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You may combine your answers to all three parts of this problem.

a) Is it valid to compare [tex]\frac{30}{70}[/tex] and [tex]\frac{20}{50}[/tex] by "cancelling" the 0s and comparing [tex]\frac{3}{7}[/tex] and [tex]\frac{2}{5}[/tex] instead? Explain your answer.

b) Is it valid to compare [tex]\frac{15}{25}[/tex] and [tex]\frac{105}{205}[/tex] by "cancelling" the 5s and comparing [tex]\frac{1}{2}[/tex] and [tex]\frac{10}{20}[/tex] instead? Explain your answer.

c) Write a paragraph discussing the distinction between your answer in (a) and your answer in (b).

Answer :

Let's go through each part of the question step-by-step:

### Part (a):

Question: Can you compare [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex] by "cancelling" the zeros and comparing [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] instead?

Answer: No, "cancelling" the zeros is not mathematically valid. What actually needs to be done is simplification of both fractions by finding the greatest common divisor (GCD) for the numerator and denominator of each fraction.

- Simplifying [tex]\(\frac{30}{70}\)[/tex]:
- Find the GCD of 30 and 70, which is 10.
- Simplify [tex]\(\frac{30}{70}\)[/tex] by dividing both the numerator and the denominator by 10:

[tex]\[
\frac{30 \div 10}{70 \div 10} = \frac{3}{7}
\][/tex]

- Simplifying [tex]\(\frac{20}{50}\)[/tex]:
- Find the GCD of 20 and 50, which is 10.
- Simplify [tex]\(\frac{20}{50}\)[/tex] by dividing both the numerator and the denominator by 10:

[tex]\[
\frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]

So, while the "cancelling" of zeros did give the same result, it's a coincidence. The proper method is simplifying using the GCD.

### Part (b):

Question: Is it valid to compare [tex]\(\frac{15}{25}\)[/tex] and [tex]\(\frac{105}{205}\)[/tex] by "cancelling" the fives and comparing [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{10}{20}\)[/tex] instead?

Answer: No, cancelling specific digits from the numbers like "5" is incorrect. Here’s how to correctly simplify:

- Simplifying [tex]\(\frac{15}{25}\)[/tex]:
- Find the GCD of 15 and 25, which is 5.
- Simplify [tex]\(\frac{15}{25}\)[/tex] by dividing both by 5:

[tex]\[
\frac{15 \div 5}{25 \div 5} = \frac{3}{5}
\][/tex]

- Simplifying [tex]\(\frac{105}{205}\)[/tex]:
- Find the GCD of 105 and 205, which is 5.
- Simplify [tex]\(\frac{105}{205}\)[/tex] by dividing both by 5:

[tex]\[
\frac{105 \div 5}{205 \div 5} = \frac{21}{41}
\][/tex]

The simplifications using the GCD show that the proper comparisons should be with [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{21}{41}\)[/tex], not [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{10}{20}\)[/tex].

### Part (c):

Discussion: The key distinction between part (a) and part (b) lies in the outcome after simplification. In part (a), even though cancelling zeros seemed to give the correct result, it is still not a valid method as it worked out coincidentally. The proper approach is through simplification using the GCD.

In part (b), cancelling the digit '5' did not yield the correct simplified form, showing that this method is flawed. In both cases, it's essential to use the GCD to ensure accurate simplification of the fractions. Always simplify fractions systematically to avoid errors and ensure precision.

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