Question

Two pumps are filling large vats of liquid. One vat is empty, and the pump is filling it at a rate of 6 gallons per minute. The other is already filled with 10 gallons, and it is continuing to fill at a rate of 4 gallons per minute. Write an equation to show when the two vats will have an equal amount of liquid. Use x as your variable. (1 point).

Answer 1

Let \( x \) be the number of minutes that have passed.

The amount of liquid in the first vat, which starts empty and fills at a rate of 6 gallons per minute, can be expressed as: \[ 6x \]

The amount of liquid in the second vat, which starts with 10 gallons and fills at a rate of 4 gallons per minute, can be expressed as: \[ 10 + 4x \]

To find the time when both vats have an equal amount of liquid, we can set these two expressions equal to each other: \[ 6x = 10 + 4x \]

This equation can be solved to find \( x \). Thus, the equation showing when the two vats will have an equal amount of liquid is: \[ 6x = 10 + 4x \]

Answer 2

A yoga studio has two participation plans. For the first plan, the cost is $8 per visit. For the second plan, the cost is $40 per month, and $2 per visit. Write an equation to show the number of visits per month that it would take for the two plans to be equal. Use x
as your variable.(1 point)

Answer 3

Let \( x \) be the number of visits per month.

For the first plan, the total cost can be expressed as: \[ C_1 = 8x \]

For the second plan, the total cost can be expressed as: \[ C_2 = 40 + 2x \]

To find the number of visits per month where the costs of the two plans are equal, we can set these two expressions equal to each other: \[ 8x = 40 + 2x \]

This is the equation that shows when the two plans will have equal costs.

Answer 4

A diner makes vanilla milkshakes and chocolate milkshakes. The vanilla milkshake machine has a 300-cup capacity, and sells 15 cups per day. The chocolate milkshake machine has a 280-cup capacity, and sells 20 cups per day. Write an equation to show when the two machines will have the same amount of milkshake left. Use x
as your variable.(1 point)

Answer 5

Let \( x \) be the number of days that have passed.

For the vanilla milkshake machine, the amount of milkshake left after \( x \) days can be expressed as: \[ 300 - 15x \]

For the chocolate milkshake machine, the amount of milkshake left after \( x \) days can be expressed as: \[ 280 - 20x \]

To find when both machines will have the same amount of milkshake left, we can set these two expressions equal to each other: \[ 300 - 15x = 280 - 20x \]

This is the equation that shows when the two machines will have the same amount of milkshake left.

Answer 6

Two pumps are filling large vats of liquid. One vat is empty, and the pump is filling it at a rate of 6 gallons per minute. The other vat already contains 28 gallons and is continuing to fill at a rate of 4 gallons per minute. When will the two vats have the same amount of liquid?(1 point)
They will have the same amount of liquid after blank
minutes.

Answer 7

Let \( x \) be the number of minutes that have passed.

For the first vat, which starts empty and fills at a rate of 6 gallons per minute, the amount of liquid after \( x \) minutes can be expressed as: \[ 6x \]

For the second vat, which starts with 28 gallons and fills at a rate of 4 gallons per minute, the amount of liquid after \( x \) minutes can be expressed as: \[ 28 + 4x \]

To find the time when both vats have the same amount of liquid, we can set these two expressions equal to each other: \[ 6x = 28 + 4x \]

Now, let's solve for \( x \):

1. Subtract \( 4x \) from both sides: \[ 6x - 4x = 28 \] \[ 2x = 28 \]

2. Divide by 2: \[ x = 14 \]

Thus, the two vats will have the same amount of liquid after 14 minutes.

Answer 8

A yoga studio has two participation plans. The first plan costs $10 per visit. The second plan costs $30 per month, and $4 per visit. How many visits would make the cost of both plans the same?(1 point)
The plans would cost the same at blank
visits.

Answer 9

Let \( x \) be the number of visits per month.

For the first plan, the total cost can be expressed as: \[ C_1 = 10x \]

For the second plan, the total cost can be expressed as: \[ C_2 = 30 + 4x \]

To find the number of visits where the costs of both plans are the same, set the two expressions equal to each other: \[ 10x = 30 + 4x \]

Now, let's solve for \( x \):

1. Subtract \( 4x \) from both sides: \[ 10x - 4x = 30 \] \[ 6x = 30 \]

2. Divide by 6: \[ x = 5 \]

Thus, the plans would cost the same at 5 visits.

Answer 10

Two machines at a factory are filling cans of soup. One machine has already filled 180 cans of soup, and fills cans at a rate of 15 per minute. The second machine has already filled 210 cans of soup, and fills cans at a rate of 12 per minute. Write an equation to show when the two machines will have filled the same number of cans of soup. Use x as your variable.(1 point)

Answer 11

Let \( x \) be the number of minutes that have passed.

For the first machine, which has already filled 180 cans and fills cans at a rate of 15 cans per minute, the total number of cans filled after \( x \) minutes can be expressed as: \[ 180 + 15x \]

For the second machine, which has already filled 210 cans and fills cans at a rate of 12 cans per minute, the total number of cans filled after \( x \) minutes can be expressed as: \[ 210 + 12x \]

To find when both machines will have filled the same number of cans, we can set these two expressions equal to each other: \[ 180 + 15x = 210 + 12x \]

This is the equation that shows when the two machines will have filled the same number of cans of soup.

Answer 12

Two machines at a factory are filling cans of soup. One machine has already filled 180 cans of soup, and fills cans at a rate of 15 per minute. The second machine has already filled 210 cans of soup, and fills cans at a rate of 12 per minute. Write an equation to show when the two machines will have filled the same number of cans of soup. Use x as your variable.(1 point)
Responses

210−12x=180−15x
210 minus 12 x equals 180 minus 15 x

12x+210=15x+180
12 x plus 210 equals 15 x plus 180

12x+180=15x+210
12 x plus 180 equals 15 x plus 210

12x−210=15x−180

Answer 13

The correct equation to show when the two machines will have filled the same number of cans of soup is:

\[ 180 + 15x = 210 + 12x \]

Now, let's rearrange this to match one of your options.

1. Starting with: \[ 180 + 15x = 210 + 12x \]

2. You can rearrange it by subtracting \( 12x \) from both sides: \[ 180 + 15x - 12x = 210 \] \[ 180 + 3x = 210 \]

3. Now, subtract 180 from both sides: \[ 3x = 210 - 180 \] \[ 3x = 30 \]

Which can then be represented in several ways, but from the options provided, the correct equation in a similar form isn't given.

However, if we adapt it to one of your options, we can manipulate it slightly or set up an alternative equation.

From the options, none exactly matches the original, but if we were to express the equation in a balanced form:

From the rearrangement \( 15x - 12x = 210 - 180 \) gives:

\[ 3x = 30 \] which could be rearranged to form something akin to: \[ 12x + 210 = 15x + 180 \]

So the second option could be seen as paralleling the foundational equation correctly.

The better mathematical expression is, however, the original setup: \[ 180 + 15x = 210 + 12x \]

This illustrates when both machines will fill the same number of cans.

Answer 14

Movie Theatre A has a popcorn machine with a 100-gallon capacity, and sells 3 gallons of popcorn per hour. Movie Theatre B has a popcorn machine with a 150-gallon capacity, and sells 5 gallons of popcorn per hour. Write an equation to show when the two popcorn machines will have the same amount of popcorn left. Use x as your variable.(1 point)

Answer 15

Let \( x \) be the number of hours that have passed.

For Movie Theatre A, which has a 100-gallon capacity and sells popcorn at a rate of 3 gallons per hour, the amount of popcorn left after \( x \) hours can be expressed as: \[ 100 - 3x \]

For Movie Theatre B, which has a 150-gallon capacity and sells popcorn at a rate of 5 gallons per hour, the amount of popcorn left after \( x \) hours can be expressed as: \[ 150 - 5x \]

To find when both popcorn machines will have the same amount of popcorn left, we can set these two expressions equal to each other: \[ 100 - 3x = 150 - 5x \]

This is the equation that shows when the two popcorn machines will have the same amount of popcorn left.

Answer 16

An ice cream shop makes vanilla and chocolate ice cream. The vanilla ice cream machine has a 180-cup capacity, and it sells 20 cups per day. The chocolate ice cream machine has a 140-cup capacity, and it sells 16 cups per day. Write and solve an equation to show when the two machines will have the same amount of ice cream left.(1 point)

Answer 17

Let \( x \) be the number of days that have passed.

For the vanilla ice cream machine, which has a 180-cup capacity and sells 20 cups per day, the amount of ice cream left after \( x \) days can be expressed as: \[ 180 - 20x \]

For the chocolate ice cream machine, which has a 140-cup capacity and sells 16 cups per day, the amount of ice cream left after \( x \) days can be expressed as: \[ 140 - 16x \]

To find when both machines will have the same amount of ice cream left, we can set these two expressions equal to each other: \[ 180 - 20x = 140 - 16x \]

Now, let's solve for \( x \):

1. Add \( 20x \) to both sides: \[ 180 = 140 - 16x + 20x \] \[ 180 = 140 + 4x \]

2. Subtract 140 from both sides: \[ 180 - 140 = 4x \] \[ 40 = 4x \]

3. Divide by 4: \[ x = 10 \]

Thus, the two machines will have the same amount of ice cream left after 10 days.

Answer 18

Two machines at a factory are filling cans of soup. One machine has filled 92 cans of soup. It fills cans at a rate of 16 cans per minute. The second machine has filled 104 cans of soup. It fills cans at a rate of 13 cans per minute. Write and solve an equation to show when the two machines will have filled the same number of cans of soup.(1 point)

Answer 19

Let \( x \) be the number of minutes that have passed.

For the first machine, which has already filled 92 cans and fills cans at a rate of 16 cans per minute, the total number of cans filled after \( x \) minutes can be expressed as: \[ 92 + 16x \]

For the second machine, which has already filled 104 cans and fills cans at a rate of 13 cans per minute, the total number of cans filled after \( x \) minutes can be expressed as: \[ 104 + 13x \]

To find when both machines will have filled the same number of cans, we can set these two expressions equal to each other: \[ 92 + 16x = 104 + 13x \]

Now, let's solve for \( x \):

1. Subtract \( 13x \) from both sides: \[ 92 + 16x - 13x = 104 \] \[ 92 + 3x = 104 \]

2. Subtract 92 from both sides: \[ 3x = 104 - 92 \] \[ 3x = 12 \]

3. Divide by 3: \[ x = 4 \]

Thus, the two machines will have filled the same number of cans of soup after 4 minutes.

Answer 20

Movie Theatre A has a popcorn machine with a 120-gallon capacity. It sells 7 gallons of popcorn per hour. Movie Theatre B has a popcorn machine with a 150-gallon capacity. It sells 9 gallons of popcorn per hour. Write and solve an equation to show when the two popcorn machines will have the same amount of popcorn left.(1 point)

Answer 21

Let \( x \) be the number of hours that have passed.

For Movie Theatre A, which has a 120-gallon capacity and sells 7 gallons of popcorn per hour, the amount of popcorn left after \( x \) hours can be expressed as: \[ 120 - 7x \]

For Movie Theatre B, which has a 150-gallon capacity and sells 9 gallons of popcorn per hour, the amount of popcorn left after \( x \) hours can be expressed as: \[ 150 - 9x \]

To find when both popcorn machines will have the same amount of popcorn left, we can set these two expressions equal to each other: \[ 120 - 7x = 150 - 9x \]

Now, let’s solve for \( x \):

1. Add \( 9x \) to both sides: \[ 120 - 7x + 9x = 150 \] \[ 120 + 2x = 150 \]

2. Subtract 120 from both sides: \[ 2x = 150 - 120 \] \[ 2x = 30 \]

3. Divide by 2: \[ x = 15 \]

Thus, the two popcorn machines will have the same amount of popcorn left after 15 hours.

Answer 22

they are all correct

Answer 23

I'm glad to hear that the responses were helpful and correct! If you have any more questions or need further assistance on any topic, feel free to ask!