Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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].
[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].
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
