Sure, let's break down and solve each question step by step.
### Question 1
Find the two-digit number that exceeds the units digit by 8 and the sum of the digits is less than 14.
Let's denote the two-digit number as represented by its tens digit [tex]\(T\)[/tex] and its units digit [tex]\(U\)[/tex]. Then, the number can be written as [tex]\(10T + U\)[/tex].
#### Conditions:
1. The number exceeds the units digit by 8: [tex]\(10T + U\)[/tex] exceeds [tex]\(U\)[/tex] by 8.
This translates to:
[tex]\[
10T + U - U = 10T = 8
\][/tex]
2. The sum of the digits is less than 14:
[tex]\[
T + U < 14
\][/tex]
Now solve this problem:
1. From the first condition, we recognize that [tex]\(T = k\)[/tex] where [tex]\(T\)[/tex] must be 8 more than [tex]\(U\)[/tex], soft [tex]\(U +8 = T\)[/tex].
2. Using the second condition [tex]\(T + U < 14\)[/tex]:
We substitute [tex]\(T = U + 8\)[/tex] into the condition and get:
[tex]\[
(U + 8) + U < 14
\][/tex]
Simplifying this, we get:
[tex]\[
2U + 8 < 14
\][/tex]
[tex]\[
2U < 6
\][/tex]
[tex]\[
U < 3
\][/tex]
Given that [tex]\(U\)[/tex] is a single-digit number (0 through 9), we find the possible values for [tex]\(U\)[/tex] under these conditions:
[tex]\[
U = 0, 1, 2
\][/tex]
Now, using the values of [tex]\(U\)[/tex] to find [tex]\(T\)[/tex]:
- If [tex]\(U = 0\)[/tex], then [tex]\(T = 0 + 8 = 8\)[/tex], resulting in the number [tex]\(80\)[/tex].
- If [tex]\(U = 1\)[/tex], then [tex]\(T = 1 + 8 = 9\)[/tex], resulting in the number [tex]\(91\)[/tex].
- If [tex]\(U = 2\)[/tex], then [tex]\(T = 2 + 8 = 10\)[/tex] which is not valid since [tex]\(T\)[/tex] should also be a single digit.
But each sum of digits:
- [tex]\(T = 8, U = 0 \rightarrow T + U = 8 + 0 = 8 < 14\)[/tex]
- [tex]\(T = 9, U = 1 \rightarrow T + U = 9 + 1 = 10 < 14\)[/tex]
So valid numbers are 80 and 91.
### Question 2
A man is 41 years old and his son is 9. In how many years will the father be three times as old as the son?
Let [tex]\(x\)[/tex] be the number of years after which the father will be three times as old as his son.
#### Current ages:
- Father's age: 41 years
- Son's age: 9 years
#### Future ages after [tex]\(x\)[/tex] years:
- Father's age: [tex]\(41 + x\)[/tex]
- Son's age: [tex]\(9 + x\)[/tex]
We need to find when the father's age will be three times the son's age:
[tex]\[
41 + x = 3(9 + x)
\][/tex]
#### Solve the equation:
[tex]\[
41 + x = 27 + 3x
\][/tex]
[tex]\[
41 - 27 = 3x - x
\][/tex]
[tex]\[
14 = 2x
\][/tex]
[tex]\[
x = \frac{14}{2}
\][/tex]
[tex]\[
x = 7
\][/tex]
Thus, in [tex]\(7\)[/tex] years, the father will be three times as old as the son.
