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Question 3

In the addition problem below, the numbers being added have unknown digits signified by the letters "A" and "B". Assume that the unknown digits represented by the same letter are the same digit. Determine which of the following choices best describes the relationship of the missing digits.

[tex]\[ 1 A B A + 32 B = 2205 \][/tex]

A. [tex]\( B - A = 1 \)[/tex]

B. [tex]\( A + B = 15 \)[/tex]

C. [tex]\( A - B = 2 \)[/tex]

D. [tex]\( A = B \)[/tex]

E. I don't know



Answer :

GinnyAnswer
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]

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