Question

The Painter siblings, Elsa, Ben and Cal paint houses. They have been in the business so long that each knows exactly how many square feet each other paints in one hour and these rates never change. For their latest job, they calculated that if Elsa and Ben did the job together, it would take exactly 11 hours. On the other hand, if Elsa and Cal did the job together, it would take exactly 9 hours. Finally, Ben and Cal could do it in exactly 9.9 hours (or if you prefer, 9 hours and 54 minutes). They decided all three would paint this particular house. Ben and Cal started the job at 8 AM and Elsa joined them at 9:00. Cal left at 1:30 and so Ben and Elsafinished the job. After deducting the supply costs (paint etc.), the brothers split the net profit based on what percentage of the total square footage each painted. Who earned the most and who earned the least for this job?

Answer 1

Ben earned the most as 6 of 16.5 as a percentage = (6*100):16.5 =36.36%

We check with Elsa and she earned the least as (5*100):16.5 =30.31%

Cal earned the average as 5.5 of 16.5 as a percentage = (5.5*100):16.5 =33.33%

Explanation:

E + B = 11hrs

E + C = 9hrs

B+C = 9.9 hrs

Lets say the house was 100ft sq

if E took 5.5 hrs

B took 5.5hrs we would have 11

C took 3.5 hrs we would have 9hrs for E+C

B took 5.5 hrs + 3.5hrs we are 0.9hrs short and see that B would need to take 0.9 more which means E works 0.9hrs less and shares 0.4 less with C having 0.5 less which gives B an extra 0.4 instead of 0.9

This equates to B: 5.9 , C: 3.9 , E: 5.1

We can change to B: 6 , C: 3.94 , E: 5.4

= E+B = 11.4 = 11

= E+C = 8.99 = 9

= B+C = 9.94 = 9.9

Percentages of fixed net pay when B: 6 , C: 3.94 , E: 5.4 = 6+ 3.94+ 5.4 = 100

Time when 15.34 hrs = 100% if working same rate changes to 16.5hrs to account for the hours given 8am-1.30pm 8am-1.30+ 9am-1.30+ = minimum 1hr more = 2pm

Cal worked 8 am till 1.30 pm = 5.5hrs = 1.66hr extra

Ben worked 8am till 2pm = 6 hrs = 0 hr extra together worked 11.5hrs

Elsa worked 9am till 2pm = 5 hrs = -0.4hr less

11.5+ 5 = 16.5 This equals 16.5hrs = 100%

Ben earned the most as 6 of 16.5 as a percentage = (6*100):16.5 =36.36%

Cal earned the average as 5.5 of 16.5 as a percentage = (5.5*100):16.5 =33.33%

We check with Elsa and she earned the least as (5*100):16.5 =30.31% is the same as 100-69.67%

Answer 2

Elsa earned the most

Ben earned the least

Explanation:

E = Elsa's speed

B = Ben's speed

C = Cal's speed

The combined speeds are:

E + B = 1/11

E + C = 1/9

B + C = 1/9.9

If we add up all three equations:

2B + 2C + 2E = 10/33

B + C + E = 5/33

If we subtract each of the first three equations from the total equation, we can find the individual speeds.

C = 5/33 − 1/11 = 2/33

B = 5/33 − 1/9 = 4/99

E = 5/33 − 1/9.9 = 5/99

Ben and Cal paint together for 1 hour. The fraction of the wall they paint is:

(1/9.9) × 1 = 10/99

Elsa, Ben, and Cal all paint together for the next 4.5 hours. The fraction of the wall they paint is:

(5/33) × 4.5 = 15/22

Ben and Elsa then finish the job. The amount of wall left is 1 − 10/99 − 15/22 = 43/198. So the time it takes is:

(43/198) / (1/11) = 43/18 hours

The fraction of the wall that Ben paints is:

(4/99) (1 + 4.5 + 43/18) = 284/891

The fraction of the wall that Cal paints is:

(2/33) (1 + 4.5) = 1/3 = 297/891

The fraction of the wall that Elsa paints is:

(5/99) (4.5 + 43/18) = 310/891