A farmer has two tractors P and Q. Tractor P, working alone, can plough a piece of land in 5 hours while tractor Q would take 1 2/3 hours less than tractor P. Determine the time the two tractors ploughing together, would take to complete the work.

Trekta P inachukua saa 5.
Trekta Q inachukua $1 \frac{2}{3}$ saa chache kuliko P.

Kiwango cha kazi cha trekta P ($R_P$) = $\frac{1}{\text{Muda wa P}} = \frac{1}{5}$ ya kazi kwa saa.
Kiwango cha kazi cha trekta Q ($R_Q$) = $\frac{1}{\text{Muda wa Q}} = \frac{1}{10/3} = \frac{3}{10}$ ya kazi kwa saa.

Kiwango cha pamoja ($R_{P+Q}$) = $R_P + R_Q = \frac{1}{5} + \frac{3}{10} = \frac{2}{10} + \frac{3}{10} = \frac{5}{10} = \frac{1}{2}$ ya kazi kwa saa.
Muda wa pamoja ($T_{P+Q}$) = $\frac{1}{\text{Kiwango cha pamoja}} = \frac{1}{1/2} = 2 \text{ saa}$.

a) Muda ambao matrekta mawili yangechukua kulima pamoja ni: $\boxed{\text{saa 2}}$

b)

Dakika 40 = $\frac{40}{60} = \frac{2}{3}$ saa.
Kazi iliyofanywa = Kiwango cha pamoja $\times$ Muda = $\frac{1}{2} \times \frac{2}{3} = \frac{1}{3}$ ya kazi.

Kazi iliyobaki = $1 - \frac{1}{3} = \frac{2}{3}$ ya kazi.
Muda wa Q kumaliza kazi iliyobaki = $\frac{\text{Kazi iliyobaki}}{\text{Kiwango cha Q}} = \frac{2/3}{3/10} = \frac{2}{3} \times \frac{10}{3} = \frac{20}{9}$ saa.

Jumla ya muda = Muda P na Q walifanya kazi pamoja + Muda Q ilifanya kazi peke yake

Ili kubadilisha sehemu ya saa kuwa dakika: $\frac{8}{9} \times 60 \approx 53.33$ dakika.

b) Jumla ya muda uliotumika kulima shamba siku hiyo ni: $\boxed{\frac{26}{9} \text{ saa (au saa 2 na dakika 53.33)}}$

c)

Muda wa pamoja = saa 1 na dakika 12 = $1 + \frac{12}{60} = 1 + \frac{1}{5} = \frac{6}{5}$ saa.
Kiwango cha pamoja ($R_{P+Q+R}$) = $\frac{1}{\text{Muda wa pamoja}} = \frac{1}{6/5} = \frac{5}{6}$ ya kazi kwa saa.

$R_{P+Q+R} = R_P + R_Q + R_R$
$\frac{5}{6} = \frac{1}{5} + \frac{3}{10} + R_R$
$\frac{5}{6} = \frac{2}{10} + \frac{3}{10} + R_R$
$\frac{5}{6} = \frac{5}{10} + R_R$
$\frac{5}{6} = \frac{1}{2} + R_R$
$R_R = \frac{5}{6} - \frac{1}{2} = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}$ ya kazi kwa saa.

Kazi iliyofanywa na R ($W_R$) = Kiwango cha R $\times$ Muda wa pamoja
$W_R = \frac{1}{3} \times \frac{6}{5} = \frac{6}{15} = \frac{2}{5}$ ya kazi.

Jumla ya thamani ya kazi = Ksh 20000.
Pesa kwa R = (Sehemu ya kazi iliyofanywa na R) $\times$ Jumla ya thamani ya kazi

c) Kiasi cha pesa alicholipwa mmiliki wa trekta R ni: $\boxed{\text{Ksh } 8000}$