word problem time

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sw87

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It takes Danny three time as long to mow a field as it takes his fauther. Together they can mow the same field in for hours. how long doesit take Danny working alone?

I know what they are asking for but I don't know how to put it into a equation.

I know it takes 4hrs together
3(fathers time) equals Danny time
 
sw87 said:
It takes Danny three time as long to mow a field as it takes his fauther. Together they can mow the same field in four hours. how long doesit take Danny working alone?

I know what they are asking for but I don't know how to put it into a equation.

I know it takes 4hrs together
3(fathers time) equals Danny time
Click to expand...

Let the time required by Danny (to mow the lawn) = D hrs

then

the time required by the Father (to mow the lawn) = D/3 hrs

then

1/D + 1/(D/3) = 1/4

Now solve for "D".....
 
I tried finding the LCD but I think i did something wrong.

4D(D/3)
i multiplied it by everything.
4(D/3)+4D=(D/3)D
Now i don't know what to do.
 
sw87 said:
I tried finding the LCD but I think i did something wrong.

4D(D/3)
i multiplied it by everything.
4(D/3)+4D=(D/3)D
Now i don't know what to do.
Click to expand...

first simplify the equation

1/D + 1/(D/3) = 1/4

1/D + 3/D = 1/4

Before starting with LCD of the "whole thing" - look at the equation a bit - see if further simplification jumps out.
 
thank you after that things made more sense. LCD is 4D

4+12=D
16=D
It takes Danny 16 hours to mow the field
 
sw87 said:
It takes Danny three time as long to mow a field as it takes his fauther. Together they can mow the same field in for hours. how long doesit take Danny working alone?

I know what they are asking for but I don't know how to put it into a equation.

I know it takes 4hrs together
3(fathers time) equals Danny time
Click to expand...

To illustrate a general type of this problem:
If it takes one person 5 hours to paint a room and another person 3 hours, how long will it take to paint the room working together?

Method 1:

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

Note - Generally speaking (if the derivation is not specifically required), if it takes one person A units of time and another person B units of time to complete a specific task working alone, the time it takes them both to complete the task working together is T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.

Given T = AB/(A+B) or T = DF/(D+F) where D = Danny and F = father
Then, 4 = 3F(F)/(3F+F) = 3F^2/4F
3F^2 = 16F
3F = 16
F = 16/3
D = 16

CHecking:
Combined time
T = 16(16/3)/[16+(16/3)]
T = (16^2/3)/(64/3)
T = (16^2)3/3(64) = 4 hours.
 
\(\displaystyle The \ above \ problem \ is \ known \ as \ a \ work \ problem.\)

\(\displaystyle Rate \ of \ working \ X \ Time \ = \ Fraction \ of \ work \ done.\)

\(\displaystyle Let \ father's \ rate \ = \ 1/x, \ then \ son's \ rate \ = \ 1/3x.\)

\(\displaystyle In \ other \ words \ if \ x \ = \ 2, \ then \ it \ would \ take \ the \ father \ 2 \ hrs \ to \ mow \ the \ field \ alone\)

\(\displaystyle while \ it \ would \ take \ the \ son \ 6 \ hrs. \ to \ mow \ the \ same \ field \ alone.\)

\(\displaystyle Now, \ together, \ they \ can \ mow \ the \ field \ in \ 4 \ hrs., \ ergo,\)

\(\displaystyle 4/x+4/3x \ = \ 1 \ \implies \ x \ = \ 16/3, \ hence \ it \ would \ take \ the \ son \ 16 \ hrs. \ to \ mow \ the \ field\)

\(\displaystyle alone \ and \ it \ would \ take \ the \ father \ 5\frac{1}{3} \ hrs. \ alone.\)
 
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