Time And Work - Study Mode

[#126] A and B can do a job in 10 days and 5 days respectively. They worked together for two days, after which B was replaced by C and the work was finished in the next three days. How long will C alone take to finish 40% of the job?
Correct Answer

(D) 12 days

Explanation

Solution: Two days work of A and B = 3 × 2 = 6 Remaining work = 10 - 6 = 4 Now as per question $$eqalign{
& frac{4}{{1 + { ext{C}}}} = 3 cr
& frac{4}{3} = 1 + { ext{C}} cr
& { ext{C}} = frac{1}{3} cr} $$ So that 40% of total work done by $${ ext{C}} = frac{{10 imes 40\% }}{{frac{1}{3}}} = 12$$

[#127] A takes 10 days less than the time taken by B to finish a piece of work. If both A and B can do it in 12 days, then the time taken by B alone to finish the work is = ?
Correct Answer

(A) 30 days

Explanation

Solution: Let B can alone finish the work = x days So, A can alone finish the work = (x - 10) days Now, one day work of A = $$frac{1}{{{ ext{x}} - 10}}$$ and one day work of B = $$frac{1}{{ ext{x}}}$$ Now, given (A + B) can finish the work = 12 day So, one day work of (A + B) = $$frac{1}{{12}}$$ $$eqalign{
& Rightarrow frac{1}{{{ ext{x}} - 10}} + frac{1}{{ ext{x}}} = frac{1}{{12}} cr
& Rightarrow frac{{x + x - 10}}{{x imes left( {x - 10}
ight)}} = frac{1}{{12}} cr
& Rightarrow frac{{2x - 10}}{{{x^2} - 10x}} = frac{1}{{12}} cr
& Rightarrow 12left( {2x - 10}
ight) = {x^2} - 10x cr
& Rightarrow 24x - 120 = {x^2} - 10x cr
& Rightarrow {x^2} - 10x - 24x + 120 = 0 cr
& Rightarrow {x^2} - 34x + 120 = 0 cr
& Rightarrow {x^2} - 30x - 4x + 120 = 0 cr
& Rightarrow xleft( {x - 30}
ight) - 4left( {x - 30}
ight) = 0 cr
& Rightarrow left( {x - 30}
ight) imes left( {x - 4}
ight) = 0 cr
& Rightarrow x = 30,,4 cr} $$ if x = 4, then A alone can finish the work = 4 - 10 = -6, which is not possible. So, x = 30 Hence, B can alone finish the work = 30 days

[#128] Dinesh and Rakesh are working on an Assignment, Dinesh takes 6 hours to type 32 pages on a computer, while Rakesh takes 5 hours to type 40 pages. How much time will they take working together on two different computers to type an assignment of 110 page ?
Correct Answer

(C) 8 hours, 15 minutes

Explanation

Solution: $$eqalign{
& { ext{Dinesh's one hour work}} cr
& = frac{{32}}{6} cr
& = frac{{16}}{3}{ ext{ pages/hour}} cr
& { ext{Rakesh's one hour work}} cr
& = frac{{40}}{5} cr
& = 8{ ext{ pages/hour}} cr
& { ext{Dinesh's and Rakesh's one hour work}} cr
& = frac{{16}}{3} + 8 cr
& = frac{{40}}{3}{ ext{ pages/hour}} cr
& { ext{They will finish the work together}} cr
& frac{{{ ext{Total work}}}}{{{ ext{Efficiency}}}} cr
& = frac{{110}}{{frac{{40}}{3}}} cr
& = 8frac{1}{4} cr
& = { ext{8 hours, 15 minutes}} cr} $$

[#129] A can do as much work as B and C together can do. A and B can together do a piece of work in 9 hours 36 minutes and C can do it in 48 hours. The time in hours that B needs to do the work alone, is ?
Correct Answer

(B) 24 hours

Explanation

Solution: 9 hours 36 minutes $$eqalign{
& = 9 + frac{{36}}{{60}},{ ext{hours}} cr
& = 9frac{3}{5} = frac{{48}}{5}{ ext{hours}} cr} $$ (A + B)’s 1 hour’s work = $$frac{5}{{48}}$$ C’s 1 hour’s work = $$frac{1}{{48}}$$ (A + B + C)’s 1 hour’s work = $$frac{5}{{48}} + frac{1}{{48}}$$ xa0 = $$frac{1}{8}$$ . . . . . . .(i) A’s 1 hour’s work = (B + C)’s 1 hour’s work . . . . . . . . (ii) From equation (i) and (ii), 2 × (A’s 1 hour’s work) = $$frac{1}{8}$$ A’s 1 hour’s work = $$frac{1}{{16}}$$ ∴ B’s 1 hour’s work $$eqalign{
& = frac{5}{{48}} - frac{1}{{16}} cr
& = frac{{5 - 3}}{{48}} cr
& = frac{1}{{24}} cr} $$ ∴ B alone will finish the work in 24 hours.

[#130] If 5 men and 3 women can reap 18 acre of crop in 4 days, 3 men and 2 women can reap 22 acre of crop in 8 days, then how many men are required to join 21 women to reap 54 acre of crop in 6 days ?
Correct Answer

(A) 5

Explanation

Solution: Acreage reaped by 5 men and 3 women in 1 day $$eqalign{
& = frac{{18}}{4} cr
& = frac{9}{2} cr} $$ Acreage reaped by 3 men and 2 women in 1 day $$eqalign{
& = frac{{22}}{8} cr
& = frac{{11}}{4} cr} $$ Suppose 1 man can reap x acres in 1 day and 1 women can reap y acres in 1 day $$eqalign{
& herefore 5x + 3y = frac{9}{2} cr
& Rightarrow 10x + 6y = 9,.....{ ext{(i)}} cr
& 3x + 2y = frac{{11}}{4} cr
& Rightarrow 9x + 6y = frac{{33}}{4},.....{ ext{(ii)}} cr
& { ext{Subtracting (ii) from (i),}} cr
& { ext{We get}}:x = 9 - frac{{33}}{4} = frac{3}{4} cr
& { ext{Putting x}} = frac{3}{4}{ ext{ in (i), we get}} cr
& Rightarrow 6y = 9 - frac{{15}}{2} cr
& Rightarrow 6y = frac{3}{2} cr
& Rightarrow y = frac{1}{4} cr} $$ Acreage reaped by 21 women in 6 days $$eqalign{
& = left( {frac{1}{4} imes 21 imes 6}
ight) cr
& = frac{{63}}{2} cr} $$ Remaining acreage to be reaped $$eqalign{
& = left( {54 - frac{{63}}{2}}
ight) cr
& = frac{{45}}{2} cr} $$ Acreage reaped by 1 men in 6 days $$eqalign{
& = left( {frac{3}{4} imes 6}
ight) cr
& = frac{9}{2} cr} $$ In 6 days, $$frac{9}{2}$$ acre is reaped by 1 man ∴ In 6 days, $$frac{{45}}{2}$$ acre is reaped by $$eqalign{
& = left( {frac{2}{9} imes frac{{45}}{2}}
ight){ ext{men}} cr
& = 5{ ext{ men}} cr} $$