Plus 1 * 1/2 This is 2 +
1, three and a half, which I can

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also write S 3.5, so that's the
decimal equivalent of 1, one

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radix .1 and listen when you're
reading out the binary

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equivalent, you don't save, just
simply point you, say radix

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point to make the difference
distinction between binary and

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decimal fractions. Let's look at
a different example and see how

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we can convert that binary
fraction into its decimal

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equivalent. So our number this
case will be one radix, .101.

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I can again build up the
binary place value table, so

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here's the radix point. I will
have two to zero.

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2 to the one, two to two, and so
on. On the left hand side and I

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would have two to the minus 1,
two to the minus, 2 two to the

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minus, three on the right hand

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side. The equivalent.

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1. 24 keep the
point half a quarter.

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And an 8 now when it comes

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to our. Binary number.

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It's very important that you
place them correctly so the

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points have to line up so the
digits I'm using is 1 radix

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.101. What it means to me? I'm
using one of the ones.

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I'm using one of the halves.

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And I'm using Zita over
the quarters and I'm

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using one of the AIDS.

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Again, 0 times anything. Just
going to give me zero and what

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I've got here is 1 + 1/2 plus
and eight, so how my adding

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together half aninat? Well,
whenever I'm adding fractions

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together I always need to make
them the common denominator. So

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what's the common denominator of
two and eight wow 8 so I know

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that half is 48 and the 18.

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Is just 118, so I've got one.

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Plus four 8 + 1 eight
altogether makes 5 eight, so

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one radix .101 in binary is
equivalent to one and 5 eights

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in decimal. Now if you want to
convert the five 8 into

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decimal.

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And look just keep it as a
fraction what you need to do.

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You need to do the division.

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So 5 / 8 is the same as 5
/ 8, so let's do the traditional

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division eight into five dozen
guy bring into O and appoint an

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borrow 0. 8 into 50 Go
6 * 6 * 8 is 48,

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so we got a two remainder borrow
0, again 18 to 20 goes twice

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2 * 8016, Four remainder,
bringing another zero an 18 to

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40 goes five times. That means
that 5 / 8 is equal to

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0.625, so that means that one
and 5 eights.

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Is also the same as 1.625?
Really depends on if you

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happy having the equivalent
instructions or you want

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them as decimal fractions.

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Let's look at another
example. So if I

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have got 111 radix

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.1001. The binary place
values again start with the

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point. I have one.

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2 four and I won't need any
higher place values because I've

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only got 3 digits and then I

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have 1/2. 1/4 on

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8. And the 16. And
again because I only have got 4

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digits in here, I only need four
place values in here. Now let's

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put the corresponding digits
under the place values. So 111

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radix .1001. So what it means to
us is that we have 1 * 4

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+ 1 * 2 + 1 * 1
+ 1 * 1/2.

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Plus 0 * 1/4
+ 0 times on

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8 + 1 *

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16 Again. Or the place
values which carry zeros will

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not be taken into account. So we
have got 4 + 2 + 1 +

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1/2 + 16. Now the whole part of
the number B 4 + 2 + 1

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seventh and the fraction part of

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the number. Is half and a
16 again to add them together? I

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need to make them into fractions
with common denominator 16 will

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be a common denominator, so 1 /
16 plus half breaking up into

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sixteens is just 8 / 16. You
might remember that a half can

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always be represented with the
numbers where the ratio top to

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bottom is one to two. So anytime

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when. Bottom part of the
fraction is double of the top

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part of the fractions. That's
always 1/2, so one 816 + 1 six

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teens make 916's, so it's 7 and
9 / 16. Now I'm going to spare

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you the long division this case
and I'm just going to pick up a

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Calculator and calculate what
this is equivalent to do. So

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this will be 7 point.

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5625 Remember when you want to
calculate the equivalent of it

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as a decimal fraction, you just
need to divide 9 by 16.

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So 1 one
1.1001 in binary

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is equivalent to
7.5625 in decimal.

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I hope you now have an idea of
how to convert binary

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fractions to decimal fractions
and had the extended place

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value system works in the
following pages you will have

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some practice questions and
the answers will follow.

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So these are the practice
questions.

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And here are the answers.