Dividing Fractions:

To solve this problem, we need to find the number by which we should multiply the fraction 2/11 to get -2/3.

Step 1: Understanding Division of Fractions

When we divide fractions, we use the reciprocal of the second fraction and multiply it with the first fraction. In other words, we multiply the first fraction by the reciprocal of the second fraction.

For example, to divide 2/3 by 4/5, we multiply 2/3 by 5/4.

Step 2: Writing the Problem as a Division

We can rewrite the problem as follows:

2/11 ÷ ? = -2/3

Here, the first fraction is 2/11 and the second fraction is ? (the number we need to find).

Step 3: Finding the Reciprocal of the Second Fraction

To find the reciprocal of a fraction, we swap the numerator and the denominator.

In this case, the reciprocal of ? is 3/-2.

So, the problem becomes:

2/11 × 3/-2 = -2/3

Step 4: Multiplying the Fractions

To multiply fractions, we multiply the numerators and the denominators separately.

When we multiply 2/11 by 3/-2, we get:

(2 × 3) / (11 × -2) = -6/(-22)

Step 5: Simplifying the Result

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD).

The GCD of -6 and -22 is 2.

When we divide -6/2 and -22/2, we get:

-6/2 ÷ -22/2 = -3/11

Step 6: Conclusion

After simplifying the fraction, we find that 2/11 ÷ 3/-2 is equal to -3/11.

Therefore, the number by which we should multiply 2/11 to get -2/3 is -3/-2, which is equivalent to 3/2.