Two independent relay teams want to qualify for the next Olympic Games. The probability that the two teams run under the qualifying time is 4/9 and 3/7 respectively. Calculate the probability that one of the relay teams will run under the qualifying time in their next race.

Foromo ea ho theoha ha boleng ka mokhoa oa ho fokotsa chelete ke:

moo $A$ e leng boleng ba hona joale, $P$ e leng theko ea pele, $i$ e leng sekhahla sa ho theoha ha boleng, 'me $n$ e le palo ea lilemo.

Re na le $A = \text{R}79\,866.96$, $P = \text{R}180\,000$, le $i = 15\% = 0.15$.

Arola mahlakore ka bobeli ka $180\,000$:

Nka logarithm ea mahlakore ka bobeli ho rarolla bakeng sa $n$:

Sandile o rekile koloi lilemo tse 5 tse fetileng.

3.1.2: Has Anil accumulated enough money in his savings account to buy Sandile's car now?

Ho tsoa ho 3.1.1, Sandile o rekile koloi lilemo tse 5 tse fetileng. Anil o ile a beha chelete "ka nako e tšoanang le eo Sandile a rekileng koloi ka eona". Kahoo, nako ea letsete la Anil ke $n = 5$ lilemo.

Foromo ea phaello e kopaneng ke:

moo $A$ e leng boleng ba nako e tlang, $P$ e leng chelete ea mantlha, $i$ e leng sekhahla sa phaello sa selemo le selemo, $m$ e leng palo ea linako tseo phaello e kopanngoang ka tsona ka selemo, 'me $n$ e le palo ea lilemo.

Re na le $P = \text{R}49\,000$, $i = 10\% = 0.10$, $m = 4$ (e kopanngoa kotara), le $n = 5$ lilemo.

Anil o bokelletse R80 292.21.

Boleng ba hona joale ba koloi ea Sandile ke R79 866.96.
Chelete e bokelletsoeng ea Anil ke R80 292.21.
Kaha R80 292.21 > R79 866.96, Anil o bokelletse chelete e lekaneng.
Karabo: E, Anil o bokelletse chelete e lekaneng.

3.2.1: How much did Jane owe immediately after making her 6th repayment?

Sekhahla sa phaello sa selemo le selemo ke $i_{annual} = 10.25\% = 0.1025$.
Phaello e kopanngoa khoeli le khoeli, kahoo sekhahla sa phaello sa khoeli le khoeli ke:

Chelete e setseng ($P_k$) ka mor'a litefo tse $k$ e fanoa ke:

moo $P$ e leng chelete ea pele ea kalimo, $x$ e leng tefo ea khoeli le khoeli, $i$ e leng sekhahla sa phaello sa khoeli le khoeli, 'me $k$ e le palo ea litefo tse entsoeng.

$P = \text{R}800\,000$
$x = \text{R}7\,853.15$
$i = \frac{0.1025}{12} \approx 0.0085416667$
$k = 6$

Jane o ne a kolota R793 736.14 hang ka mor'a ho etsa tefo ea hae ea bo6.

3.2.2: Calculate Jane's increased monthly payment in order to settle the loan in the original 20 years.

Chelete e setseng ka mor'a tefo ea bo6 e ne e le $P_6 = \text{R}793\,736.14$.
Jane o ile a fosa litefo tsa likhoeli tsa 7, 8, le 9. Nakong ena, phaello e ile ea bokellana holim'a chelete e setseng.
Chelete e setseng ka mor'a likhoeli tse 9 ($P_9$) e tla ba $P_6$ e kopanngoa bakeng sa likhoeli tse 3:

Chelete e setseng ka mor'a likhoeli tse 9 ke R814 270.81.

Nako ea pele ea kalimo e ne e le lilemo tse 20, e leng $20 \times 12 = 240$ likhoeli.
Jane o se a fetile likhoeli tse 9 (litefo tse 6 li entsoe, tse 3 li fosetsoe).
Palo e setseng ea litefo ke $n_{rem} = 240 - 9 = 231$ likhoeli.

Re hloka ho fumana tefo e ncha ea khoeli le khoeli ($x_{new}$) e tla lefa chelete e setseng $P_9$ ka holim'a likhoeli tse setseng tsa $n_{rem}$ ka sekhahla sa phaello sa khoeli le khoeli $i$.
Foromo ea boleng ba hona joale ba annuity ke $P_n = \frac{x[1-(1+i)^{-n}]}{i}$.
Ho rarolla bakeng sa $x$:

Tefo e eketsehileng ea khoeli le khoeli ea Jane e tla ba R8 135.69.

QUESTION 4: Calculate the probability that one of the relay teams will run under the qualifying time in their next race.

E re $P(T_1)$ e be monyetla oa hore Sehlopha sa 1 se tla matha ka tlas'a nako e tšoanelehang.

E re $P(T_2)$ e be monyetla oa hore Sehlopha sa 2 se tla matha ka tlas'a nako e tšoanelehang.

Sena se bolela hore Sehlopha sa 1 se atleha HAHOLO Sehlopha sa 2 se hloleha, KAPA Sehlopha sa 1 se hloleha HAHOLO Sehlopha sa 2 se atleha. Kaha liketsahalo li ikemetse, re ka atisa menyetla ea tsona.

Monyetla oa hore sehlopha se le seng sa lihlopha tsa relay se tla matha ka tlas'a nako e tšoanelehang ke $\boxed{\frac{31}{63}}$.