Question

1. If 6 typists typing 8 hours a day (at equal speed) take 20 days to complete a manuscript. If the work is to be completed in $12$ days, how many more typists should be employed if each typist works for $5$ hours a day?

2.If $2$ men and $8$ women can finish a piece of work in $15$ days whereas $8$ men and $16$ women can finish the same work in $6$ days then-
a.According to the amount of work done, $3$ men are equivalent to how many women?
b.Find the time taken by $9$ men and $4$women to complete the same work.

Answer 1

Hint: 1.First use the formula-
The total work will be =number of typists× number of hours ×number of days
Put the given values and calculate the work done when there are $6$ typists.
Then assume the total number of typists to be x then put the values given from the second statement in the formula and equate both the values of the equation. Solve the obtained equation for x then subtract the number of typists already employed to get the number of typists to be employed.
2.First we have to find the number of women equivalent to $3$ men so first we will count the women and men required for one day to complete the work according to the first statement of the question then equate both equations formed. Solve the equation to get the answer.
Secondly, we have to find the time taken by $9$ men and $4$women to complete the same work. So let us assume that one man’s one day’s work=x and one woman’s one day work =y then form two equations based on the statement and solve for x and y. Then put the value of x and y to calculate the work done by $9$ men and $4$women. They take its reciprocal to find the time.

Complete step-by-step answer:
(1)Given $6$ typists typing $8$ hours a day (at equal speed) take $20$ days to complete a manuscript.
We have to find the number of typists to be employed if each typist works for $5$ hours a day to complete the work in $12$ days.
When number of typists=$6$
The number of working hours a day=$8$ hours
And number of days to complete the work=$20$ days
$ \Rightarrow $ The total work will be =number of typists× number of hours ×number of days {because Work =time ×number of people doing the work}
Now on putting the given values we get-
 $ \Rightarrow $ The total work=$6 \times 8 \times 20$ --- (i)
Now let the number of typists be x then the number of working hours a day =$5$ hours
And the number of days to complete the work =$12$ days
$ \Rightarrow $ The total work will be =number of typists× number of hours ×number of days {because Work =time ×number of people doing the work}
On putting the given values, we get-
$ \Rightarrow $ The total work=$x \times 5 \times 12$ -- (ii)
Now since the work is the same then on equating eq. (i) and (ii), we get-
$ \Rightarrow x \times 5 \times 12 = 6 \times 8 \times 20$
On re-arranging, we get-
$ \Rightarrow x = \dfrac{{6 \times 8 \times 20}}{{5 \times 12}}$
On simplifying, we get-
$ \Rightarrow $ x=$\dfrac{{960}}{{60}}$
On division, we get-
$ \Rightarrow $ x=$16$
Since there are already $6$ typists employed then the number of typist to be employed=$16 - 6$
On solving, we get-
Answer-The number of typists to be employed=$10$
(2)
a. Given, $2$ men and $8$ women can finish a piece of work in $15$ days.
Then we can write,
In $15$ days, the number of men and women required=$2$ men +$8$ women
Then in one day, the number of men and women required=$15$ ($2$ men +$8$ women)
On solving, we get-
In one day, the number of men and women required= $30$ men +$120$ women-- (i)
Now, it is given that$8$ men and $16$ women can finish the same work in $6$ days.
So we can write,
In one day, the number of men and women required=$6$ ($8$ men +$16$ women)
On solving, we get-
In one day, the number of men and women required=$48$ men +$96$ women-- (ii)
We have to find the number of women equivalent to $3$ men according to the amount of work done.
On equating eq. (i) and (ii), we get-
$ \Rightarrow $ $30$ Men +$120$ women=$48$ men +$96$ women
On taking the value of number of men on one side and value of number of women on the other side, we get-
$ \Rightarrow $ $120$Women-$96$ women =$48$ men -$30$ men
On solving, we get-
$ \Rightarrow 24$ Women=$18$ men
Then $1$ men=$\dfrac{{24}}{{18}}$ women
On solving, we get-
$ \Rightarrow $ $1$ Men=$\dfrac{4}{3}$ women
Then $3$ men=$\dfrac{4}{3} \times 3$ women
On solving we get-
Answer-$3$ Men=$4$ women
b. Given, $2$ men and $8$ women can finish a piece of work in $15$ days. It is given that$8$ men and $16$ women can finish the same work in $6$ days.
Let one man’s one day’s work=x and one woman’s one day work =y
Then we get-
$ \Rightarrow 2x + 8y = \dfrac{1}{{15}}$ -- (i)
And $8x + 16y = \dfrac{1}{6}$ -- (ii)
On multiplying $2$ in eq. (i) and subtracting eq. (ii) from eq. (i), we get-
$ \Rightarrow - 4x = \dfrac{2}{{15}} - \dfrac{1}{6}$
On solving, we get-
$ \Rightarrow 4x = \dfrac{1}{6} - \dfrac{2}{{15}}$
On taking LCM we get,
$ \Rightarrow 4x = \dfrac{{5 - 4}}{{30}} = \dfrac{1}{{30}}$
On further solving, we get-
$ \Rightarrow x = \dfrac{1}{{120}}$
On substituting this value in eq. (i), we get-
$ \Rightarrow \dfrac{2}{{120}} + 8y = \dfrac{1}{{15}}$
On solving, we get-
$ \Rightarrow \dfrac{{2 + 960y}}{{120}} = \dfrac{1}{{15}}$
On further solving, we get-
$ \Rightarrow 2 + 960y = 8$
$ \Rightarrow 960y = 6$
$ \Rightarrow y = \dfrac{1}{160}$
The work done by $9$ men and $4$ women to complete the same work=$9x + 4y$
On putting value of x and y we get,
The work done by $9$ men and $4$ women to complete the same work=$\dfrac{9}{{120}} + \dfrac{4}{160} = \dfrac{1}{10}$
The time taken by $9$ men and $4$ women to complete the same work = 10 days.

Note: The student may mistake the value of x for the answer. Here remember that the x we assumed in (I) is the total number of typists needed to complete the work in $12$ days. Some of the typists are already employed from the total typists typing hence we subtract the number of typists already employed from the total number of typists.