Bo- Using Unit Conversions in Problem Solving

"The Children of Israel journeyed from Rameses to Succoth, about six hundred thousand on foot, the men, aside from the children." ~Shemot 12:37

Rashi here explains that the distance from Rameses to Succoth was 120 mils, and that Hashem made their travel there as if in an instant.

To gain a better understanding of this explanation in today's measurements, 1 mil = 2,000 amot, and 1 amah is estimated between 18-24 inches. Based on this estimate, 1 mil would be anywhere from 3,000-4,000 ft.

Rashi also explains that the estimated number that is given here is men from twenty years and older, possibly draft age, which would be men 20-60 years old.

As I was reading this, I wondered how much distance a group of this size would cover when they're traveling as a whole group (ie, how much land would the whole group cover at once?). If Hashem hadn't brought them immediately through a miracle, would the first person have reached Succoth before the last people left Rameses?

To begin to figure this out, we have 2 calculations to consider:
1) What would be today's equivalent of the distance they needed to travel?
2) What would the full population estimate be for the group?

Distance Traveled:
Based on Rashi's information above, they traveled 120 mils, where each mil is anywhere from 3,000-4,000 ft. For an easier calculation, you can choose either the lower estimate or the upper estimate. I'm going to work through both calculations and offer a range.
To calculate, we will multiply the distance in feet for 1 mil times the number of mils that was their travel distance (120).
For the lower estimate, that means 3,000 (ft) x 120 (mils) = 360,000 ft
For the upper estimate, that means 4,000 (ft) x 120 (mils) = 480,000 ft

So, the distance from Rameses to Succoth was anywhere from 360,000 - 480,000 ft.

Another level of unit conversion to consider:
If we know that 5,280 ft = 1 mile, we can convert this range into miles. Do we multiply or divide here? This question will stump many students. Let's think, we are trying to convert a number of feet into miles. We know that every group of 5,280 ft together make 1 mile. We are taking big chunks of feet and changing them each out for 1 mile. To make this conversion, we are finding out how many chunks of 5,280 ft can be fit into, say, 360,000 ft. This means division.

So, 360,000 ft ÷ 5,280 ft = approximately 68.18 miles 
and 480,000 ft ÷ 5,280 ft = approximately 90.91 miles

So, the distance from Rameses to Succoth was anywhere from 68.18 - 90.91 miles.

Population Estimate:
If we use the estimate that there were approximately 600,000 men between the ages of 20-60, it's probably a fair estimate to say that this is 1/3 of the male population (men younger than 20 and men older than 60 making up the other 2/3 of the male population). We can also say that the male population is only half of the total population, assuming that the population was roughly 1/2 male and 1/2 female. 

To calculate how many men there were, we can multiply 600,000 x 3 = 1,800,000 men.
To calculate the total population, we can double this number: 1,800,000 x 2 = 3,600,000 total estimated population. 

So, we can safely estimate that there were more than 3.5 million people who left Egypt and traveled from Rameses to Succoth.

Final Calculations:
Although the numbers are larger to calculate with, I'd like to use the distance measured in feet to finalize our calculations. One important aspect of understanding the information and numbers that you're working with is recognizing which numbers make the most sense to use for a calculation. If we want to estimate how much space a person needs, it's easier to estimate this number using a measurement of feet rather than miles. 
If we estimate, conservatively, that each person needs 2 feet of space, then with a distance of 360,000 ft, you could have a line of 360,000 ÷ 2 = 180,000 people, at most.

With the same conservative estimate, then with a distance of 480,000 ft, you could have a line of 480,000 ÷ 2 = 240,000 people, at most.

So, with little space in front or back of them, you could have a line of between 180,000 - 240,000 people reaching all the way from Rameses to Succoth. 

But we have an estimate of 3,600,000 people!?
3,600,000 (population) ÷ 180,000 (people in each line) = 20 lines of people 
or
3,600,000 (population) ÷ 240,000 (people in each line) = 15 lines of people 

So, without consideration for all the animals and possessions that the Israelites left with, if they lined up almost front to back with each other, there would have been 15-20 lines of people spanning the entire distance from Rameses to Succoth.

An Extra Look:
In Judaism, we often speak of a person's "daled amot" or 4 amot of personal space. With this measurement in mind, we could use the distance from Rameses to Succoth and calculate how much space each person would have needed if we allowed each person 4 amot.

Let's go back to the first equivalency above: 1 mil = 2,000 amot
We know that there were 120 mil, so that would be 
120 x 2,000 = 240,000 amot as the distance between Rameses and Succoth.

If we want to allow 4 amot each to 3,600,000 people, we need to calculate 
3,600,000 x 4 = 14,400,000 amot needed.

Ok, so now we need to figure out how many times the distance from Rameses to Succoth would be needed in order to accommodate all the space needed by the population.

14,400,000 amot (population) ÷ 240,000 amot (distance) = 60 lines of people.

Another way to look at the same calculation is to follow the same calculation process that we did above for feet.
The distance was 240,000 amot ÷ 4 amot for each person = 60,000 people would fill a line reaching from Rameses to Succoth.

Our population of 3,600,000 ÷ 60,000 people in each line = 60 lines of people.

We just confirmed the same answer through 2 different calculation approaches!

So, again, if each person had 4 amot of space to travel (not accounting for animals or possessions), they would have had 60 lines of people lined all the way from Rameses to Succoth.