Lesson Solving rate of work problem by reducing to a system of linear equations

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<H2>Solving rate of work problem by reducing to a system of linear equations</H2>

<H3>Problem 1</H3>A piece of work can be done by &nbsp;6&nbsp; men and &nbsp;5&nbsp; women in &nbsp;6&nbsp; days or &nbsp;3&nbsp; men and &nbsp;4&nbsp; women in &nbsp;10&nbsp; days.
In how many days can it be done by &nbsp;9&nbsp; men and &nbsp;15&nbsp; women?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Assume that all the men have the same qualification and, therefore, the same rate of work. &nbsp;Similarly, all the women have the same rate of work, &nbsp;perhaps,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;different from &nbsp;(or distinct of)&nbsp; that of the men.

<B>Solution</B>

Let &nbsp;<B>m</B>&nbsp; be the the man's rate of work and &nbsp;<B>w</B>&nbsp; be the woman's rate of work.

Then we have the linear system of two equations in two unknowns, &nbsp;in accordance with the given data:

{{{system(6*(6m + 5w) = 1,
10*(3m + 4w) = 1)}}},

{{{system(36m + 30w = 1,
30m + 40w = 1)}}}.

To solve it, &nbsp;subtract the second equation of the last system from the first equation. &nbsp;You will get

6m = 10w, &nbsp;&nbsp;&nbsp;&nbsp;or &nbsp;&nbsp;&nbsp;&nbsp;w = {{{6/10}}}{{{m}}} = {{{3/5}}}{{{m}}}.

Next, &nbsp;substitute it into the first equation of the system, &nbsp;and you will get

w = {{{1/90}}}, &nbsp;&nbsp;and then &nbsp;&nbsp;m = {{{1/54}}}.

Very good. &nbsp;Now, &nbsp;calculate the value of the expression &nbsp;&nbsp;9m + 15w, &nbsp;&nbsp;which is the subject of the problem's question. &nbsp;It is

9m + 15w = {{{9}}}.{{{1/54}}} + {{{15}}}.{{{1/90}}} = {{{9/54}}} + {{{15/90}}} = {{{1/6}}} + {{{1/6}}} = {{{1/3}}}.

Hence, &nbsp;9&nbsp; men and &nbsp;15&nbsp; women will complete the work in &nbsp;3&nbsp; days.

<H3>Problem 2</H3>8 &nbsp;women and &nbsp;12&nbsp; girls can paint a large mural in &nbsp;10&nbsp; hours. &nbsp;6 &nbsp;women and &nbsp;8 &nbsp;girls can paint the same mural in &nbsp;14 &nbsp;hours.

How long it would take to paint the mural one woman? &nbsp;How long it would take to paint the mural one girl?

<B>Solution</B>

Let &nbsp;w = number of hours it would take to paint the mural one woman.
Let &nbsp;g = number of hours it would take to paint the mural one girl.

Then one woman paints &nbsp;{{{1/w}}} &nbsp;of the mural per hour. &nbsp;8 &nbsp;women paint &nbsp;{{{8/w}}} &nbsp;of the mural per one hour.

Next, &nbsp;one girl paints &nbsp;{{{1/g}}} &nbsp;of the mural per hour. &nbsp;12 &nbsp;girls paint &nbsp;{{{12/g}}} &nbsp;of the mural per one hour.

In this way you get the system of two equations

{{{8/w}}} + {{{12/g}}} = {{{1/10}}}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

{{{6/w}}} + {{{8/g}}} = {{{1/14}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

for two unknowns &nbsp;w &nbsp;and &nbsp;g. &nbsp;To solve it, &nbsp;multiply equation &nbsp;(1) &nbsp;by &nbsp;2 &nbsp;(both sides) &nbsp;and equation &nbsp;(2) &nbsp;by &nbsp;(-3), &nbsp;then add.

Thus you exclude the variable &nbsp;g and get a single equation for the variable &nbsp;w &nbsp;only:

{{{16/w - 18/w}}} = {{{2/10 - 3/14}}}, &nbsp;&nbsp;&nbsp;&nbsp;or

{{{-2/w}}} = {{{14/70 - 15/70}}} = {{{-1/70}}}.

It gives &nbsp;w = 140. &nbsp;Hence, it takes &nbsp;140 &nbsp;hours for 1 woman to paint the mural.

Now please complete the solution yourself.

<H3>Problem 3</H3>If &nbsp;5&nbsp; men and &nbsp;3&nbsp; boys can reap &nbsp;23&nbsp; acres in &nbsp;4&nbsp; days and &nbsp;3&nbsp; men and &nbsp;2&nbsp; boys can reap 7&nbsp; acres in &nbsp;2&nbsp; days,

how many boys must assist &nbsp;7&nbsp; men in order that they may reap &nbsp;45&nbsp; acres in &nbsp;6&nbsp; days?

<B>Solution</B>

Let &nbsp;m&nbsp; be &nbsp;(an unknown)&nbsp; rate of work for one man, measured in &nbsp;{{{acres/day}}}, 
and let &nbsp;b&nbsp; be &nbsp;(an unknown)&nbsp; rate of work for one boy, &nbsp;measured in same units.

Then we have a system of two linear equations in two unknowns &nbsp;m&nbsp; and &nbsp;b

4*(5m + 3b) = 23, &nbsp;&nbsp;&nbsp;&nbsp;(1)
2*(3m + 2b) = &nbsp;&nbsp;7. &nbsp;&nbsp;&nbsp;&nbsp;(2)

20m + 12b = 23,     &nbsp;&nbsp;&nbsp;&nbsp;(1')
&nbsp;&nbsp;6m + &nbsp;&nbsp;4b = &nbsp;7.     &nbsp;&nbsp;&nbsp;&nbsp;(2')

To solve it, &nbsp;multiply &nbsp;(2')&nbsp; by &nbsp;3&nbsp; and then distract it from &nbsp;(1'). &nbsp;You will get

20m - 18m = 23 - 3*7, &nbsp;&nbsp;&nbsp;&nbsp;or

2m = 2, &nbsp;&nbsp;hence, &nbsp;m = 1.

Substitute it into &nbsp;(2'), &nbsp;and you will get &nbsp;4b = 7 - 6 = 1, &nbsp;hence, &nbsp;b = {{{1/4}}}.

Thus you obtained that &nbsp;1&nbsp; man can reap &nbsp;1&nbsp; acr in &nbsp;1&nbsp; day, &nbsp;while &nbsp;1&nbsp; boy can reap &nbsp;{{{1/4}}} &nbsp;of acr in &nbsp;1&nbsp; day.

Then &nbsp;7&nbsp; men in &nbsp;6&nbsp; days can reap &nbsp;7*6*1 = 42  &nbsp;acres.

How many boys must assist them to reap remaining &nbsp;45-42 = 3 &nbsp;acres acres in &nbsp;6&nbsp; days?

Their number is &nbsp;{{{3/((1/4)*6)}}} = {{{4/2}}} = 2.

<B>Answer</B>. &nbsp;Two boys.

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