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Answer :
Sure! Let's break down the problem and work through it step by step.
### Part a: Comparing Fractions by Cancelling Zeros
The problem asks whether it is valid to compare the fractions [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex] by "cancelling" the zeros to compare [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] instead. Let's analyze this step by step.
#### Step 1: Simplify the Original Fractions
First, we should simplify the original fractions [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex] to see if cancelling zeros is a valid operation.
- [tex]\(\frac{30}{70}\)[/tex]
- Simplifying by their greatest common divisor (GCD), which is 10:
- [tex]\(\frac{30 \div 10}{70 \div 10} = \frac{3}{7}\)[/tex]
- [tex]\(\frac{20}{50}\)[/tex]
- Simplifying by their GCD, which is 10:
- [tex]\(\frac{20 \div 10}{50 \div 10} = \frac{2}{5}\)[/tex]
So, [tex]\(\frac{30}{70} = \frac{3}{7}\)[/tex], and [tex]\(\frac{20}{50} = \frac{2}{5}\)[/tex].
#### Step 2: Compare the Simplified Fractions
Now, let's compare [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex]:
- [tex]\(\frac{3}{7} \approx 0.42857\)[/tex]
- [tex]\(\frac{2}{5} = 0.4\)[/tex]
Since 0.42857 (which is [tex]\(\frac{3}{7}\)[/tex]) is greater than 0.4 (which is [tex]\(\frac{2}{5}\)[/tex]), we conclude:
- [tex]\(\frac{3}{7} > \frac{2}{5}\)[/tex]
#### Step 3: Conclusion on Cancelling Zeros
In the provided fractions, cancelling the zeros did not change the outcome because the fractions [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex] simplify to [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex], respectively.
However, this method ([tex]\(\frac{30}{70} \rightarrow \frac{3}{7}\)[/tex] and [tex]\(\frac{20}{50} \rightarrow \frac{2}{5}\)[/tex]) does not hold for all fractions.
If we cancel zeros, both numerators and denominators need to be divided by a common factor, retaining the equivalency between the fractions:
- Example: [tex]\(\frac{20}{40} \neq \frac{2}{4}\)[/tex] (cancelling zeros directly would be wrong because [tex]\(\frac{20}{40} = \frac{1}{2} \neq \frac{1}{2} = \frac{2}{4}\)[/tex])
### Final Explanation
In conclusion, in this specific case, comparing [tex]\(\frac{30}{70}\)[/tex] with [tex]\(\frac{20}{50}\)[/tex] by reducing to [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] is valid because both fractions simplify correctly. However, in general, cancelling zeros is not a valid method unless it simplifies both the numerator and the denominator by the same factor through correct fraction reduction.
### Part a: Comparing Fractions by Cancelling Zeros
The problem asks whether it is valid to compare the fractions [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex] by "cancelling" the zeros to compare [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] instead. Let's analyze this step by step.
#### Step 1: Simplify the Original Fractions
First, we should simplify the original fractions [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex] to see if cancelling zeros is a valid operation.
- [tex]\(\frac{30}{70}\)[/tex]
- Simplifying by their greatest common divisor (GCD), which is 10:
- [tex]\(\frac{30 \div 10}{70 \div 10} = \frac{3}{7}\)[/tex]
- [tex]\(\frac{20}{50}\)[/tex]
- Simplifying by their GCD, which is 10:
- [tex]\(\frac{20 \div 10}{50 \div 10} = \frac{2}{5}\)[/tex]
So, [tex]\(\frac{30}{70} = \frac{3}{7}\)[/tex], and [tex]\(\frac{20}{50} = \frac{2}{5}\)[/tex].
#### Step 2: Compare the Simplified Fractions
Now, let's compare [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex]:
- [tex]\(\frac{3}{7} \approx 0.42857\)[/tex]
- [tex]\(\frac{2}{5} = 0.4\)[/tex]
Since 0.42857 (which is [tex]\(\frac{3}{7}\)[/tex]) is greater than 0.4 (which is [tex]\(\frac{2}{5}\)[/tex]), we conclude:
- [tex]\(\frac{3}{7} > \frac{2}{5}\)[/tex]
#### Step 3: Conclusion on Cancelling Zeros
In the provided fractions, cancelling the zeros did not change the outcome because the fractions [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex] simplify to [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex], respectively.
However, this method ([tex]\(\frac{30}{70} \rightarrow \frac{3}{7}\)[/tex] and [tex]\(\frac{20}{50} \rightarrow \frac{2}{5}\)[/tex]) does not hold for all fractions.
If we cancel zeros, both numerators and denominators need to be divided by a common factor, retaining the equivalency between the fractions:
- Example: [tex]\(\frac{20}{40} \neq \frac{2}{4}\)[/tex] (cancelling zeros directly would be wrong because [tex]\(\frac{20}{40} = \frac{1}{2} \neq \frac{1}{2} = \frac{2}{4}\)[/tex])
### Final Explanation
In conclusion, in this specific case, comparing [tex]\(\frac{30}{70}\)[/tex] with [tex]\(\frac{20}{50}\)[/tex] by reducing to [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] is valid because both fractions simplify correctly. However, in general, cancelling zeros is not a valid method unless it simplifies both the numerator and the denominator by the same factor through correct fraction reduction.
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