Stan and Hilda can mow the lawn in 60 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone?

Question

Stan and Hilda can mow the lawn in 60 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone?

 

A pilot flew a jet from City A to City B, a distance of 1800 mi. On the return trip, the average speed was 20% faster than the outbound speed. The round-trip took 9 h 10 min. What was the speed from City A to City B?

 

Answer 1

Hi Elhonna,

 

We'll go step-by-step in solution to each problem.

 

 

Problem #1:

 

Let's begin by expressing each person's time to do the job alone:

    x = Hilda's time 

  2x = Stan's time (twice as long as Hilda's)

 

Next, let's express each person's portion of the completed job working together:

    60/x   = Hilda's portion of the completed job

    60/2x = Stan's portion of the completed job

 

Next, let's write an equation to express the sum of both portions as being equal to the whole job and then solve for x to determine each person's time:

    60/x + 60/2x = 1

    (2x)(60/x) + (2x)(60/2x) = (2x)(1)

    120 + 60 = 2x

    2x = 180

      x = 180/2

      x = 90 minutes (Hilda's time)

    2x = (2)(90)

    2x = 180 minutes (Stan's time)

 

Finally, let's plug both our answers into our equation and see if the whole is equal to the sum of the parts:

    60/x + 60/2x = 1

    60/90 + 60/(2)(90) = 1

    60/90 + 60/180 = 1

    2/3 + 1/3 = 1

    1 = 1 (whole is equal to the sum of the parts)

 

Since the whole has been proven equal to the sum of the parts with our answer, we are confident that our answers are correct.

 

 

Problem #2:

 

Let's begin by recalling the distance formula and then get an expression to calculate the average rate based upon the total distance and the total time:

    total distance = average rate x total time

    average rate = total distance / total time

 

Next, let's determine the total distance and

the total time:

    total distance = 1800 (1-way) x 2 = 3600 miles

    total time = 9 hours and 10 minutes (round trip)

    total time = 9 and 10/60 = 9 and 1/6 hours

 

Next, let's express the average rate in terms of the outbound rate:

    average rate = (outbound rate + return rate) / 2

    average rate = (x + 1.2x) /2

    average rate = 2.2x/2

    average rate = 1.1x

    

Next, let's plug in our total distance, total time, and  expression for the average rate in terms of the outbound rate - all into our average rate equation and solve it for x (the outbound rate):

    average rate = total distance / total time

    1.1x = 3600 / (9 and 1/6)

    1.1x = 3600/[(9 x 6 + 1)/6]

    1.1x = 3600/(55/6)

    1.1x = 3600(6/55)

    55(1.1x) = 3600(6)

    60.5x = 21600

        x = 21600/60.5

        x = 43200/121 mph (exact outbound speed)

        x ≈ 357 mph (rounded outbound speed)

 

Finally, we can plug in our exact answer into our equation and see if both sides balance:

    1.1x = 3600 / (9 and 1/6) 

    1.1(43200/121) = 3600/(55/6)

    4320/121 = 3600(6/55)

    4320/11 = 21600/55

    4320/11 = 4320/11 (both sides balance)

 

Since both sides balance, we are confident that our answer is correct.

 

Thanks for submitting these problems & glad to help.

 

God bless, Jordan.

Answer 2

t represents the time it takes Hilda alone.

2t represents the time it takes Stan alone. (He works half a fast, therefore twice as long).

 

60/t + 60/2t = 1          multiply both sides by 2t

120 + 60 = 2t              combine like terms

180 = 2t                      divide both sides by 2

 

Working alone Stan would need 180 minutes, three hours.

 

Check:

60/180 + 60/90

1/3 + 2/3

1