16. Assume that there are 20,000 runners in the New York City Marathon. Each runner runs a distance of 26 miles. If you add together the total number of miles for all the runners, how many times around the globe would the marathon runners have gone? Consider the circumference of the earth to be 2.5 x 104 miles.
I understand how to get the total number of miles for all runners -- 26 x 20,000 which equals 520,000 miles.
But why does the answer then require you to divide by the scientific number?
This really confuses me - when to multiply by the scientific number and when to divide.
Thank you
The scientific number 2.5 x 10^4 is not that large and can be changed to a standard number
2.5 x 10^4 = 25000
So now just divide 520,000 by 25,000, which is 20.8
or ,
you could change the 520000 to a scientific number which would be
5.2 x 10^5
so now we divide (5.2 x 10^5) by (2.5 x 10^4)
= (5.2 x 10^5) / (2.5 x 10^4)
= (5.2/2.5) (10^5/10^4)
= 2.08 x 10
= 20.8
I assume you know how to change a scientific to a standard number and vice versa.
I understand that, but what I don't understand is when you multiply by the scientific number and when you divide by the scientific number
any scientific number consists of two factors, the first is a real number with the decimal place after the first digit, and a power of 10
e.g. 4.56 x 10^9
When you multiply two of them, multiply the decimal parts, and multiply the powers of 10 by adding their exponents
e.g.
(3.45 x 10^5)(6.56 x 10^12) = 22.632 x 10^17 = 2.2632 x 10^18
notice I had to change the first factor so it fits the definition of a single digit followed by the decimal
(1.2 x 10^-6)(8.12 x 10^10) = 9.744 x 10^4
when dividing, just divide the first parts, then the powers of 10
e.g.
6.32 x 10^13 ÷ (4.5 x 10^4) = 1.404 x 10^9
3.56 x 10^-4 ÷ (9.23 x 10^5) = .386 x 10^-9 = 3.86 x 10^-10
Some people get confused in the last step. Count how many places you had to move the decimal place, the either add/subtract that count to the exponent of 10
If you are not sure whether to add or subtract, think of it this way:
.386 ---> 3.86 , so I made that one larger, so I have to make the power of 10 smaller and 10^-9 ----> 10^-10
Yes, I understand this, but what I'm not clear about is when reading a problem how do I know whether to multiply or divide by the scientific number??
For example, the example I originally gave, you divided by the scientific number, but how do I know I was supposed to divide by that, and not multiply by that number?
To calculate how many times around the globe the marathon runners would have gone, you need to compare the total number of miles they ran with the circumference of the earth.
You correctly found that the total number of miles for all runners is 520,000 miles (26 miles per runner multiplied by 20,000 runners).
To find out how many times around the globe this distance is equivalent to, you need to divide the total number of miles by the circumference of the earth.
In this case, the circumference of the earth is given as 2.5 x 10^4 miles (which means 2.5 multiplied by 10 raised to the power of 4, or 25,000 miles).
So, to calculate how many times around the globe the marathon runners would have gone, you would divide the total number of miles (520,000 miles) by the circumference of the earth (25,000 miles):
520,000 miles / 25,000 miles
To divide by a scientific number, you can simply divide as you normally would (in this case, 520,000 divided by 25,000), and then move the decimal point in the result to match the position specified by the exponent of the scientific notation.
In this case, the exponent is 4, so you would move the decimal point four places to the left:
520,000 / 25,000 = 20.8
20.8 x 10^4
Therefore, the marathon runners would have gone around the globe approximately 20.8 times.