Answer :
Let's analyze each statement one by one based on the given conditions:
1. The equation [tex]\( y = x \)[/tex], where [tex]\( y \)[/tex] is the number of dollars and [tex]\( x \)[/tex] is the number of calls, represents Elena's situation.
Since Elena earns [tex]$1 for each call she makes, for \( x \) calls, she would earn \( y = x \) dollars. Therefore, this statement is true. 2. Diego needs to wash 35 windows to make as much money as Elena makes for 40 calls. First, calculate how much Elena makes for 40 calls. Since Elena earns $[/tex]1 per call:
[tex]\[ \text{Elena's earnings for 40 calls} = 40 \times 1 = 40 \text{ dollars} \][/tex]
Next, find the rate at which Diego gets paid per window. Using the table, we can see that Diego makes [tex]$30 for 27 windows, $[/tex]50 for 45 windows, and [tex]$90 for 81 windows. The rate per window is consistent: \[ \text{Diego's rate per window} = \frac{90}{81} = \frac{10}{9} \text{ dollars per window} \] Now, calculate how much Diego would make for 35 windows: \[ \text{Diego's earnings for 35 windows} = 35 \times \frac{10}{9} \approx 38.89 \text{ dollars} \] Elena's 40 call earnings ($[/tex]40) is not equal to Diego's 35 window earnings (approximately [tex]$38.89). Therefore, this statement is false. 3. Elena makes the same amount of money for 20 calls as Diego makes for 18 windows. Calculate Elena's earnings for 20 calls: \[ \text{Elena's earnings for 20 calls} = 20 \times 1 = 20 \text{ dollars} \] Calculate Diego's earnings for 18 windows: \[ \text{Diego's earnings for 18 windows} = 18 \times \frac{10}{9} = 20 \text{ dollars} \] Since both amounts are equal, this statement is true. 4. Diego makes more money for washing each window than Elena for making each call. Compare their rates: \[ \text{Elena's rate per call} = 1 \text{ dollar per call} \] \[ \text{Diego's rate per window} = \frac{10}{9} \text{ dollars per window} \approx 1.11 \text{ dollars per window} \] Since $[/tex]1.11 > [tex]$1, Diego makes more money for each window than Elena does for each call. Therefore, this statement is true. 5. Elena makes more money for making 10 calls than Diego makes for washing 10 windows. Calculate Elena's earnings for 10 calls: \[ \text{Elena's earnings for 10 calls} = 10 \times 1 = 10 \text{ dollars} \] Calculate Diego's earnings for 10 windows: \[ \text{Diego's earnings for 10 windows} = 10 \times \frac{10}{9} \approx 11.11 \text{ dollars} \] Since $[/tex]10 < $11.11, Diego makes more for 10 windows than Elena does for 10 calls. Therefore, this statement is false.
6. The equation [tex]\( y = \frac{9}{10} x \)[/tex], where [tex]\( y \)[/tex] is the number of dollars and [tex]\( x \)[/tex] is number of windows, represents Diego's situation.
According to Diego's rate per window:
[tex]\[ \text{Diego's rate per window} = \frac{10}{9} \text{ dollars per window} \][/tex]
The correct equation representing Diego's situation would be [tex]\( y = \frac{10}{9} x \)[/tex], not [tex]\( y = \frac{9}{10} x \)[/tex]. Therefore, this statement is false.
Thus, the statements that are true are:
1. The equation [tex]\( y = x \)[/tex], where [tex]\( y \)[/tex] is the number of dollars and [tex]\( x \)[/tex] is the number of calls, represents Elena's situation.
3. Elena makes the same amount of money for 20 calls as Diego makes for 18 windows.
4. Diego makes more money for washing each window than Elena for making each call.
1. The equation [tex]\( y = x \)[/tex], where [tex]\( y \)[/tex] is the number of dollars and [tex]\( x \)[/tex] is the number of calls, represents Elena's situation.
Since Elena earns [tex]$1 for each call she makes, for \( x \) calls, she would earn \( y = x \) dollars. Therefore, this statement is true. 2. Diego needs to wash 35 windows to make as much money as Elena makes for 40 calls. First, calculate how much Elena makes for 40 calls. Since Elena earns $[/tex]1 per call:
[tex]\[ \text{Elena's earnings for 40 calls} = 40 \times 1 = 40 \text{ dollars} \][/tex]
Next, find the rate at which Diego gets paid per window. Using the table, we can see that Diego makes [tex]$30 for 27 windows, $[/tex]50 for 45 windows, and [tex]$90 for 81 windows. The rate per window is consistent: \[ \text{Diego's rate per window} = \frac{90}{81} = \frac{10}{9} \text{ dollars per window} \] Now, calculate how much Diego would make for 35 windows: \[ \text{Diego's earnings for 35 windows} = 35 \times \frac{10}{9} \approx 38.89 \text{ dollars} \] Elena's 40 call earnings ($[/tex]40) is not equal to Diego's 35 window earnings (approximately [tex]$38.89). Therefore, this statement is false. 3. Elena makes the same amount of money for 20 calls as Diego makes for 18 windows. Calculate Elena's earnings for 20 calls: \[ \text{Elena's earnings for 20 calls} = 20 \times 1 = 20 \text{ dollars} \] Calculate Diego's earnings for 18 windows: \[ \text{Diego's earnings for 18 windows} = 18 \times \frac{10}{9} = 20 \text{ dollars} \] Since both amounts are equal, this statement is true. 4. Diego makes more money for washing each window than Elena for making each call. Compare their rates: \[ \text{Elena's rate per call} = 1 \text{ dollar per call} \] \[ \text{Diego's rate per window} = \frac{10}{9} \text{ dollars per window} \approx 1.11 \text{ dollars per window} \] Since $[/tex]1.11 > [tex]$1, Diego makes more money for each window than Elena does for each call. Therefore, this statement is true. 5. Elena makes more money for making 10 calls than Diego makes for washing 10 windows. Calculate Elena's earnings for 10 calls: \[ \text{Elena's earnings for 10 calls} = 10 \times 1 = 10 \text{ dollars} \] Calculate Diego's earnings for 10 windows: \[ \text{Diego's earnings for 10 windows} = 10 \times \frac{10}{9} \approx 11.11 \text{ dollars} \] Since $[/tex]10 < $11.11, Diego makes more for 10 windows than Elena does for 10 calls. Therefore, this statement is false.
6. The equation [tex]\( y = \frac{9}{10} x \)[/tex], where [tex]\( y \)[/tex] is the number of dollars and [tex]\( x \)[/tex] is number of windows, represents Diego's situation.
According to Diego's rate per window:
[tex]\[ \text{Diego's rate per window} = \frac{10}{9} \text{ dollars per window} \][/tex]
The correct equation representing Diego's situation would be [tex]\( y = \frac{10}{9} x \)[/tex], not [tex]\( y = \frac{9}{10} x \)[/tex]. Therefore, this statement is false.
Thus, the statements that are true are:
1. The equation [tex]\( y = x \)[/tex], where [tex]\( y \)[/tex] is the number of dollars and [tex]\( x \)[/tex] is the number of calls, represents Elena's situation.
3. Elena makes the same amount of money for 20 calls as Diego makes for 18 windows.
4. Diego makes more money for washing each window than Elena for making each call.
