Question

Enrique has $50 in his lunch account and spends $5 per day from the account. Maya has $46 in her lunch account and spends $4 per day from the account. Which equations model the situation?50 – 5x = y and 46 – 4x = y5x + 50 = y and 4x + 46 = y5x – 50 = y and 4x – 46 = y5 – 50x = y and 4 – 46x = y

Answer 1

To model the situation with equations, we can denote \( x \) as the number of days that pass.

For Enrique:

• He starts with $50 and spends $5 each day.
• The equation that models his remaining amount would be: \[ y = 50 - 5x \]

For Maya:

• She starts with $46 and spends $4 each day.
• The equation that models her remaining amount would be: \[ y = 46 - 4x \]

So the correct equations that model the situation are:

1. \( y = 50 - 5x \)
2. \( y = 46 - 4x \)

Thus, the first option—\( 50 – 5x = y \) and \( 46 – 4x = y \)—is the correct representation of the situation.

Answer 2

Danae is choosing between two jobs. One job pays an annual bonus of $1,500 plus $120 per day worked. The second job pays an annual bonus of $2,500 plus $110 per day worked. Which equation can be solved to determine after how many days, d, Danae would make the same amount of money regardless of the job she chooses?

120d + 110d = 1,500 + 2,500
120 + 110 = 1,500d + 2,500d
120d + 1,500 = 110d + 2,500
120d + 2,500 = 110d + 1,500

Answer 3

To find out after how many days Danae would make the same amount of money regardless of the job she chooses, we can set up equations for the total earnings from both jobs.

For the first job:

• Annual bonus: $1,500
• Daily pay: $120
• Total earnings after \(d\) days: \[ \text{Earnings from Job 1} = 120d + 1,500 \]

For the second job:

• Annual bonus: $2,500
• Daily pay: $110
• Total earnings after \(d\) days: \[ \text{Earnings from Job 2} = 110d + 2,500 \]

To find the number of days \(d\) when the earnings from both jobs are equal, we set the two expressions equal to each other: \[ 120d + 1,500 = 110d + 2,500 \]

Thus, the correct equation that can be solved to determine after how many days Danae would make the same amount of money from either job is: \[ 120d + 1,500 = 110d + 2,500 \]

This is the correct option.

Answer 4

A factory has fixed costs of $1,275 per month. The cost of producing each unit of its product is $2.50. Each unit sells for $10.

The equations that model the cost and revenue for producing x units are shown.

Cost: C(x) = 1275 + 2.5x

Revenue: R(x) = 10x

If the company sells every unit it produces, how many units must it sell in order to break even?

units___

Answer 5

To determine the break-even point, we need to find the number of units, \( x \), at which the total cost equals total revenue.

The cost equation is given by: \[ C(x) = 1275 + 2.5x \] The revenue equation is given by: \[ R(x) = 10x \]

To find the break-even point, we set the cost equal to revenue: \[ 1275 + 2.5x = 10x \]

Now, we will solve for \( x \):

1. First, subtract \( 2.5x \) from both sides: \[ 1275 = 10x - 2.5x \] \[ 1275 = 7.5x \]

2. Next, divide both sides by \( 7.5 \): \[ x = \frac{1275}{7.5} \]

3. Calculate \( x \): \[ x = 170 \]

Thus, the company must sell 170 units in order to break even.

Answer 6

Bryan’s monthly electric bill is determined by adding a flat administration fee to the product of the number of kilowatt hours of electricity used and the cost per kilowatt hour. When he uses 1,100 kilowatt hours of electricity, his bill is $113. When he uses 1,500 kilowatt hours of electricity, his bill is $153. What is the monthly administration fee?

$

Answer 7

Let \( F \) be the flat administration fee, and let \( C \) be the cost per kilowatt hour.

From the information provided, we can set up the following equations based on Bryan's electric bill:

1. When Bryan uses 1,100 kilowatt hours: \[ F + 1100C = 113 \quad \text{(1)} \]

2. When he uses 1,500 kilowatt hours: \[ F + 1500C = 153 \quad \text{(2)} \]

Now, we can solve these two equations simultaneously.

First, let's isolate \( F \) in both equations:

From equation (1): \[ F = 113 - 1100C \quad \text{(3)} \]

From equation (2): \[ F = 153 - 1500C \quad \text{(4)} \]

Now, we can set equations (3) and (4) equal to each other: \[ 113 - 1100C = 153 - 1500C \]

Now, we'll solve for \( C \):

1. Rearranging the equation gives us: \[ 1500C - 1100C = 153 - 113 \] \[ 400C = 40 \]

2. Dividing both sides by 400: \[ C = \frac{40}{400} = 0.1 \]

Now that we have the cost per kilowatt hour \( C \), we can substitute \( C \) back into one of the equations to find the administration fee \( F \). We'll use equation (3): \[ F = 113 - 1100 \times 0.1 \] \[ F = 113 - 110 \] \[ F = 3 \]

Thus, the monthly administration fee is $3.

Answer 8

Dalia flies an ultralight plane with a tailwind to a nearby town in 1/3 of an hour. On the return trip, she travels the same distance in 3/5 of an hour. What is the average rate of speed of the wind and the average rate of speed of the plane?



Initial trip: 18 miles 1/3 hours

Return trip: 18 miles 3/5 hours

Let x be the average airspeed of the plane.

Let y be the average wind speed.

Initial trip: 18 = (x + y) 1/3

Return trip: 18 = (x – y) 3/5

Dalia had an average airspeed of___
miles per hour.

The average wind speed was___
miles per hour.

Answer 9

Let's set up the equations based on the information given.

Let:

• \( x \) = average airspeed of the plane (in miles per hour).
• \( y \) = average wind speed (in miles per hour).

Initial Trip: With a tailwind, the effective speed is \( (x + y) \) and the time taken is \( \frac{1}{3} \) hours. The distance traveled is 18 miles, so we can write the equation: \[ 18 = (x + y) \cdot \frac{1}{3} \]

Multiplying both sides by 3 to eliminate the fraction gives: \[ 54 = x + y \quad \text{(1)} \]

Return Trip: Against the wind, the effective speed is \( (x - y) \) and the time taken is \( \frac{3}{5} \) hours. The distance for the return trip is also 18 miles, so we write: \[ 18 = (x - y) \cdot \frac{3}{5} \]

Multiplying both sides by \( \frac{5}{3} \) gives: \[ 30 = x - y \quad \text{(2)} \]

Now we have a system of equations:

1. \( x + y = 54 \) (equation 1)
2. \( x - y = 30 \) (equation 2)

Now we can solve these equations simultaneously.

Step 1: Add the two equations to eliminate \( y \): \[ (x + y) + (x - y) = 54 + 30 \] This simplifies to: \[ 2x = 84 \] So: \[ x = 42 \]

Step 2: Substitute \( x = 42 \) back into equation (1) to find \( y \): \[ 42 + y = 54 \] Subtracting 42 from both sides gives: \[ y = 12 \]

Conclusion:

• The average airspeed of the plane is 42 miles per hour.
• The average wind speed is 12 miles per hour.

Answer 10

Charlotte’s weekly paycheck is based on the number of hours worked during the week and on the weekend. If she works 13 hours during the week and 14 hours on the weekend, she earns $250.90. If she works 15 hours during the week and 8 hours on the weekend, she earns $204.70. How much more does Charlotte earn per hour on the weekends than she earns during the week? Round to the nearest cent.

$0.40
$1.90
$2.30
$8.10

Answer 11

To find how much more Charlotte earns per hour on the weekends than she does during the week, we can set up a system of equations based on the information provided.

Let:

• \( w \) = hourly wage during the week
• \( e \) = hourly wage on the weekend

From the information given, we can create the following equations:

1. For the first scenario (working 13 hours during the week and 14 hours on the weekend, earning $250.90): \[ 13w + 14e = 250.90 \quad (1) \]

2. For the second scenario (working 15 hours during the week and 8 hours on the weekend, earning $204.70): \[ 15w + 8e = 204.70 \quad (2) \]

Now, we can solve this system of equations.

Step 1: Solve one equation for one variable

Let's solve equation (1) for \( e \): \[ 14e = 250.90 - 13w \] \[ e = \frac{250.90 - 13w}{14} \quad (3) \]

Step 2: Substitute into the second equation

Now substitute equation (3) into equation (2): \[ 15w + 8\left(\frac{250.90 - 13w}{14}\right) = 204.70 \]

Multiply the entire equation by 14 to eliminate the fraction: \[ 14 \cdot 15w + 8(250.90 - 13w) = 14 \cdot 204.70 \] \[ 210w + 2007.2 - 104w = 2865.8 \]

Combine like terms: \[ 106w + 2007.2 = 2865.8 \]

Step 3: Solve for \( w \)

Subtract \( 2007.2 \) from both sides: \[ 106w = 2865.8 - 2007.2 \] \[ 106w = 858.6 \]

Now, divide by 106: \[ w = \frac{858.6}{106} \approx 8.10 \]

Step 4: Substitute back to find \( e \)

Now, substitute \( w \) back into equation (3) to find \( e \): \[ e = \frac{250.90 - 13(8.10)}{14} \] Calculate: \[ e = \frac{250.90 - 105.30}{14} \] \[ e = \frac{145.60}{14} \approx 10.40 \]

Step 5: Find the difference

Now, to find how much more she earns on weekends than during the week, calculate: \[ e - w = 10.40 - 8.10 = 2.30 \]

Thus, Charlotte earns $2.30 more per hour on the weekends than she earns during the week.