Question

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?

Answer 1

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`
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`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`
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`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`
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`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"`

Answer 2

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