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The image asks: *"Kuriuo atveju į langelį reikia įrašyti neigiamą skaičių, kad lygybė būtų teisinga?"* 

Translation: *"In which case should a negative number be entered in the box for the equality to be correct?"*

The three options given are:

1. \( \frac{6}{7} \div (-3) = -\frac{2}{\square} \)
2. \( -\frac{1}{6} \div 2 = \frac{5}{12} \)
3. \( -\frac{2}{3} \div \left( -\frac{1}{6} \right) = \square \)

Let's solve these step by step to see in which option a negative number is required.

### Option 1:
\[
\frac{6}{7} \div (-3) = -\frac{2}{\square}
\]
Dividing:
\[
\frac{6}{7} \div (-3) = \frac{6}{7} \times \frac{-1}{3} = \frac{-6}{21} = \frac{-2}{7}
\]
Thus, the left side becomes \(\frac{-2}{7}\), so the equation is:
\[
\frac{-2}{7} = -\frac{2}{\square}
\]
Clearly, \(\square = 7\). No negative number is needed here.

### Option 2:
\[
-\frac{1}{6} \div 2 = \frac{5}{12}
\]
Dividing:
\[
-\frac{1}{6} \div 2 = -\frac{1}{6} \times \frac{1}{2} = -\frac{1}{12}
\]
So the equation becomes:
\[
-\frac{1}{12} = \frac{5}{12}
\]
This is clearly incorrect, so this option is not valid.

### Option 3:
\[
-\frac{2}{3} \div \left( -\frac{1}{6} \right) = \square
\]
Dividing:
\[
-\frac{2}{3} \div \left( -\frac{1}{6} \right) = -\frac{2}{3} \times -6 = 4
\]
Thus, \(\square = 4\). No negative number is needed here either.

### Conclusion:
None of the options require a negative number to be entered in the box for the equality to be true. However, **Option 1** does provide the correct answer without requiring a negative number.

Would you like further details or have any questions?

Here are 5 related questions you can explore:
1. How do you divide fractions involving negative numbers?
2. What are the rules for multiplying and dividing negative numbers?
3. Can you explain why multiplying two negative numbers gives a positive result?
4. How do you convert complex divisions into multiplications?
5. How can you check if an equation with fractions is balanced?

**Tip**: Always remember that dividing by a negative number flips the sign of the result!

Kuriuo atveju į langelį reikia įrašyti neigiamą skaičių, kad lygybė būtų teisinga?

Solve a problem where you need to determine which case requires a negative number to be entered for the equality to hold true. This exercise involves the division and multiplication of fractions with negative numbers, suitable for students in Grades 6-8.

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