Question

A deck of 320 punched cards can be read by one reader in 20 minutes. A second reader can read the same deck in 12 minutes. How many minutes would it take both readers working together to read a deck? How long would it take to process the deck of cards in example one if the first reader stopped working
after 5 minutes?

Answer 1

Denote \(R_1, R_2\) to be the average reading speed of each reader, then, we have:
\[
R_1=\frac{320}{20}=16\\
R_2=\frac{320}{12}=\;\frac{80}{3}
\]So, if both work in perfect parallel motion, we have that their combined reading speed is:
\[
16+\frac{80}{3}=\frac{128}{3}
\]So, it would take a total of:
\[
\frac{320}{\frac{128}{3}}=320\cdot\frac{3}{128}=7.5\text{ minutes}
\]To read the whole deck.

Answer 2

If one of them stopped working after 5 minutes, then, we have that they would have read a combined total of:
\[
16\cdot5+\frac{80}{3}\cdot5=80+\frac{400}{3}=213+\frac{1}{3}\text{ cards}
\]So, we get a total of
\[
320-(213+\frac{1}{3})=146+\frac{2}{3}
\]Remaining cards.
We only have the second machine working, so we get:
\[
\frac{146+\frac{2}{3}}{\frac{128}{3}}=(146+\frac{2}{3})\cdot\frac{3}{128}=3.4375\text{ minutes}
\]To finish the deck, giving us a combined total of:
\[
5+3.4375=8.4375\text{ minutes}
\]
Wow... I sure hope I didn't screw up any of the calculations.

Answer 3

Wait, wait, wait, wait, yes I did...
For that last part, it would take:
\[
\frac{146+\frac{2}{3}}{\frac{80}{3}}=5.5\text{ minutes}
\]More, so, we get that it would take:
\[
5.5+5=10.5\text{ minutes}
\]