(a) Elsa divides a two-digit number by another two-digit number. She gets the answer 0.15. She says that there is only one possible pair of numbers that will give this answer. Is she correct? Show how you decide.



Answer :

Certainly! Let's go through the problem step-by-step to see if there is indeed only one possible pair of two-digit numbers that results when one is divided by the other and yields 0.15.

1. Understanding the problem: Elsa divides a two-digit number [tex]\( A \)[/tex] by another two-digit number [tex]\( B \)[/tex] and gets the result 0.15. This can be expressed as:
[tex]\[ \frac{A}{B} = 0.15 \][/tex]
which implies:
[tex]\[ A = 0.15 \times B \][/tex]

2. Defining the range: Since [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are both two-digit numbers, both must be between 10 and 99 inclusive.

3. Substituting values: For each two-digit number [tex]\( B \)[/tex], we will calculate [tex]\( A \)[/tex] as [tex]\( A = 0.15 \times B \)[/tex]. We need to check if [tex]\( A \)[/tex] is also a two-digit integer within the range from 10 to 99.

4. Finding the possible pairs:
- Suppose [tex]\( B = 80 \)[/tex]:
[tex]\[ A = 0.15 \times 80 = 12 \][/tex]
Since 12 is a two-digit number, this is a possible pair [tex]\((A, B) = (12, 80)\)[/tex].

5. Checking for other pairs:
- If [tex]\( B \)[/tex] takes values different from 80, we calculate [tex]\( 0.15 \times B \)[/tex]. If the result is not a two-digit integer between 10 and 99, we discard it.
- For [tex]\( B \)[/tex] values from 10 to 99, the only value that when multiplied by 0.15 results in a two-digit number is [tex]\( B = 80 \)[/tex].

6. Conclusion: After checking all possible values for [tex]\( B \)[/tex] in the range from 10 to 99, we find that the only pair [tex]\((A, B)\)[/tex] that satisfies [tex]\( \frac{A}{B} = 0.15 \)[/tex] with [tex]\( A \)[/tex] and [tex]\( B \)[/tex] being two-digit numbers is:
[tex]\[ (12, 80) \][/tex]
Therefore, Elsa is correct. There is only one possible pair of numbers that give the answer 0.15 when a two-digit number is divided by another two-digit number. This pair is [tex]\( (12, 80) \)[/tex].