Problem Solving

BTGmoderatorDC wrote:Twenty-four men can complete a work in sixteen days.Thirty-two women can complete the same work in twenty-four days.Sixteen men and sixteen women started working for twelve days.How many more men are to be added to complete the work remaining work in 2 days?

A .16
B. 24
C. 36
D. 48
E. 54

Since 24 men can complete a work in 16 days, the rate of 24 men is 1/16, and hence the rate of 1 man is (1/16)/24 = 1/384.

Similarly, 32 women can complete the same work in 24 days, so the rate of 32 women is 1/24, and hence the rate of 1 woman is (1/24)/32 = 1/768.

Thus, the rate for 16 men and 16 women is 16 x (1/384) + 16 x (1/768) = 1/24 + 1/48 = 2/48 + 1/48 = 3/48 = 1/16.

Since these 16 men and 16 women worked for 12 days, they completed (1/16) x 12 = 12/16 = 3/4 of the job.

Thus, 1/4 of the job is left to be done. To find the number of additional men needed to complete the remaining work in 2 days, we can let n = the additional men needed and create the following equation (keep in mind that the rate of each man is 1/384 and the rate of the existing 16 men and 16 women is 1/16):

n(1/384)(2) + (1/16)(2) = 1/4

n/192 + 1/8 = 1/4

n/192 = 1/8

8n = 192

n = 24

Alternate Solution:

If 24 men can do the job in 16 days, then 16 men, which is 2/3 of 24 men, can do the job in 16/(2/3) = 24 days.

Similarly, since 32 women can do the same job in 24 days, 16 women, which is half of 32 women, can do the same job in 24/(1/2) = 48 days.

Thus, in one day, 1/24 of the job is done by men and 1/48 of the job is done by women; thus 1/24 + 1/48 = 3/48 = 1/16 of the job is done in total. Therefore, with 16 men and 16 women working, the job will be finished in 16 days. Since 16 men and 16 women have already worked for 12 days, the remaining job would have been completed by 16 men and 16 women in 16 - 12 = 4 days.

Since 16 men complete the job in exactly half the time compared to 16 women, the men work twice as fast as women; i.e. 1 man can do the same job done by 2 women. Thus, the work done by 16 men and 16 women is equivalent to the work done by 16 + 16/2 = 24 men.

Now, if we want the remaining job to be completed in 2 days instead of 4; in other words, if we want the remaining job to be completed in half the time, we should double the number of men. Since 16 men and 16 women is equivalent to 24 men, we should add 24 more men to get the remaining job done in 2 days.

Answer: B