Let's walk through the solution for the question about Sanjaya's monthly fee, which he wrote in the expanded form:
[tex]\[ 4 \times 5^4 + 0 \times 5^3 + 2 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 \][/tex]
### Part a: Writing the short form of his fee in quinary number.
Quinary, also known as base-5, uses digits from 0 to 4. To convert the expanded form into quinary, observe how each term corresponds to its place in the quinary number:
- [tex]\( 4 \times 5^4 \)[/tex] contributes 4 at the [tex]\( 5^4 \)[/tex] place.
- [tex]\( 0 \times 5^3 \)[/tex] contributes 0 at the [tex]\( 5^3 \)[/tex] place.
- [tex]\( 2 \times 5^2 \)[/tex] contributes 2 at the [tex]\( 5^2 \)[/tex] place.
- [tex]\( 0 \times 5^1 \)[/tex] contributes 0 at the [tex]\( 5^1 \)[/tex] place.
- [tex]\( 0 \times 5^0 \)[/tex] contributes 0 at the [tex]\( 5^0 \)[/tex] place.
Summarizing these values, the short form of Sanjaya's monthly fee in quinary number is:
[tex]\[ 40200_5 \][/tex]
### Part b: Converting his monthly fee into binary numbers.
First, we need to compute the value in decimal form from the expanded notation. Substituting the values, we get:
[tex]\[ 4 \times 5^4 + 0 \times 5^3 + 2 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 \][/tex]
[tex]\[ = 4 \times 625 + 0 \times 125 + 2 \times 25 + 0 \times 5 + 0 \times 1 \][/tex]
[tex]\[ = 2500 + 0 + 50 + 0 + 0 \][/tex]
[tex]\[ = 2550 \][/tex]
Now, convert the decimal number 2550 to a binary number. The process involves repeatedly dividing the decimal number by 2 and recording the remainder:
[tex]\[ 2550 \div 2 = 1275 \text{ remainder } 0 \][/tex]
[tex]\[ 1275 \div 2 = 637 \text{ remainder } 1 \][/tex]
[tex]\[ 637 \div 2 = 318 \text{ remainder } 1 \][/tex]
[tex]\[ 318 \div 2 = 159 \text{ remainder } 0 \][/tex]
[tex]\[ 159 \div 2 = 79 \text{ remainder } 1 \][/tex]
[tex]\[ 79 \div 2 = 39 \text{ remainder } 1 \][/tex]
[tex]\[ 39 \div 2 = 19 \text{ remainder } 1 \][/tex]
[tex]\[ 19 \div 2 = 9 \text{ remainder } 1 \][/tex]
[tex]\[ 9 \div 2 = 4 \text{ remainder } 1 \][/tex]
[tex]\[ 4 \div 2 = 2 \text{ remainder } 0 \][/tex]
[tex]\[ 2 \div 2 = 1 \text{ remainder } 0 \][/tex]
[tex]\[ 1 \div 2 = 0 \text{ remainder } 1 \][/tex]
Listing the remainders from bottom to top, we find that the binary representation of 2550 is:
[tex]\[ 100111110110_2 \][/tex]
### Summary:
- The short form of Sanjaya's fee in the quinary number system is: [tex]\( 40200_5 \)[/tex].
- His monthly fee converted to binary is: [tex]\( 100111110110_2 \)[/tex].
