Monday, 22 April 2013

Maths and Movies

Let's read some facts about Motion Picture Films:

When we watch a movie, we think we are seeing people and objects moving across a lighted screen. The term “movie” is short for moving pictures. But the motion is an illusion. What is really happening is that a series of still pictures are being projected on the screen in rapid succession—fast enough that our eyes and brain are tricked into thinking we see movement.

Almost all movies are shot using photographic film, using a technique invented over 100 years ago. The film is a long strip of plastic consisting of a series of individual pictures called “frames.” Motion picture film is not much different from film used in ordinary cameras except that it is much longer.

Image from 123RF

Film serves two purposes in the motion picture industry. First, it is used to shoot the scenes that make up the movie. An editor selects the best shots and splices them together to create a master copy. Then the master is copied onto other rolls of film, and the copies are sent to movie theaters.

If you’ve ever seen a film projector as it works, you may think that the film is moving through the projector continuously. If you look closer, you will notice that the film seems to be shaking or vibrating. In fact, a special mechanism slides each frame into position behind the lens and then stops it while a flash of light is projected through the film and onto the screen. To allow the film to move precisely, a series of holes are punched in the sides of the film.

Most feature films and many TV shows are shot on 35 mm film that has 64 holes per foot. There are 4 holes on each side of each frame.

Now that you know the facts, let's surf the net and try to answer the following questions. 

1. The earliest commercial movies did not have sound. They were called silent movies, even though there was usually a band playing music when the film was shown. Most silent moves were projected using a film speed of one foot per second. How many frames per second were displayed in a silent movie?

2. When sound was introduced in 1929, the motion picture frame rate was increased to 24 frames per second. The same rate is still used in movies today. How fast does the film move at this rate, in feet per second?

3. The movie Toy Story 2 runs for 92 minutes. How long is the projector film for this movie? Try estimating the answer first.

4. How many frames were in the movie Toy Story 2?

5. In a live action movie, only a small fraction of the film that is shot makes it into the final movie. Each scene may be shot several times (“Scene 6, take 3!”), and only portions of each shot are used. Suppose only 1/20th of the film that is shot is eventually used. How much film must be purchased to shoot a 90 minute movie? If a 400 foot roll of camera film cost $220 and developing costs $0.20 per foot, how much
money would you put in your budget for film and developing?

6. Documentaries, many independent films, and some TV shows are shot using 16 mm film. The smaller size means smaller, more portable cameras and lower costs for copies. How much do you think a 400 foot roll of 16 mm film would cost? Hint: Think about what might affect the cost of the film (area, length, labor hours).

Write your answers into the Comments box. I will soon provide you with the correct answer key sheet.

Image from FilmEducation

Source: PBS Teachers


  1. 1. Silent films where shot at variable speeds from 12 to 22 frames per second being the average speed of around about 16 frames per second.
    2. It's a direct proportion: the original speed of 1 feet per second divided by 16 frames and multiplied by the actual 24 frames produces a result of 1.5 feet per second.
    3. 1.5 ft/sec · 60 sec/min · 92 min = 8280 feet
    4. 24 frames / sec · 60 sec/min· 92 min = 132480 frames

    1. Thanks a lot, Bea, we will soon find out the answers to this 'mystery'!

    2. 5. 8100·20 = 162000 feet of film must be purchased

    3. OK, Bea, keep up the good work!

  2. About Maths and Movies:

    An approach to the solution could be the following:

    1- Silent movies film speed = 1 foot per second. ! foot = 64 holes. 8 holes/frame (4 each side) 8 frames per second

    2- 24 frames/second mean 3 feet per second

    3- Toy Story 2 92 minutes, 3 feet per second = 180 feet per minute x 92 = 16,560 feet

    4- 132,480 frames

    5- Only 1/20th of the fil is eventually used . 90 minute movie means 14,600 feet. If it is just the 20th part of the necessary length it means that 292,000 feet must be purchased.

    6- I don’t have accurate information to answer this question. As far I understand 16 mm film cost the same for processing and transferring than 35 mm film. Also, some factors as area, labour hour costs and so forth increase the final cost enough to make 35 mm film competitive with 16 mm film.
    María tomás

  3. Monserrat Menéndez19 May 2013 at 20:31

    Here my calcultations:

    1. If the speed of the film is 1 ft/s, and every ft has 64 holes, there will be 64 holes/s.
    If there are 4 holes per frame, in 64 holes (1 s) there will be 16 frames. So the answer is 16 frames per second.

    2. 24 frames/s are equivalent to 96 holes/s.
    If there are 64 holes per foot, the speed will be 96/64= 1,5 ft/s.

    3. 92 minutes are 92*60 = 4620 s
    As the speed of the film is 1,5 ft/s, the film will be 6900 feet long.

    4. If the film runs for 4620 s and the speed is 96 holes per second, the film has 96*4620 = holes. As a frame has 4 holes, there will be 96*4620/4 = 96*1155 = 110880 frames.

    5. The length of a 90 minutes film is 90*60*1.5 = 8100 feet.
    If only 1/20th of the film purchased is used to shoot, the film purchased must be 8100*20 = 162000 feet.

    On the other hand:
    If a roll costs $220, and the film need 162000/400 = 405 rolls, they will cost 405*220 = 89100 $
    To develop the film will cost 0,20*162000 = $32400

    The overall cost then will be = 32400+89100 = $121500

    6. In this case, labor hours are the same if the film is shooted with a 35 or a 16 mm film, but the roll cost changes and it will be at about 16/35th of the cost of a 35 mm film.

    So according to previous exercise, for a 400 foot roll, the labor hours will be 400*0.2 = 80 $, and the estimated cost of the film 16/35*220 = $100.57.

    In resume, a 16 mm film will be at about (80 +100.57) / [220*(1+0.2)] = 76 % of the cost of a 35 mm film.


  4. 1. In a silent movie there are 8 frames per second.
    2. The film does 192 frame per foot per second.
    3. Toy story 2 has 168249.6 cm proyector film.
    4. 44160 frames.
    5. 1800. 300 dollars
    6. 182.86 dollars

  5. 1.
    Old films ran at anywhere from 12 to 22 frames per second, with 16-20 fps being about average up through the early 1920s.
    Sound film speed ....24 FPS, or 90 feet per minute.
    8,280 feet and as 1 foot = 0.3048 meters, then 2.500 meters
    92 min * 60 sec * 24 frames per second = 132,480 frames
    Another example.......Final Fantasy has got 149, 246 frames
    400 foot roll = 200 $ + (400 * 0.20) = 300 $per foot
    90 minutes + 20 times more= 1800 minutes
    90 feet per minute
    1800 * 90 = 162000 feet
    162000 *300 = 48 millions
    6mm could cost $142 to $146 per 400’ roll (11 minutes). Then, a normal film of 90 minutes would cost $1200. But if you have to repeat the scenes could be 10 or 20 times more. Then, $25.000

  6. Solutions
    1. Silent films ran at 16 frames per second (64 holes/foot divided by 4 holes per frame times one foot per
    2. 1.5 ft/sec or 1 1/2 ft per sec. (the original speed of 1 ft/sec times 24 frames divided by 16 frames).
    3. Estimate 8100 feet (1.5 ft/sec times 60 second per minute = 90 ft per minute. 90 ft per minute times 90
    minutes). Exact answer: 8280 feet (1.5 times 60 times 92). Of course, the actual film is longer because of
    previews of coming attractions, cute announcements not to smoke or talk in the theater, and a section of
    blank film to allow the projectionist to thread the camera. Teachers might ask students to estimate how
    much length all this adds.
    4. 132480 (24 frames per second times 60 seconds/minute times 92 minutes). Teachers might note that
    Toy Story 2 was made using computers, but in older animated movies like Snow White or Fantasia, each
    frame had to be drawn and colored separately. That took an army of artists.
    5. You would need at least 162,000 feet of film (8100 feet times 20). That is 405 rolls (162,000/400).
    Your budget for film and developing needs to be at least $89,100 ($220/roll times 405 rolls). Your budget
    for developing needs to be at least $32,400 (162,000 feet times $0.20 per foot). The total is $121,500.
    Another way to solve this problem is to calculate the cost of film per foot ($220/400 = $0.55) to get a total
    cost of $0.75 per foot for film and developing. Then multiply the total cost per foot by the 162,00 foot
    length. An extra credit answer would take into account the fact that some film would be wasted at the end
    of each roll.
    6. There is no "right" answer. One estimate would be to assume that the cost is prrportional to the total
    area of film. We know that a 400 foot roll of 35 mm film costs $220. So one estimate of the cost for a 16
    mm roll of the same length would be $101 (220 times 16/35). But there are other costs that do not change
    much like perforating the guide holes, packaging and paying for the building, machinery and the workers
    salaries. The actual cost for the same type of film is $126.56, based on the Kodak catalog at


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