A team of two men, three women, four boys and six girls are to collect some lemons from an orchard. Each team member could shake X times as fast as he or she could pick, where X is a certain positive number.
- One woman can shake lemons exactly as fast as four boys can gather them.
- Two of the boys can shake 69 lemons off the trees as fast as three women and one boy can gather 50 lemons.
- Two of the girls can shake 70 lemons off the trees as fast as three girls and one woman can collectively gather 50 lemons.
- One man can shake 141 lemons from the trees as fast as a man, woman and girl can gather 100 lemons.
- Everyone worked the whole time, either shaking or gathering lemons.
- The group payment received by the team was $1240.
If the money is to be divided up based on how fast each member could pick and gather lemons, how much money should each team member get?
|
|
Submitted by K Sengupta
|
|
Rating: 2.0000 (1 votes)
|
|
|
Solution:
|
(Hide)
|
The required amounts to be received by each man, each woman , each boy and each girl are $124.04, $100.02, $81.09 and $61.25 respectively.
EXPLANATION:
I solved this problem using a lengthy algebraic proceedure. Charlie's solution utilising Excel techniques precisely corresponds to the result obtained by my own algebraic method. Consequently, the said solution is provided hereunder.
EXCEL SOLUTION (Submitted By Charlie):
Let r be the ratio of each person's speed at gathering lemons to that of picking lemons.
Let the speed of the girls at picking (shaking) be 1, and the picking speeds of the women, boys and men be w, b and m respectively.
We have:
w*r = 4*b
2*b*r = 1.38*(3*w+b)
2*r = 1.4*(3+w)
m*r = 1.41*(m+1+w)
I used Excel to place each variable in a cell, and set up cells with formulas that would result in zero as each equation held, such as =1.41*(m+1+w)-rat*m for the last equation above. (Excel wouldn't let me name a cell r, so I used rat.) Then I used Tools... solver to adjust rat, w, b and m so that these cells would come out to zero.
Solver came up with r=3.24310714087661, w=1.6330102012523, b=1.32400676120142 and m=2.02527408299242.
Presumably each person should be paid proportionally to these numbers. As there are 2 men, 3 women, 4 boys and 6 girls, 2*m+3*w+4*b+6 units of work were done, so each unit of work (which is one girl's pay for the day) is worth 1240/(2*m+3*w+4*b+6) = 61.24785849. This is multiplied by w, b and m to get the daily pay for each woman, boy and man respectively. These come out to 100.02 for each woman, 81.09 for each boy and 124.04 for each man. The girls' rate rounds to 61.25. Thus the two men together make 248.08; the three women, 300.06; the four boys, 324.36 and the six girls, 367.50. This adds up to the 1240 available.
|
