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The problem presented involves Sorin calculating the sum of a two-digit number, where he uses **double the tens digit** of the number and adds the **units digit**.

### Problem breakdown:

#### a) Can the result of the sum be 109? Justify your answer!

We can assume the number is represented as \(10a + b\), where:
- \(a\) is the tens digit (so it is an integer between 1 and 9),
- \(b\) is the units digit (so it is an integer between 0 and 9).

The calculation that Sorin makes is:
\[
S = 2a + b
\]
Now, let's determine if this sum can equal 109:
\[
2a + b = 109
\]
For this equation to hold:
1. \(2a\) would need to be greater than or equal to 109, but since \(a\) is a digit (from 1 to 9), the maximum value for \(2a\) is \(2 \times 9 = 18\).
2. Even if \(b\) takes its maximum value (9), the sum \(2a + b\) can only reach \(18 + 9 = 27\).

Since 109 is far beyond the possible maximum sum of 27, **the result cannot be 109**.

#### b) If the result is 120, determine the two-digit number Sorin used in his calculation.

We use the same calculation:
\[
2a + b = 120
\]
From the analysis earlier, we know that the maximum possible value for \(2a + b\) is 27. Therefore, it is impossible for this calculation to give 120 because the maximum values of \(a\) and \(b\) do not support such a large result.

There seems to be a mistake in assuming the result could be 120 given the constraints, or the problem may require some additional interpretation or context.

Would you like further details on this, or have any questions? Here are 5 related questions you might find helpful:

1. How can we check the possible maximum sum of \(2a + b\) for different two-digit numbers?
2. What are the possible values for the sum \(2a + b\) when \(a\) and \(b\) take different digit values?
3. Can the result exceed 27 if additional factors were involved in the calculation?
4. What kind of numbers could produce results close to 120 under different rules?
5. How does the relationship between digits \(a\) and \(b\) affect the final sum?

**Tip:** Always double-check the range and constraints of numbers, especially when dealing with digit-based problems!

Sorin calculează suma dintre un număr de două cifre, dublul cifrei zecilor acelui număr și cifra unităților acelui număr. a) Poate fi rezultatul sumei egal cu 109? Justifică răspunsul dat. b) Dacă rezultatul calculului făcut de Sorin este egal cu 120, determină numărul de două cifre folosit în calculul pe care l-a făcut.

Explore Sorin's problem where the sum of a two-digit number's doubled tens and units is calculated. Includes analysis of whether the result can be 109 or 120, using simple algebraic expressions. Suitable for Grades 6-8.

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