A man starts on a 290 mile trip at 10 a. m. and drives at an average speed of 40 mph until he stops for lunch. After he stops one-half hour for lunch, he finds that he must drive the rest of the way at an average speed of 50mph if he is to arrive by 5pm. When did he stop for lunch? Please show work, trying to understand how to work this problem.

1:30pm

Step-by-step explanation:

Start Time: 10:00am

Start Speed: 40MPH

Time for Lunch: 30minutes

Arrival Time: 5:00pm

Ending Speed: 50MPH

Total Miles: 290miles

Find Total Time Taken:

5pm - 10am = 7 hours

7 hours - 30 minutes (lunch) = 6.5 hours of travel

Create Equations:

Distance = Speed x Time

Finding Total Numbers of Miles Traveled

Miles = SpeedBefore x TimeBefore + SpeedAfter x TimeAfter

290(Miles) = 40(mph) x E + 50(mph) x Y

290 = 40E + 50A

Finding Time Traveled Before and After the lunch break

Note: It is based on duration in hours

Total Time = Time Before + Time After + Lunch Time

7 hours = E + Y + .5 hours

7 = E + Y + .5

Solve for Y

7 = Y + E + .5

6.5 = Y + E

6.5 - E = Y

Use Substitution in second equation to solve for E

290 = 40E + 50Y

290 = 40E + 50(6.5 - E)

290 = 40E + (325 - 50E)

290 = 40E + 325 - 50E

290 = -10E + 325

-35 = -10E

10E = 35

E = 3.5 hours, time traveled before lunch

Note: The following is unnecessary

Solve for Y, Again

Y = 6.5 - E

Y = 6.5 - 3.5

Y = 3 hours, time traveled after lunch

So the man traveled for 3.5 hours and then ate lunch. Then he traveled for 3 more hours.

So...

10am + 3.5 hours = 1:30pm - the time when he ate lunch

1:30pm

Step-by-step explanation:

From the question, we can derive the following equation:

1. Total time for the Journey was 7hours(10am-5pm) but the effective hours for the journey is 6.5hours(when we take out the lunch time).

So the combined time spent - before and after lunch, is x+y=6.5

x=time spent before lunch

y=time spent after lunch

2. The assumed distance traveled before lunch is 40x while assumed distance after lunch is 50y; but the total distance should equal 290.

So,

40x+50y=290

This gives us a simultaneous equation:

x+y=6.5

40x+50y=290

(Using substitution)

40x+50(6.5-x)=290(curled from x+y=6.5)

40x+325-50x=290(solving the parenthesis first)

325-50x+40x=290

325-10x=290 (add 10x to both sides)

325-10x+10x=290+10x

325=290+10x (subtract 290 from both sides)

325-290=290+10x-290

35=10x (divide both sides by 10)

3.5=x

Since x is the time spent before lunch(in hours), we can then add 3 and half hours to 10am

10am +3.5hours= 1:30pm

Hence, the man stopped for lunch at 1:30pm