 # Problem of the Week Problem B and Solution Screen Size, Now and Then

## Problem

Flat screen TVs usually have a screen ratio of $$16:9$$. This means that if the screen is $$16$$ units wide, then it will be $$9$$ units high. If the screen is $$32$$ units wide, then since $$32 = 16\times 2$$, it will be $$9\times 2 = 18$$ units high, and so on.

1. Starting in the bottom-left corner of a grid that is $$20$$ units wide and $$10$$ units high, use a ruler to draw a flat screen TV screen that is $$16$$ units wide and $$9$$ units high. 2. Older TVs had a screen ratio of $$4:3$$. If an older TV was $$9$$ units high, how many units wide would it be?

3. Draw the TV screen from part (b) on the same grid used in part (a), also starting in the bottom-left corner.

4. How many more square units of area does the flat screen TV screen have compared to the older TV screen, if they both have a height of $$9$$ units?

5. A 4K flat screen TV has $$3840\times 2160$$ pixels. If the screen is $$122$$ cm wide by $$69$$ cm high, how many pixels per cm$$^2$$ are there? Round to the nearest whole number.

## Solution

1. The drawing of the flat screen TV screen on the grid is shown in part (c).

2. The screen ratio of an older TV is $$4:3$$, so if the the height is $$9$$ units, that means we have multiplied the $$3$$ in our screen ratio by $$3$$ to get $$9$$. So the width would be $$4\times 3=12$$ units.

3. The grid below shows the flat screen TV with a dashed blue line and the older TV with a solid red line. 4. We can count the squares on our grid that are part of the flat screen TV but not the older TV. We notice that the flat screen TV has $$4$$ more squares of width, and since the height is $$9$$ units for both TVs, there are $$4\times 9=36$$ more square units of area in the flat screen TV.

5. There are $$3840\times 2160=8\,294\,400$$ pixels in total, and the area of the TV is $$122 \times 69 =8418$$ cm$$^2$$. Thus there are $$8\,294\,400 \div 8418 = 985$$ pixels per cm$$^2$$.