Let's analyze the given problem step by step.
We have the equation:
[tex]\[ 1ABA + 32B = 2205 \][/tex]
Where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are unknown digits.
We need to determine which of the given relationships between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is correct:
1. [tex]\( B - A = 1 \)[/tex]
2. [tex]\( A + B = 15 \)[/tex]
3. [tex]\( A - B = 2 \)[/tex]
4. [tex]\( A = B \)[/tex]
5. I don't know
First, let's extract what we know about the form of the numbers:
- [tex]\( 1ABA \)[/tex] is a four-digit number where [tex]\( 1 \)[/tex] is the thousands place, [tex]\( A \)[/tex] is the hundreds and units place, and [tex]\( B \)[/tex] is the tens place. This can be written as:
[tex]\[
1ABA = 1000 + 100A + 10B + A
\][/tex]
Simplifying, we get:
[tex]\[
1ABA = 1000 + 101A + 10B
\][/tex]
- [tex]\( 32B \)[/tex] is a three-digit number where [tex]\( 3 \)[/tex] is the hundreds place and [tex]\( 2 \)[/tex] is the tens place, and [tex]\( B \)[/tex] is the units place. This can be written as:
[tex]\[
32B = 320 + B
\][/tex]
Next, let's sum these transformed numbers:
[tex]\[
1000 + 101A + 10B + 320 + B = 2205
\][/tex]
Combining like terms, we get:
[tex]\[
1000 + 320 + 101A + B + 10B = 2205
\][/tex]
Simplifying further:
[tex]\[
1320 + 101A + 11B = 2205
\][/tex]
Now, isolate the terms involving [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
101A + 11B = 885
\][/tex]
We now test each of the given relationships:
### 1. [tex]\( B - A = 1 \)[/tex]
Substitute [tex]\( B = A + 1 \)[/tex] into the equation:
[tex]\[
101A + 11(A + 1) = 885
\][/tex]
[tex]\[
101A + 11A + 11 = 885
\][/tex]
[tex]\[
112A + 11 = 885
\][/tex]
[tex]\[
112A = 874
\][/tex]
[tex]\[
A = \frac{874}{112}
\][/tex]
Calculating the division, [tex]\( A \)[/tex] is not an integer.
### 2. [tex]\( A + B = 15 \)[/tex]
Substitute [tex]\( B = 15 - A \)[/tex] into the equation:
[tex]\[
101A + 11(15 - A) = 885
\][/tex]
[tex]\[
101A + 165 - 11A = 885
\][/tex]
[tex]\[
90A + 165 = 885
\][/tex]
[tex]\[
90A = 720
\][/tex]
[tex]\[
A = \frac{720}{90} = 8
\][/tex]
Then [tex]\( B = 15 - 8 = 7 \)[/tex].
Let's check by substituting [tex]\( A = 8 \)[/tex] and [tex]\( B = 7 \)[/tex] back into the original numbers:
[tex]\[
1ABA = 1 \cdot 1000 + 100 \cdot 8 + 10 \cdot 7 + 8 = 1788
\][/tex]
[tex]\[
32B = 320 + 7 = 327
\][/tex]
[tex]\[
1788 + 327 = 2115 \neq 2205
\][/tex]
### 3. [tex]\( A - B = 2 \)[/tex]
Substitute [tex]\( B = A - 2 \)[/tex] into the equation:
[tex]\[
101A + 11(A - 2) = 885
\][/tex]
[tex]\[
101A + 11A - 22 = 885
\][/tex]
[tex]\[
112A - 22 = 885
\][/tex]
[tex]\[
112A = 907
\][/tex]
[tex]\[
A = \frac{907}{112}
\][/tex]
Calculating the division, [tex]\( A \)[/tex] is not an integer.
### 4. [tex]\( A = B \)[/tex]
Substitute [tex]\( B = A \)[/tex] into the equation:
[tex]\[
101A + 11A = 885
\][/tex]
[tex]\[
112A = 885
\][/tex]
[tex]\[
A = \frac{885}{112}
\][/tex]
Calculating the division, [tex]\( A \)[/tex] is not an integer.
Since we tried all given relationships and none fit correctly, the best description is:
5. I don't know.
Therefore, the best choice to describe the relationship of the missing digits is:
[tex]\[ \text{Oldon't know.} \][/tex]
