### Video Transcript

Alaine is tiling a T-shaped section
of wall in her kitchen. The section of wall is shown in the
picture along with its dimensions in millimeters. Each tile is a rectangle with
dimensions of 30 centimeters by 25 centimeters. Alaine uses exactly 12 whole tiles
to tile the T-shaped section of wall. Show in a labelled sketch exactly
how Alaine tiled the wall.

Now the first thing that we notice
in this question is that the dimensions of the wall are given in millimeters, but
the dimensions of the tiles are given in centimeters. And so our first step in solving
this problem can be to convert these dimensions so that they’re in the same
units. We know that one centimeter is
equal to 10 millimeters. If we divide both sides here by 10,
we get that one over 10 centimeters is equal to one millimeter. And we can use this conversion in
order to convert the dimensions of the wall, which are in millimeters, to
centimeters. So we simply divide each length by
10. The length of 1200 millimeters
becomes 120 centimeters. The length of 500 millimeters
become 50 centimeters. The length of 300 millimeters
become 30 centimeters. And the length of 600 millimeters
becomes 60 centimeters.

Now that we’ve got the dimensions
of the wall in centimeters and the question gives us the dimensions of the tiles in
centimeters, we have one standard unit for all of the measurements in this
question. And that unit is centimeters. So we’re ready to answer the
question. Let’s consider our tiles with
dimensions of 30 centimeters by 25 centimeters. Let’s look at the top left of the
wall and consider the different ways in which Alaine could have placed a tile
here. She could have placed a tile with
the 30- centimeter side vertical, which would’ve looked like this. We can see that the height of this
section of the wall on the left is 50 centimeters. And in this orientation, the height
of the tile is 30 centimeters. So this means that the height of
the gap which is left below the tile is 50 minus 30, which is also equal to 20
centimeters. And since this is smaller than any
of the dimensions on the tiles, Alaine would not be able to fit a tile in this
gap. And the question tells us that
Alaine used exactly 12 whole tiles. So this means that Alaine didn’t
cut up any of the tiles.

So we can conclude the Alaine did
not place the tile in this orientation in this corner. Let’s see what happens if we put
the tile with the 30- centimeter side horizontal. Now we can calculate the height of
the gap below the tile with the tile in this orientation. The height of the wall on the left
is still 50 centimeters. The height of the tile is 25
centimeters. And so the height of the bit of
wall left below the tile will be 50 minus 25, which is also equal to 25
centimeters. And now we can see that one of the
dimensions of our tile is also 25 centimeters. So we’ll be able to fit another
tile below this tile, and this tile will be in the same orientation as the one above
it. The 30- centimeter side will be
horizontal, and the 25-centimeter side will be vertical. And since we have tried both
possible orientations of the tiles, we know that this is the only possible way in
which we can fit tiles in this left-hand side of the T. Since the T-shape is symmetrical
down the horizontal axis, we know that the tiles on the opposite side of the T must
be arranged the same way.

Now we have fitted four tiles into
our T-shaped wall. Let’s now consider the section of
wall which we have remaining. This section of wall has got a
width of 60 centimeters. And its height is 50 centimeters
plus 50 centimeters, which is also equal to 100 centimeters. Now let’s try and fit the tiles in
this rectangular section so that the 30-centimeter side is vertical. What this means is at the height of
the tiles is going to be a multiple of 30. The height of the wall in this
rectangular section is 100 centimeters. So let’s see how many times 30
centimeters goes into 100 centimeters. So we divide 100 by 30, and we can
cancel a multiple of 10 from the top and bottom. So we simply cross the zero out,
leaving us with 10 over three. Now we notice that we have an
improper or top-heavy fraction. And we know that three times three
is equal to nine. So we can write this improper
fraction as nine plus one all over three. Then we can separate the addition
in the fraction into two separate fractions to give us nine over three plus one over
three. Then nine divided by three is
simply three. So we’re left with three plus
one-third, which we can write as a mixed number. And that mixed number is three and
a third.

Now what this tells us is it three
and a third tiles will fit vertically in this rectangle when the 30- centimeter side
is vertical. So we can draw three tiles with the
30-centimeter side vertical on our rectangle in the middle. However, this leaves us with a gap
at the bottom. And the length of this gap is
one-third the height of the tile in this orientation, or one- third times 30, which
is also equal to 30 over three or 10 centimeters. And since 10 centimeters is too
small to fit another tile in, this orientation of the tiles will not work. Next we can try to fit the tiles
with the 30-centimeter side horizontal. What this means is that the
25-centimeter side will be the vertical side. Let’s see how many times this
height of 25 centimeters of the tile will fit in the height of the rectangle of the
remaining wall, so that’s the height of 100 centimeters.

In order to do this, we divide the
height of the wall, so that’s 100 centimeters, by the height of the tile in this
orientation, so that’s 25 centimeters. So we have 100 over 25. And first we notice that both the
top and bottom of the fraction are multiples of five. So we can divide both the top and
bottom by five, leaving us with 20 divided by five. And again both the top and bottom
of the fraction are multiples of five. So we will divide both numerator
and denominator by five. This gives us an answer of four
over one, which is also equal to four since anything divided by one is simply
itself. What this tells us is that in this
orientation, we can fit four tiles above one another inside this rectangle. So let’s draw these four tiles
in.

Now we’ve figured out how many of
the tiles we can fit vertically in this orientation, let’s quickly calculate and
check that we can fit another tile next to this one. We have that the width of our tile
in this orientation is 30 centimeters, and the width of the rectangle is 60
centimeters. So therefore, the width of the gap
remaining will be 60 minus 30, which is also equal to 30 centimeters. And since this matches the width of
the tile in this orientation, this tells us that we can fit another four tiles in
the same orientation in the remaining space. So let’s draw them in. And now we can see the all 12 tiles
fit perfectly on the T-shaped section of wall. And so this must be exactly how
Alaine tiled the wall. And therefore, this is the solution
to this question.