Answer :
Given the condition that [tex]\(\frac{x}{y} < 1\)[/tex] where [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are positive integers, we need to examine the given options to determine which ones result in a value greater than 1.
We will analyze the following options:
1. [tex]\(\frac{x+1}{y}\)[/tex]
2. [tex]\(\frac{x}{y+1}\)[/tex]
3. [tex]\(\frac{x}{y} + 1\)[/tex]
4. [tex]\(\frac{y}{x}\)[/tex]
### Option 1: [tex]\(\frac{x+1}{y}\)[/tex]
- To understand if [tex]\(\frac{x+1}{y}\)[/tex] is greater than 1, we need to check:
[tex]\[ \frac{x+1}{y} > 1 \quad \text{which implies} \quad x+1 > y \][/tex]
- Since [tex]\(\frac{x}{y} < 1\)[/tex], we know [tex]\(x < y\)[/tex].
- However, increasing the numerator by 1 (i.e., [tex]\(x+1\)[/tex]) might still be less than or equal to [tex]\(y\)[/tex], depending on the actual values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. For example, if [tex]\(x = 1\)[/tex] and [tex]\(y = 2\)[/tex], [tex]\(\frac{1+1}{2} = 1\)[/tex], which is not greater than 1. Therefore, [tex]\(\frac{x+1}{y}\)[/tex] is not consistently greater than 1.
### Option 2: [tex]\(\frac{x}{y+1}\)[/tex]
- To understand if [tex]\(\frac{x}{y+1}\)[/tex] is greater than 1, we need to check:
[tex]\[ \frac{x}{y+1} > 1 \quad \text{which implies} \quad x > y+1 \][/tex]
- Since [tex]\(\frac{x}{y} < 1\)[/tex], we know [tex]\(x < y\)[/tex], which means [tex]\(x\)[/tex] is already less than [tex]\(y\)[/tex], so it cannot be greater than [tex]\(y+1\)[/tex]. Hence, [tex]\(\frac{x}{y+1}\)[/tex] is not greater than 1.
### Option 3: [tex]\(\frac{x}{y} + 1\)[/tex]
- We need to check if this expression is greater than 1:
[tex]\[ \frac{x}{y} + 1 > 1 \quad \text{which simplifies to} \quad \frac{x}{y} > 0 \][/tex]
- Given that [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are positive integers, [tex]\(\frac{x}{y}\)[/tex] is undoubtedly positive but less than 1. Therefore, adding 1 to [tex]\(\frac{x}{y}\)[/tex] results in a value greater than 1. For example, if [tex]\(\frac{x}{y} = 0.5\)[/tex], then [tex]\(0.5 + 1 = 1.5\)[/tex]. Hence, [tex]\(\frac{x}{y} + 1\)[/tex] is greater than 1.
### Option 4: [tex]\(\frac{y}{x}\)[/tex]
- To check if [tex]\(\frac{y}{x}\)[/tex] is greater than 1, we need to see:
[tex]\[ \frac{y}{x} > 1 \][/tex]
- Since [tex]\(x < y\)[/tex] for [tex]\(\frac{x}{y} < 1\)[/tex], it directly follows that [tex]\(\frac{y}{x} > 1\)[/tex]. For example, if [tex]\(x = 1\)[/tex] and [tex]\(y = 2\)[/tex], [tex]\(\frac{2}{1} = 2\)[/tex]. Thus, [tex]\(\frac{y}{x}\)[/tex] is indeed greater than 1.
### Conclusion
Among the given options, the expressions that are greater than 1 are:
- [tex]\(\frac{x}{y} + 1\)[/tex]
- [tex]\(\frac{y}{x}\)[/tex]
So, the correct options are 3 and 4.
We will analyze the following options:
1. [tex]\(\frac{x+1}{y}\)[/tex]
2. [tex]\(\frac{x}{y+1}\)[/tex]
3. [tex]\(\frac{x}{y} + 1\)[/tex]
4. [tex]\(\frac{y}{x}\)[/tex]
### Option 1: [tex]\(\frac{x+1}{y}\)[/tex]
- To understand if [tex]\(\frac{x+1}{y}\)[/tex] is greater than 1, we need to check:
[tex]\[ \frac{x+1}{y} > 1 \quad \text{which implies} \quad x+1 > y \][/tex]
- Since [tex]\(\frac{x}{y} < 1\)[/tex], we know [tex]\(x < y\)[/tex].
- However, increasing the numerator by 1 (i.e., [tex]\(x+1\)[/tex]) might still be less than or equal to [tex]\(y\)[/tex], depending on the actual values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. For example, if [tex]\(x = 1\)[/tex] and [tex]\(y = 2\)[/tex], [tex]\(\frac{1+1}{2} = 1\)[/tex], which is not greater than 1. Therefore, [tex]\(\frac{x+1}{y}\)[/tex] is not consistently greater than 1.
### Option 2: [tex]\(\frac{x}{y+1}\)[/tex]
- To understand if [tex]\(\frac{x}{y+1}\)[/tex] is greater than 1, we need to check:
[tex]\[ \frac{x}{y+1} > 1 \quad \text{which implies} \quad x > y+1 \][/tex]
- Since [tex]\(\frac{x}{y} < 1\)[/tex], we know [tex]\(x < y\)[/tex], which means [tex]\(x\)[/tex] is already less than [tex]\(y\)[/tex], so it cannot be greater than [tex]\(y+1\)[/tex]. Hence, [tex]\(\frac{x}{y+1}\)[/tex] is not greater than 1.
### Option 3: [tex]\(\frac{x}{y} + 1\)[/tex]
- We need to check if this expression is greater than 1:
[tex]\[ \frac{x}{y} + 1 > 1 \quad \text{which simplifies to} \quad \frac{x}{y} > 0 \][/tex]
- Given that [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are positive integers, [tex]\(\frac{x}{y}\)[/tex] is undoubtedly positive but less than 1. Therefore, adding 1 to [tex]\(\frac{x}{y}\)[/tex] results in a value greater than 1. For example, if [tex]\(\frac{x}{y} = 0.5\)[/tex], then [tex]\(0.5 + 1 = 1.5\)[/tex]. Hence, [tex]\(\frac{x}{y} + 1\)[/tex] is greater than 1.
### Option 4: [tex]\(\frac{y}{x}\)[/tex]
- To check if [tex]\(\frac{y}{x}\)[/tex] is greater than 1, we need to see:
[tex]\[ \frac{y}{x} > 1 \][/tex]
- Since [tex]\(x < y\)[/tex] for [tex]\(\frac{x}{y} < 1\)[/tex], it directly follows that [tex]\(\frac{y}{x} > 1\)[/tex]. For example, if [tex]\(x = 1\)[/tex] and [tex]\(y = 2\)[/tex], [tex]\(\frac{2}{1} = 2\)[/tex]. Thus, [tex]\(\frac{y}{x}\)[/tex] is indeed greater than 1.
### Conclusion
Among the given options, the expressions that are greater than 1 are:
- [tex]\(\frac{x}{y} + 1\)[/tex]
- [tex]\(\frac{y}{x}\)[/tex]
So, the correct options are 3 and 4.
