need help w/ applications/prop.: How long to work alone?

[jonadkins]

Here it is:

Beth can build a cabinet 3 times as fast as Ashley. If they work together, it takes them 4 hrs to build the cabinet. How long would it take each to work alone?

Please reply back quickly. Thank you!

 

[stapel]

Try using the standard rate table:

. . .time (in hours) to complete job:
. . . . .beth: b
. . . . .ashley: 3b
. . . . .together: 4

. . .completed per time unit (hour):
. . . . .beth: 1/b
. . . . .ashley: 1/??
. . . . .toghether: 1/4

Complete the table.

Add their labors, set equal to what they accomplish in an hour together, and solve the resulting equation for Beth's time. Back-solve for Ashley's time.

If you get stuck, please reply showing how far you have gotten. Thank you.

Eliz.

 

[jonadkins]

i'm not sure how to set it from the table

 

[TchrWill]

Beth can build a cabinet 3 times as fast as Ashley. If they work together, it takes them 4 hrs to build the cabinet. How long would it take each to work alone?

Problems of this type are solvable by either of the following methods.

<< If it takes me 2 hours to paint a room and you 3 hours, ow long will it take to paint it together? >>

Method 1:

1--A can paint the house in 5 hours.
2--B can paint the house in 3 hours.
3--A's rate of painting is 1 house per A hours (5 hours) or 1/A (1/5) houses/hour.
4--B's rate of painting is 1 house per B hours (3 hours) or 1/B (1/3) houses/hour.
5--Their combined rate of painting is 1/A + 1/B (1/5 + 1/3) = (A+B)/AB (8/15) houses /hour.
6--Therefore, the time required for both of them to paint the 1 house is 1 house/(A+B)/AB houses/hour = AB/(A+B) = 5(3)/(5+3) = 15/8 hours = 1 hour-52.5 minutes.

Note - T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.

Method 2:

Consider the following diagram -

.........._______________ _________________
..........I B /............................/\
..........I..*.................../..............................I
..........I.....*............../................................I
..........Iy.......*........./.................................I
..........I................./...................................{
..........I*****x****** ....................................{
..........I............./....*................................(c)
..........I(c-y)..../.........*...............................{
..........I......../...............*...........................I.
..........I....../....................*........................I
..........I..../.........................*.....................I
..........I../.............................*...................{
.........I./___________________* ________\/__
A

1--Let c represent the area of the house to be painted.
2--Let A = the number of hours it takes A to paint the house.
3--Let B = the number of hours it takes B to paint the house.
4--A and B start painting at the same point but proceed in opposite directions around the house.
5--Eventually they meet in x hours, each having painted an area proportional to their individual painting rates.
6--A will have painted y square feet and B will have painted (c-y) square feet.
7--From the figure, A/c = x/y or Ay = cx.
8--Similarly, B/c = x/(c-y) or by = bc - cx.
9--From 7 & 8, y = cx/a = (bc - cx)/b from which x = AB/(A+B), one half of the harmonic mean of A and B.

I think this should give you enough of a clue as to how to solve your particular problem.

 

[stapel]

i'm not sure how to set it from the table

Read off the amounts that each does per hour (from the "completed per hour" part, the "beth" and "ashley" lines), add them (put a "plus" sign between the two terms), set this sum (the two terms with the "plus" sign between) equal to (put an "equals" sign followed by) the total they do per hour (read this off the table, from the "completed per hour" part, the "together" line):

. . . . .(beth/hour) + (ashley/hour) = (together/hour)

Then solve this equation for the value of the variable "b", which stands for the number of hours that Beth takes. Once you have this value, multiply it by "3" to find the number of hours that Ashley takes.

I apologize for any confusion.

Eliz.