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Which equation can be used to determine the time for each leg?

600t = 480t
600t = 18 – t
600t = 480 – t
600t = 480(18 – t)

Rate (mi/h) | Time (h) | Distance (mi)
-------------------------------------
Trip to Seattle | 600 | t | 600t
-------------------------------------
Trip to Tokyo | 480 | 18 - t | 480(18 - t)
Summary
The image is a table comparing the rate, time, and distance for two trips: one to Seattle and one to Tokyo. The rate for the trip to Seattle is 600 miles per hour, with the time variable represented as ‘t’, leading to a distance of 600t miles. The rate for the trip to Tokyo is 480 miles per hour, with the time as ‘18 - t’, resulting in a distance of 480(18 - t) miles.

An airplane flew at an average rate of 600 miles per hour from Tokyo to Seattle. Because of a strong headwind, the average rate of the return trip was only 480 mph. The round trip took 18 hours.

11 answers

To determine the time for each leg of the trip, we need to account for the total time of the round trip, which is 18 hours. We have the time for the trip to Seattle as \( t \) hours and the time for the return trip to Tokyo as \( 18 - t \) hours.

Given that the distance is the same for both legs of the trip, we can set up the equations based on the definition of distance:

  1. For the trip to Seattle: \[ \text{Distance} = \text{Rate} \times \text{Time} = 600 \times t \]

  2. For the trip to Tokyo: \[ \text{Distance} = \text{Rate} \times \text{Time} = 480 \times (18 - t) \]

Since both distances are equal (the distance to Seattle is the same as the distance back from Tokyo), we can equate these two expressions: \[ 600t = 480(18 - t) \]

Thus, the correct equation to determine the time for each leg of the trip is: \[ 600t = 480(18 - t) \]

Use the table to write and solve an equation to determine the approximate amount of time it would take Sanjay and Felipe to do the puzzle if they were working together.

0.8 hours
1.7 hours
3.4 hours
7 hours

Sanjay and Felipe are putting together a 500-piece puzzle. Sanjay could do the entire puzzle by himself in 6 hours, while it would take Felipe 8 hours to do the puzzle alone.

Rate (part/h) | Time (h) | Part of puzzle completed
Sanjay | t | t
Felipe | t | t

Summary
The image is a table comparing two individuals, Sanjay and Felipe, on three parameters: Rate (parts per hour), Time (hours), and the Part of the puzzle completed. The values for all parameters are represented by the variable ( t ).

To determine the amount of time it would take Sanjay and Felipe to complete the puzzle if they worked together, we first need to calculate their individual rates of working on the puzzle, then combine those rates to find the total rate when they work together.

  1. Calculate Rates:

    • Sanjay's Rate: He completes the 500 piece puzzle in 6 hours. \[ \text{Rate of Sanjay} = \frac{500 \text{ pieces}}{6 \text{ hours}} \approx 83.33 \text{ pieces/hour} \]

    • Felipe's Rate: He completes the 500 piece puzzle in 8 hours. \[ \text{Rate of Felipe} = \frac{500 \text{ pieces}}{8 \text{ hours}} = 62.5 \text{ pieces/hour} \]

  2. Combined Rate: When working together, their combined rate is the sum of their individual rates: \[ \text{Combined Rate} = 83.33 + 62.5 = 145.83 \text{ pieces/hour} \]

  3. Total Time Working Together: To find the time \( t \) it takes for both to complete the puzzle working together, use the formula: \[ \text{Total Time} = \frac{\text{Total Pieces}}{\text{Combined Rate}} = \frac{500 \text{ pieces}}{145.83 \text{ pieces/hour}} \approx 3.43 \text{ hours} \]

Now we look at the options you provided:

  • 0.8 hours
  • 1.7 hours
  • 3.4 hours
  • 7 hours

The approximate amount of time it would take Sanjay and Felipe to complete the puzzle if they worked together is approximately 3.4 hours.

Use the scenario and table to determine which statements are true. Check all that apply.

Armand and Jill will each travel the same distance to meet.
Armand and Jill will travel the same amount of time to meet.
The equation used to find the time, t, it will take for them to meet is 50t + 60t = 247.5.
It will take 4.5 hours for them to meet.
Jill will travel 135 miles to meet Armand.

Armand and Jill are 247.5 miles apart, and driving toward each other. Armand is driving at a rate of 50 miles per hour and Jill is driving at a rate of 60 miles per hour.

Image to Text
Rate (mph) Time (h) Distance (mi)
Armand 50 t
Jill 60 t
Summary
The image is a table comparing the rates, times, and distances traveled by two individuals, Armand and Jill. Armand travels at a rate of 50 mph, and Jill travels at a rate of 60 mph. Both travel for the same amount of time, denoted by “t” hours. The distances they travel are expressed as 50t and 60t miles, respectively.

Let's analyze the statements based on the scenario provided about Armand and Jill traveling toward each other.

  1. Armand and Jill will each travel the same distance to meet.

    • False. Armand drives at 50 mph and Jill drives at 60 mph. Since they travel for the same amount of time, they will cover different distances. The distance Armand travels is \( 50t \) and the distance Jill travels is \( 60t \).

  2. Armand and Jill will travel the same amount of time to meet.

    • True. They are both traveling toward each other until they meet, so they will travel for the same amount of time, which we denote as \( t \).

  3. The equation used to find the time, t, it will take for them to meet is \( 50t + 60t = 247.5 \).

    • True. Using the distance formula, the total distance traveled by both Armand and Jill will equal the distance between them (247.5 miles). Their combined distances yield the equation \( 50t + 60t = 247.5 \), or \( 110t = 247.5 \).

  4. It will take 4.5 hours for them to meet.

    • False. To find \( t \): \[ 110t = 247.5 \implies t = \frac{247.5}{110} \approx 2.25 \text{ hours} \]

  5. Jill will travel 135 miles to meet Armand.

    • True. We can determine the distance Jill travels: \[ \text{Distance} = 60t = 60 \times 2.25 = 135 \text{ miles} \]

Summary of True Statements:

  • Armand and Jill will travel the same amount of time to meet.
  • The equation used to find the time, \( t \), it will take for them to meet is \( 50t + 60t = 247.5 \).
  • Jill will travel 135 miles to meet Armand.

Final Answer:

So, the true statements are:

  • Armand and Jill will travel the same amount of time to meet.
  • The equation used to find the time, t, it will take for them to meet is \( 50t + 60t = 247.5 \).
  • Jill will travel 135 miles to meet Armand.
Roni and Allie are mowing the grass at the soccer field. Roni has a riding lawn mower and can mow the field in 30 minutes. Allie is pushing a lawn mower and can mow the field in 75 minutes.

If Roni and Allie work together to mow the field, what part of the field would Roni mow?

about 0.52 of the field
about 0.64 of the field
about 0.71 of the field
about 0.87 of the field

To find out what part of the field Roni would mow while working together with Allie, we first need to calculate their rates of mowing the field:

  1. Calculate the rates of Roni and Allie:

    • Roni's rate (in fields per minute): \[ \text{Rate of Roni} = \frac{1 \text{ field}}{30 \text{ minutes}} = \frac{1}{30} \text{ fields/minute} \]
    • Allie's rate (in fields per minute): \[ \text{Rate of Allie} = \frac{1 \text{ field}}{75 \text{ minutes}} = \frac{1}{75} \text{ fields/minute} \]

  2. Calculate their combined rate: \[ \text{Combined Rate} = \text{Rate of Roni} + \text{Rate of Allie} = \frac{1}{30} + \frac{1}{75} \] To add these fractions, we need a common denominator. The least common multiple of 30 and 75 is 150.

    Convert each rate: \[ \frac{1}{30} = \frac{5}{150} \] \[ \frac{1}{75} = \frac{2}{150} \]

    Now, adding the two rates: \[ \text{Combined Rate} = \frac{5}{150} + \frac{2}{150} = \frac{7}{150} \text{ fields/minute} \]

  3. Calculate the total time to mow the field together: If they work together, the time \( t \) to mow one field can be calculated as: \[ t = \frac{1 \text{ field}}{\text{Combined Rate}} = \frac{1}{\frac{7}{150}} = \frac{150}{7} \approx 21.43 \text{ minutes} \]

  4. Calculate the amount of time Roni mows: During this time, Roni mows: \[ \text{Part of the field Roni mows} = \text{Rate of Roni} \times \text{Time} \] \[ = \frac{1}{30} \times \frac{150}{7} = \frac{150}{210} = \frac{5}{7} \approx 0.7143 \]

Conclusion:

Roni mows approximately 0.71 of the field.

So the correct answer is:

  • about 0.71 of the field.
What was the distance of the first part of Yuna's swim? Round to the nearest tenth as needed.

The distance in the first part of her swim was
miles.

In training for a triathlon, Yuna swam 1.9 miles in the open ocean, which took her 2 hours. For the first part of her swim, she averaged 1.4 miles per hour. For the second part of her swim, she swam against the current and started to tire, so her average speed decreased to 0.8 mph.

Image to Text
Rate (mi/h) | Time (h) | Distance (mi)
Part 1 | 1.4 | t | 1.4t
Part 2 | 0.8 | 2 - t | 0.8(2 - t)

Summary
The image is a table that presents information about two parts of a journey. It shows the rate, time, and distance for each part. Part 1 has a rate of 1.4 mi/h, time represented by t, and distance represented by 1.4t. Part 2 has a rate of 0.8 mi/h, time represented by 2 - t, and distance represented by 0.8(2 - t).

To find the distance of the first part of Yuna's swim, we can set up an equation based on the total distance and the distances covered in each part of her swim.

  1. Define the distances:

    • Distance for the first part of her swim: \( D_1 = 1.4t \)
    • Distance for the second part of her swim: \( D_2 = 0.8(2 - t) \)

  2. Total Distance: The total distance Yuna swam is 1.9 miles. Therefore, we have: \[ D_1 + D_2 = 1.9 \] Substituting the expressions for \( D_1 \) and \( D_2 \): \[ 1.4t + 0.8(2 - t) = 1.9 \]

  3. Simplifying the equation: Expanding the equation: \[ 1.4t + 1.6 - 0.8t = 1.9 \] Combining like terms: \[ (1.4t - 0.8t) + 1.6 = 1.9 \] \[ 0.6t + 1.6 = 1.9 \]

  4. Isolating \( t \): Subtract 1.6 from both sides: \[ 0.6t = 1.9 - 1.6 \] \[ 0.6t = 0.3 \] Dividing both sides by 0.6 gives: \[ t = \frac{0.3}{0.6} = 0.5 \text{ hours} \]

  5. Finding the distance of the first part of the swim: Now, we substitute \( t \) back into the distance formula for the first part: \[ D_1 = 1.4t = 1.4 \times 0.5 = 0.7 \text{ miles} \]

Conclusion:

The distance of the first part of Yuna's swim was 0.7 miles.

It takes 12 minutes to fill an entire bathtub using both the cold and hot water. If just the cold water is used, it takes 18 minutes to fill the bathtub. How long would it take to fill the bathtub if just the hot water were used?

The equation
can be used to solve for the rate, r, for the hot water alone to fill the bathtub.

The hot water can fill
of the tub in 1 minute.

It would take
minutes for the hot water to fill the bathtub.

To find the time it takes for the hot water to fill the bathtub by itself, we will first define the rates for each type of water.

  1. Define the rates:

    • Let \( r_c \) be the rate at which the cold water fills the tub. If it takes 18 minutes for just the cold water, then: \[ r_c = \frac{1 \text{ bathtub}}{18 \text{ minutes}} = \frac{1}{18} \text{ bathtubs per minute} \]

    • Let \( r_h \) be the rate at which the hot water fills the tub. If it takes \( t \) minutes for just the hot water, then: \[ r_h = \frac{1 \text{ bathtub}}{t \text{ minutes}} = \frac{1}{t} \text{ bathtubs per minute} \]

  2. Combined rate with both faucets open: When both the cold and hot water are used together, they can fill the tub in 12 minutes: \[ r_c + r_h = \frac{1 \text{ bathtub}}{12 \text{ minutes}} = \frac{1}{12} \text{ bathtubs per minute} \]

  3. Set up the equation: We can now substitute the expression for \( r_c \) and \( r_h \) into the equation: \[ \frac{1}{18} + \frac{1}{t} = \frac{1}{12} \]

  4. Solve for \( t \): To solve this equation, we will first find a common denominator. The least common multiple of 18, \( t \), and 12 is 36t. Multiplying through by this common denominator gives us: \[ 36t \left( \frac{1}{18} \right) + 36t \left( \frac{1}{t} \right) = 36t \left( \frac{1}{12} \right) \] This simplifies to: \[ 2t + 36 = 3t \] Rearranging the equation: \[ 36 = 3t - 2t \] \[ 36 = t \]

  5. Find the rate of hot water: Now that we know it takes 36 minutes for the hot water to fill the bathtub, we can find the rate for just the hot water: \[ r_h = \frac{1}{t} = \frac{1}{36} \text{ bathtubs per minute} \] In 1 minute, the hot water fills \( \frac{1}{36} \) of the tub.

Conclusion:

  • The equation used to solve for the rate for the hot water alone to fill the bathtub is: \[ \frac{1}{18} + \frac{1}{t} = \frac{1}{12} \]

  • The hot water can fill \(\frac{1}{36}\) of the tub in 1 minute.

  • It would take 36 minutes for the hot water to fill the bathtub.