A farmer can plow the field in 8 days. After working for 3 days, his son joins him and together they plow the field in 3 more days. How many days will it require for the son to plow the field alone?

2 Answers
Apr 30, 2017

Solution 1 of 2
Go to solution 2 of 2 to see what it should look like.
12 days

#color(brown)("This is more a tutorial on how to handle units of measurement")#

Explanation:

Method uses rates or work measured in units of 1 field per day

Let the rate of work per day for the father be #W_f#
Let the rate of work per day for the sun be #W_s#

Let the total work done by the father be #T_f#
Let the total work done by the son be #T_s#

Let the unit identifying counts of a field be #f#
Let the unit identifying counts of days be #d#

Example #2d->2 days and 1/3f->1/3 of 1 field#
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#color(blue)("Determine the rate of work for the father")#

Modeling the number of days worked for completion of one field

#W_fxxT_(f1)=1f" "->" "W_fxx8d=1f#

So the amount of work done in 1 day #->W_f=1/8 f/d#

Where #f/d# represents 'fields per day'
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#color(blue)("Determine the total amount of work done by the father ")#

Works for 3 days without the son #" "...........->3dW_f#
Then works a further 3 days but with the son #ul(->3dW_f)larr" Add"#
Total work done by the father:#" ".......................6dW_f#

Do not forget the #d# is a unit of measurement. As is #f#
But #W_f=1/8f/d# giving:

#T_(f2)=6dW_f" "=" "6cancel(d) xx 1/8 f/(cancel(d))" "=" "3/4f#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the total amount of work done by the son")#

The father has done #3/4f# so the son did:
#1f-3/4f" "=" "1/4f#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Determine the work rate of the son")#

The son worked for 3 days so we have the model:

#3dW_s=1/4f#
Do not forget the #d# is a unit of measurement. As is #f#

#3dW_s=1/4f" "->" "#

divide both sides by #3d#

#(3d)/(3d) xx W_s=f/4xx1/(3d)" "->" "W_s=1/12 f/d#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Determine son's time for 1 field")#

#1f=W_s xx T_s " "->1f=(1f)/(12d) xx T_s#

Divide both sides by #(1f)/(12d)# giving:

#T_s=(12d)/(1cancel(f color(white)(.))) xx 1 cancel(fcolor(white)(.))#

#T_s=12d" "->" "12" days"#

Apr 30, 2017

Solution 2 of 2
This more efficient calculated is done without explaining what to do with units and how they work.

12 days

Explanation:

Fathers rate of work: 1 field in 8 days #=>1/8# fields per day

When working with his son the father completed #6 xx 1/8 = 3/4# fields

So his son completed #1-3/4=1/4# fields in 3 days

Thus the sons rate of work is #1/4-:3 =1/12# fields per day

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Thus the number of days (d) it would take for the son to do one field is:

#d->d xx 1/12=1 " "=>" " d=12# days