Questions: Find a counterexample to show that the following statement is false. Adding the same number to both the numerator and the denominator (top and bottom) of a fraction does not change the fraction's value. Choose the correct answer below. (1+0)/(2+0) = 1/2 ; the fraction 1/2 is equal to 1/2. (1 * 3)/(2 * 3) = 3/6; the fraction 3/6 is equal to 1/2. (3+4)/(3+4) = 7/7; the fraction 7/7 is not equal to 3/3. (1+1)/(2+1) = 2/3; the fraction 2/3 is not equal to 1/2.

Solution Steps

The statement claims that adding the same number to both the numerator and the denominator of a fraction does not change the fraction's value. To disprove this, we need to find a counterexample where adding the same number to both the numerator and denominator results in a fraction with a different value.

We evaluate each option to see if it serves as a valid counterexample:

1. \(\frac{1+0}{2+0} = \frac{1}{2}\); this does not disprove the statement because the fraction remains unchanged.
2. \(\frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}\); this involves multiplication, not addition, so it is irrelevant to the statement.
3. \(\frac{3+4}{3+4} = \frac{7}{7}\); this simplifies to 1, which is not equal to \(\frac{3}{3}\) (also 1), so it does not disprove the statement.
4. \(\frac{1+1}{2+1} = \frac{2}{3}\); this fraction is not equal to \(\frac{1}{2}\), so it serves as a valid counterexample.

The fourth option, \(\frac{1+1}{2+1} = \frac{2}{3}\), demonstrates that adding the same number (1) to both the numerator and denominator changes the fraction's value from \(\frac{1}{2}\) to \(\frac{2}{3}\). This disproves the original statement.

Final Answer