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Answer :
- Analyze each proportion to determine if it is true.
- Identify that $\frac{12}{15}=\frac{40}{\pi}$ is the only proportion that contains $\pi$.
- Conclude that $\frac{12}{15}=\frac{40}{\pi}$ does not belong with the other three because it contains an irrational number, while the others involve only rational numbers.
- The proportion that does not belong is $\boxed{\frac{12}{15}=\frac{40}{\pi}}$.
### Explanation
1. Analyze the Problem
We are given four proportions and asked to identify the one that doesn't belong with the others. To do this, we will simplify each proportion and determine if it is a true statement.
2. Analyze Proportion 1
Let's analyze the first proportion: $\frac{15}{10}=\frac{50}{100}$. Simplifying both sides, we get $\frac{3}{2} = \frac{1}{2}$, which is $1.5 = 0.5$. This is false.
3. Analyze Proportion 2
Now let's analyze the second proportion: $\frac{12}{15}=\frac{40}{\pi}$. Simplifying the left side, we get $\frac{4}{5} = 0.8$. For the proportion to be true, $\pi$ would have to be equal to $\frac{40}{0.8} = 50$. Since $\pi \approx 3.14$, this proportion is false. Also, this proportion involves $\pi$, which is an irrational number, while the other proportions involve only rational numbers or variables that would represent rational numbers.
4. Analyze Proportion 3
Let's analyze the third proportion: $\frac{15}{25}=\frac{p}{100}$. Simplifying the left side, we get $\frac{3}{5}$. Solving for $p$, we have $p = \frac{3}{5} \times 100 = 60$. So, the proportion is $\frac{15}{25}=\frac{60}{100}$, which simplifies to $\frac{3}{5} = \frac{3}{5}$. This is true.
5. Analyze Proportion 4
Now let's analyze the fourth proportion: $\frac{a}{20}=\frac{35}{100}$. Simplifying the right side, we get $\frac{7}{20}$. Solving for $a$, we have $a = \frac{7}{20} \times 20 = 7$. So, the proportion is $\frac{7}{20}=\frac{35}{100}$, which simplifies to $\frac{7}{20} = \frac{7}{20}$. This is true.
6. Conclusion
The first and second proportions are false, while the third and fourth are true. However, the second proportion, $\frac{12}{15}=\frac{40}{\pi}$, is the only one that involves the irrational number $\pi$. Therefore, it is the proportion that does not belong with the other three.
### Examples
Proportions are used in everyday life to scale recipes, calculate discounts, and understand maps. For example, if a recipe calls for 2 cups of flour for 4 servings, you can use a proportion to determine how much flour you need for 10 servings. Understanding proportions helps in making accurate adjustments and informed decisions in various practical situations.
- Identify that $\frac{12}{15}=\frac{40}{\pi}$ is the only proportion that contains $\pi$.
- Conclude that $\frac{12}{15}=\frac{40}{\pi}$ does not belong with the other three because it contains an irrational number, while the others involve only rational numbers.
- The proportion that does not belong is $\boxed{\frac{12}{15}=\frac{40}{\pi}}$.
### Explanation
1. Analyze the Problem
We are given four proportions and asked to identify the one that doesn't belong with the others. To do this, we will simplify each proportion and determine if it is a true statement.
2. Analyze Proportion 1
Let's analyze the first proportion: $\frac{15}{10}=\frac{50}{100}$. Simplifying both sides, we get $\frac{3}{2} = \frac{1}{2}$, which is $1.5 = 0.5$. This is false.
3. Analyze Proportion 2
Now let's analyze the second proportion: $\frac{12}{15}=\frac{40}{\pi}$. Simplifying the left side, we get $\frac{4}{5} = 0.8$. For the proportion to be true, $\pi$ would have to be equal to $\frac{40}{0.8} = 50$. Since $\pi \approx 3.14$, this proportion is false. Also, this proportion involves $\pi$, which is an irrational number, while the other proportions involve only rational numbers or variables that would represent rational numbers.
4. Analyze Proportion 3
Let's analyze the third proportion: $\frac{15}{25}=\frac{p}{100}$. Simplifying the left side, we get $\frac{3}{5}$. Solving for $p$, we have $p = \frac{3}{5} \times 100 = 60$. So, the proportion is $\frac{15}{25}=\frac{60}{100}$, which simplifies to $\frac{3}{5} = \frac{3}{5}$. This is true.
5. Analyze Proportion 4
Now let's analyze the fourth proportion: $\frac{a}{20}=\frac{35}{100}$. Simplifying the right side, we get $\frac{7}{20}$. Solving for $a$, we have $a = \frac{7}{20} \times 20 = 7$. So, the proportion is $\frac{7}{20}=\frac{35}{100}$, which simplifies to $\frac{7}{20} = \frac{7}{20}$. This is true.
6. Conclusion
The first and second proportions are false, while the third and fourth are true. However, the second proportion, $\frac{12}{15}=\frac{40}{\pi}$, is the only one that involves the irrational number $\pi$. Therefore, it is the proportion that does not belong with the other three.
### Examples
Proportions are used in everyday life to scale recipes, calculate discounts, and understand maps. For example, if a recipe calls for 2 cups of flour for 4 servings, you can use a proportion to determine how much flour you need for 10 servings. Understanding proportions helps in making accurate adjustments and informed decisions in various practical situations.
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