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Randy was playing "Double Heads," a game in which he flipped two coins at a time. He recorded the number of times he flipped two heads at the same time. He flipped the coins 50 times, and there were 20 times when both coins landed heads up, which he recorded as [tex]\frac{20}{50}[/tex].

a. How does [tex]\frac{20}{50}[/tex] compare to [tex]50\%[/tex]?

b. Randy claims, "[tex]\frac{20}{50}[/tex] is equivalent to [tex]\frac{70}{100}[/tex] because if I add 50 to the denominator to get 100 and add 50 to the numerator to get 70, the fraction becomes [tex]\frac{70}{100}[/tex]." Explain to Randy why [tex]\frac{20}{50}[/tex] and [tex]\frac{70}{100}[/tex] are not equivalent fractions.

Answer :

Certainly! Let's go through the question step-by-step:

Part a: Compare [tex]\(\frac{20}{50}\)[/tex] to 50%.

1. First, we need to convert the fraction [tex]\(\frac{20}{50}\)[/tex] to a percentage. To do this, you divide the numerator by the denominator and then multiply by 100.

[tex]\[
\frac{20}{50} = 0.4
\][/tex]
[tex]\[
0.4 \times 100 = 40\%
\][/tex]

2. Now, compare 40% to 50%. Clearly, 40% is less than 50%. So, [tex]\(\frac{20}{50}\)[/tex] corresponds to 40%, which is less than 50%.

Part b: Explain why [tex]\(\frac{20}{50}\)[/tex] and [tex]\(\frac{70}{100}\)[/tex] are not equivalent portions.

1. Let's simplify each fraction and see if they are equal.

2. Simplify [tex]\(\frac{20}{50}\)[/tex]:
- The greatest common divisor (GCD) of 20 and 50 is 10.
- Divide both the numerator and the denominator by 10.
[tex]\[
\frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]

3. Simplify [tex]\(\frac{70}{100}\)[/tex]:
- The greatest common divisor (GCD) of 70 and 100 is 10.
- Divide both the numerator and the denominator by 10.
[tex]\[
\frac{70}{100} = \frac{70 \div 10}{100 \div 10} = \frac{7}{10}
\][/tex]

4. Now, compare the simplified fractions [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{7}{10}\)[/tex]. They are not the same because [tex]\(\frac{2}{5}\)[/tex] means 0.4, and [tex]\(\frac{7}{10}\)[/tex] means 0.7. These values clearly differ.

Thus, [tex]\(\frac{20}{50}\)[/tex] and [tex]\(\frac{70}{100}\)[/tex] are not equivalent. Randy's method of adding the same number to the numerator and denominator doesn't produce an equivalent fraction.

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