Alright, let's work through the problem step-by-step.
### Part (a):
1. Setting Up the Variables:
- Let the smaller number be [tex]\( b \)[/tex].
- The larger number is [tex]\( 5 \)[/tex] times the smaller number, so we can represent it as [tex]\( a = 5b \)[/tex].
2. Formulating the Equation with the Condition:
- According to the problem, if we add 24 to both numbers, the larger new number becomes twice the smaller new number. Mathematically, this can be expressed as:
[tex]\[
(a + 24) = 2(b + 24)
\][/tex]
3. Substituting for [tex]\( a \)[/tex]:
- We know that [tex]\( a = 5b \)[/tex]. Substituting this into the equation gives:
[tex]\[
(5b + 24) = 2(b + 24)
\][/tex]
4. Solving for [tex]\( b \)[/tex]:
- Expand and simplify the equation:
[tex]\[
5b + 24 = 2b + 48
\][/tex]
- Subtract [tex]\( 2b \)[/tex] from both sides:
[tex]\[
3b + 24 = 48
\][/tex]
- Subtract 24 from both sides:
[tex]\[
3b = 24
\][/tex]
- Divide by 3:
[tex]\[
b = 8
\][/tex]
5. Finding [tex]\( a \)[/tex]:
- Substitute [tex]\( b = 8 \)[/tex] back into the equation [tex]\( a = 5b \)[/tex]:
[tex]\[
a = 5 \times 8 = 40
\][/tex]
So, the numbers are:
- The smaller number, [tex]\( b \)[/tex], is [tex]\( 8 \)[/tex].
- The larger number, [tex]\( a \)[/tex], is [tex]\( 40 \)[/tex].
Thus, the two numbers are 8 and 40.
### Part (b):
You mentioned "One of the digits of a 2-digit numb" but did not complete the sentence or provide a clear question. If you can provide the complete question, I'll be happy to help solve it!
