To solve the problem, we need to find an unknown number given the essen (sum) and the core (average) of two numbers.
Here's what we know:
- The essen (sum) of two numbers is [tex]\(\frac{-3}{5}\)[/tex].
- The core (average) of the numbers is [tex]\(\frac{-9}{20}\)[/tex].
Let's denote the two numbers as [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. Given:
[tex]\[ a + b = \frac{-3}{5} \][/tex]
The core (average) of the two numbers is given by:
[tex]\[ \frac{a + b}{2} = \frac{-9}{20} \][/tex]
Substituting the known sum into the average formula, we get:
[tex]\[ \frac{\frac{-3}{5}}{2} = \frac{-9}{20} \][/tex]
This confirms that our initial values are consistent. Now, we need to determine the value of the other number.
Let's denote the unknown number by [tex]\( x \)[/tex]. So, we have:
[tex]\[ a = \frac{-3}{5} \][/tex]
[tex]\[ b = x \][/tex]
Given the core (average):
[tex]\[ \frac{\frac{-3}{5} + x}{2} = \frac{-9}{20} \][/tex]
To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[ \frac{\frac{-3}{5} + x}{2} = \frac{-9}{20} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ \frac{-3}{5} + x = \frac{-9}{10} \][/tex]
Now, let's isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-9}{10} - \frac{-3}{5} \][/tex]
Next, we need a common denominator to perform the subtraction. The fraction [tex]\(\frac{-3}{5}\)[/tex] can be rewritten with a denominator of 10:
[tex]\[ \frac{-3}{5} = \frac{-6}{10} \][/tex]
Now, we can subtract:
[tex]\[ x = \frac{-9}{10} - \frac{-6}{10} \][/tex]
[tex]\[ x = \frac{-9 + 6}{10} \][/tex]
[tex]\[ x = \frac{-3}{10} \][/tex]
Therefore, the other number is [tex]\( \frac{-3}{10} \)[/tex].
So, the two numbers are:
[tex]\[ \frac{-3}{5} \quad \text{and} \quad \frac{-3}{10} \][/tex]
