Alright, let's decipher and solve this mathematical problem step-by-step.
Let's denote the two unknown numbers as [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step 1: Understanding the Problem Statement
1. Translate into mathematical terms:
- Suma de dos numeros en 6 a [tex]$39 \frac{5}{10}$[/tex]:
The sum of two numbers is equal to [tex]\( 6a \)[/tex] plus [tex]\( 39.5 \)[/tex].
This can be written as:
[tex]\[
x + y = 6a + 39.5
\][/tex]
2. Interpreting "1 meno a 26":
- [tex]$\frac{5}{6}$[/tex] de su diferencia son 1 meno a 26:
This means that [tex]\( \frac{5}{6} \)[/tex] of the difference of these numbers, minus 1, is equal to 26.
This can be written as:
[tex]\[
\frac{5}{6} |y - x| - 1 = 26
\][/tex]
### Step 2: Solve for the Sum
Let's assume [tex]\( a = 1 \)[/tex] in our equation for simplicity:
[tex]\[
x + y = 6 \cdot 1 + 39.5
\][/tex]
[tex]\[
x + y = 6 + 39.5
\][/tex]
[tex]\[
x + y = 45.5
\][/tex]
### Step 3: Solve for the Difference
From the second statement, let's isolate [tex]\( |y - x| \)[/tex]:
[tex]\[
\frac{5}{6} |y - x| - 1 = 26
\][/tex]
Add 1 to both sides:
[tex]\[
\frac{5}{6} |y - x| = 27
\][/tex]
Multiply both sides by [tex]\( \frac{6}{5} \)[/tex]:
[tex]\[
|y - x| = 27 \times \frac{6}{5}
\][/tex]
[tex]\[
|y - x| = 32.4
\][/tex]
Normally, the absolute value [tex]\( |y - x| \)[/tex] can be positive or negative, so:
[tex]\[
y - x = 32.4 \quad \text{or} \quad y - x = -32.4
\][/tex]
### Step 4: Find [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
We have two key equations:
[tex]\[
x + y = 45.5
\][/tex]
[tex]\[
y - x = 32.4 \quad \text{or} \quad y - x = -32.4
\][/tex]
#### Case 1: [tex]\( y - x = 32.4 \)[/tex]
Add the equations together:
[tex]\[
(x + y) + (y - x) = 45.5 + 32.4
\][/tex]
[tex]\[
2y = 77.9
\][/tex]
[tex]\[
y = 38.95
\][/tex]
Substitute [tex]\( y \)[/tex] back into [tex]\( x + y = 45.5 \)[/tex]:
[tex]\[
x + 38.95 = 45.5
\][/tex]
[tex]\[
x = 6.55
\][/tex]
#### Case 2: [tex]\( y - x = -32.4 \)[/tex]
Add the equations together:
[tex]\[
(x + y) - (x - y) = 45.5 - 32.4
\][/tex]
[tex]\[
2x = 13.1
\][/tex]
[tex]\[
x = 6.55
\][/tex]
Substitute [tex]\( x \)[/tex] back into [tex]\( x + y = 45.5 \)[/tex]:
[tex]\[
6.55 + y = 45.5
\][/tex]
[tex]\[
y = 38.95
\][/tex]
### Conclusion
Both cases lead to:
[tex]\[
x = 6.55 \quad \text{and} \quad y = 38.95
\][/tex]
Thus, the two numbers are [tex]\( 6.55 \)[/tex] and [tex]\( 38.95 \)[/tex]. Checking these values will confirm that they satisfy the initial conditions given in the problem statement accurately.
