https://file.aiquickdraw.com/mathbot/file/1727032915083gwi5azh4.jpg

In this image, the problem involves arithmetic in base 5 (denoted by the subscript "five").

The given equation is:

\[
100_{\text{five}} + \Box_{\text{five}} = 1000_{\text{five}}
\]

To solve this, we first need to convert the numbers from base 5 to base 10, perform the arithmetic in base 10, and then convert the result back to base 5.

### Step 1: Convert \(100_{\text{five}}\) to base 10
In base 5, \(100_{\text{five}}\) represents:

\[
100_{\text{five}} = 1 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 = 25
\]

So, \(100_{\text{five}} = 25_{\text{ten}}\).

### Step 2: Convert \(1000_{\text{five}}\) to base 10
In base 5, \(1000_{\text{five}}\) represents:

\[
1000_{\text{five}} = 1 \times 5^3 + 0 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 = 125
\]

So, \(1000_{\text{five}} = 125_{\text{ten}}\).

### Step 3: Solve for the missing value
Now the equation becomes:

\[
25_{\text{ten}} + \Box_{\text{ten}} = 125_{\text{ten}}
\]

To find \(\Box_{\text{ten}}\), we subtract 25 from 125:

\[
\Box_{\text{ten}} = 125 - 25 = 100
\]

### Step 4: Convert 100 (base 10) to base 5
Now, we convert \(100_{\text{ten}}\) to base 5. We do this by repeatedly dividing 100 by 5 and recording the remainders:

- \(100 \div 5 = 20\) remainder 0
- \(20 \div 5 = 4\) remainder 0
- \(4 \div 5 = 0\) remainder 4

Reading the remainders in reverse, \(100_{\text{ten}} = 400_{\text{five}}\).

### Final Answer:
The missing value is \( \Box_{\text{five}} = 400_{\text{five}}\).

Would you like more details or have any questions? Here are five related questions you might explore:

1. How do you convert numbers between other bases, such as base 2 and base 10?
2. What are the advantages of using different bases in various applications?
3. How does the carryover process work in base 5 addition?
4. Can you explain subtraction in base 5 using the same method?
5. What are some examples of other base systems used in computing and mathematics?

**Tip**: When converting between bases, always remember to align the powers of the base to the correct digits (starting from zero for the rightmost digit).

Solve for the missing number in the equation 100_five + □_five = 1000_five

Learn how to solve the equation 100_five + □_five = 1000_five by converting between base 5 and base 10. This example covers key number system concepts, including base conversion and arithmetic in non-decimal bases. Suitable for students in Grades 5-7.

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