The Johnson family is planning a summer road trip to meet up with some friends who live $730$ miles away from them. The family takes a scenic route through the winding roads of the countryside. They expect to complete the road in trip in three days. The distances and average speeds for the first two days of the trip are shown. $\newline$ - Day $1$ : $5$ hours at an average speed of $45$ miles per hour $\newline$ - Day $2$ : $8$ hours at an average speed of $35$ miles per hour $\newline$ The average speed on the third day is $45$ miles per hour. $\newline$ Write an equation using letters and numbers to determine how many more hours it will take the Johnson family to reach their destination. Explain your reasoning.

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Q. The Johnson family is planning a summer road trip to meet up with some friends who live

730

miles away from them. The family takes a scenic route through the winding roads of the countryside. They expect to complete the road in trip in three days. The distances and average speeds for the first two days of the trip are shown.

\newline

- Day

1

5

hours at an average speed of

45

miles per hour

\newline

- Day

2

8

hours at an average speed of

35

miles per hour

\newline

The average speed on the third day is

45

miles per hour.

\newline

Write an equation using letters and numbers to determine how many more hours it will take the Johnson family to reach their destination. Explain your reasoning.

Define Variables and Values: Let's define the variables and the known values. We know the total distance of the trip is $730$ miles. On Day $1$ , the family travels for $5$ hours at an average speed of $45$ miles per hour. On Day $2$ , they travel for $8$ hours at an average speed of $35$ miles per hour. We need to find the time (let's call it ' $t$ ' hours) they will travel on Day $3$ at an average speed of $45$ miles per hour to complete the remaining distance.
Calculate Day $1$ Distance: First, we calculate the distance covered on Day $1$ by multiplying the time traveled by the average speed: $\text{Distance} = \text{Time} \times \text{Speed}$ . So, for Day $1$ , the distance is $5$ hours $\times 45$ miles per hour.
Calculate Day $2$ Distance: Performing the calculation for Day $1$ : $5$ hours $\times$ $45$ miles per hour = $225$ miles.
Add Day $1$ and Day $2$ Distances: Next, we calculate the distance covered on Day $2$ using the same formula: Distance = Time $\times$ Speed. For Day $2$ , the distance is $8$ hours $\times$ $35$ miles per hour.
Find Remaining Distance: Performing the calculation for Day $2$ : $8$ hours $\times$ $35$ miles per hour = $280$ miles.
Set Up Equation for Day $3$ : Now, we add the distances from Day $1$ and Day $2$ to find the total distance covered in the first two days: $225$ miles + $280$ miles.
Solve for Time 't': Adding the distances gives us a total of $225$ miles + $280$ miles = $505$ miles covered in the first two days.
Solve for Time 't': Adding the distances gives us a total of $225$ miles + $280$ miles = $505$ miles covered in the first two days.To find the remaining distance to be covered on Day $3$ , we subtract the total distance covered in the first two days from the total trip distance: $730$ miles - $505$ miles.
Solve for Time 't': Adding the distances gives us a total of $225$ miles + $280$ miles = $505$ miles covered in the first two days.To find the remaining distance to be covered on Day $3$ , we subtract the total distance covered in the first two days from the total trip distance: $730$ miles - $505$ miles.Performing the subtraction gives us the remaining distance: $730$ miles - $505$ miles = $225$ miles to be covered on Day $3$ .
Solve for Time 't': Adding the distances gives us a total of $225$ miles + $280$ miles = $505$ miles covered in the first two days.To find the remaining distance to be covered on Day $3$ , we subtract the total distance covered in the first two days from the total trip distance: $730$ miles - $505$ miles.Performing the subtraction gives us the remaining distance: $730$ miles - $505$ miles = $225$ miles to be covered on Day $3$ .Now we can set up the equation to find the time 't' it will take to cover the remaining $225$ miles on Day $3$ at an average speed of $45$ miles per hour. The equation is: Distance = Speed × Time, or $225$ miles = $45$ miles per hour × $280$ $2$ .
Solve for Time 't': Adding the distances gives us a total of $225$ miles + $280$ miles = $505$ miles covered in the first two days.To find the remaining distance to be covered on Day $3$ , we subtract the total distance covered in the first two days from the total trip distance: $730$ miles - $505$ miles.Performing the subtraction gives us the remaining distance: $730$ miles - $505$ miles = $225$ miles to be covered on Day $3$ .Now we can set up the equation to find the time 't' it will take to cover the remaining $225$ miles on Day $3$ at an average speed of $45$ miles per hour. The equation is: Distance = Speed × Time, or $225$ miles = $45$ miles per hour × $280$ $2$ .To solve for 't', we divide both sides of the equation by $45$ miles per hour: $280$ $4$ .
Solve for Time 't': Adding the distances gives us a total of $225$ miles + $280$ miles = $505$ miles covered in the first two days.To find the remaining distance to be covered on Day $3$ , we subtract the total distance covered in the first two days from the total trip distance: $730$ miles - $505$ miles.Performing the subtraction gives us the remaining distance: $730$ miles - $505$ miles = $225$ miles to be covered on Day $3$ .Now we can set up the equation to find the time 't' it will take to cover the remaining $225$ miles on Day $3$ at an average speed of $45$ miles per hour. The equation is: Distance = Speed $280$ $0$ Time, or $225$ miles = $45$ miles per hour $280$ $3$ .To solve for 't', we divide both sides of the equation by $45$ miles per hour: $280$ $5$ .Performing the division gives us the time 't': $280$ $6$ hours.
Solve for Time 't': Adding the distances gives us a total of $225$ miles + $280$ miles = $505$ miles covered in the first two days.To find the remaining distance to be covered on Day $3$ , we subtract the total distance covered in the first two days from the total trip distance: $730$ miles - $505$ miles.Performing the subtraction gives us the remaining distance: $730$ miles - $505$ miles = $225$ miles to be covered on Day $3$ .Now we can set up the equation to find the time 't' it will take to cover the remaining $225$ miles on Day $3$ at an average speed of $45$ miles per hour. The equation is: Distance = Speed $280$ $0$ Time, or $225$ miles = $45$ miles per hour $280$ $3$ .To solve for 't', we divide both sides of the equation by $45$ miles per hour: $280$ $5$ .Performing the division gives us the time 't': $280$ $6$ hours.We have found that it will take the Johnson family $280$ $7$ more hours to reach their destination on the third day at an average speed of $45$ miles per hour.