Would you rather have $200 base 10 or $1000 base on 5? What would be the base for $200. Which I would be better of the two choices
To determine which choice is better, we can convert both amounts to the same base (i.e., the base in which $200 is represented) and then compare their values. Let's go step by step:
1. To find out the base of $200, we follow these steps:
a. Divide $200 by the smallest possible power of the base, which is 10^0 or simply 1. In this case, $200 divided by 1 is $200.
b. If the quotient in the previous step is greater than or equal to the base, divide it by the base again. In our case, $200 is not greater than or equal to any base lower than itself.
c. The base is given by the divisor used in the last step. Therefore, the base for $200 is 1.
2. Now, let's convert $1000 to base 1.
a. Since the base is 5, we need to express $1000 in terms of powers of 5. We can start by finding the highest power of 5 that is less than or equal to $1000, which is 5^4 or 625.
b. Divide $1000 by 625. The quotient is 1, and we are left with a remainder of $375.
c. Now, we need to find the highest power of 5 less than or equal to $375, which is 5^3 or 125.
d. Divide $375 by 125. The quotient is 3, and we are left with a remainder of $0.
3. So, $1000 in base 5 is represented as $1300.
Now, let's compare the values in both bases:
- In base 1, $200 is represented as $200.
- In base 5, $1000 is represented as $1300.
Comparing $200 in base 1 to $1300 in base 5, we can conclude that $200 in base 10 is the better choice, as it has a higher value than $1000 in base 5.