0s and that in general, this procedure works for simplifying fractions. Explain whether or not the statement is true using the rational number \frac{75}{95}. Choose the correct answer below. A. The statement is not true in general. In this example \frac{75}{95} \neq \frac{7}{9} because the cross products, 95 and 90, are not equal. B. The statement is true. In this example \frac{75}{95} = \frac{7}{9}. C. The statement is never true. The faction \frac{70}{90} does not simplify by crossing out the 0’s. D. The statement is mostly true. It works in every case except the case given in the problem statement.
A student says that \frac{70}{90} can be simplified by crossing out the 0s and that in general, this procedure works for simplifying fractions. Explain whether or not the statement is true using the rational number \frac{75}{95}. Choose the correct answer below. A. The statement is not true in general. In this example \frac{75}{95} \neq \frac{7}{9} because the cross products, 95 and 90, are not equal. B. The statement is true. In this example \frac{75}{95} = \frac{7}{9}. C. The statement is never true. The faction \frac{70}{90} does not simplify by crossing out the 0’s. D. The statement is mostly true. It works in every case except the case given in the problem statement.
The Correct Answer and Explanation is:
Correct Answer: A. The statement is not true in general. In this example, 75⁄95 ≠ 7⁄9 because the cross products, 95 and 90, are not equal.
Explanation:
The student’s idea of simplifying a fraction by “crossing out the 0s” is a misunderstanding of how fractions work. Proper simplification involves dividing both the numerator and the denominator by their greatest common factor (GCF), not removing digits arbitrarily.
Let’s look at the fraction 75⁄95.
To simplify this correctly:
- Find the GCF of 75 and 95.
- The GCF of 75 and 95 is 5.
- So, divide both numerator and denominator by 5:
7595=75÷595÷5=1519\frac{75}{95} = \frac{75 ÷ 5}{95 ÷ 5} = \frac{15}{19}9575=95÷575÷5=1915
This is the correct simplified form. Now, let’s consider what happens if we “cross out” digits from 75 and 95. Removing the digit 5 from both would give us 7⁄9. But:7595=1519and79\frac{75}{95} = \frac{15}{19} \quad \text{and} \quad \frac{7}{9}9575=1915and97
These two fractions are not equal. To test this, use cross multiplication:
- 75 × 9 = 675
- 7 × 95 = 665
Since 675 ≠ 665, we can conclude that 75⁄95 ≠ 7⁄9.
Similarly, the earlier example of 70⁄90 simplifying to 7⁄9 appears to work, but it is only a coincidence. In this case:
- 70⁄90 = 7⁄9 because both 70 and 90 are divisible by 10.
However, this does not mean crossing out zeros is a valid method. It only works when both numbers end in zero and are divisible by 10.
Therefore, the correct interpretation is A: the student’s method is not a valid general rule, and the example using 75⁄95 clearly disproves it.
