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.

Question

Asked on August 18, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-08-18T17:56:43.456Z

3.1.1: How many years ago did Sandile buy the car? Step 1: Ngola foromo ea ho theoha ha boleng. Foromo ea ho theoha ha boleng ka mokhoa oa ho fokotsa chelete ke: A = P(1 - i)^n 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. Step 2: Kenya boleng bo fanoeng ka har'a foromo. Re na le A = R79\,866.96, P = R180\,000, le i = 15\% = 0.15. 79\,866.96 = 180\,000(1 - 0.15)^n 79\,866.96 = 180\,000(0.85)^n Step 3: Rarolla bakeng sa n. Arola mahlakore ka bobeli ka 180\,000: (79\,866.96)/(180\,000) = (0.85)^n 0.443705333... = (0.85)^n Nka logarithm ea mahlakore ka bobeli ho rarolla bakeng sa n: (0.443705333...) = n (0.85) n = ((0.443705333...))/((0.85)) n ≈ 5 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? Step 1: Fumana nako ea letsete la Anil. 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. Step 2: Ngola foromo ea phaello e kopaneng. Foromo ea phaello e kopaneng ke: A = P(1 + (i)/(m))^n × m 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. Step 3: Kenya boleng le ho bala chelete e bokelletsoeng ea Anil. Re na le P = R49\,000, i = 10\% = 0.10, m = 4 (e kopanngoa kotara), le n = 5 lilemo. A = 49\,000(1 + (0.10)/(4))^5 × 4 A = 49\,000(1 + 0.025)^20 A = 49\,000(1.025)^20 A ≈ 49\,000(1.63861644) A ≈ 80\,292.21 Anil o bokelletse R80 292.21. Step 4: Bapisa chelete e bokelletsoeng ea Anil le boleng ba hona joale ba koloi ea Sandile. 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? Step 1: Bala sekhahla sa phaello sa khoeli le khoeli. 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: i = (0.1025)/(12) Step 2: Ngola foromo ea chelete e setseng ea kalimo. Chelete e setseng (P_k) ka mor'a litefo tse k e fanoa ke: P_k = P(1+i)^k - (x((1+i)^k - 1))/(i) 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. Step 3: Kenya boleng le ho bala chelete e setseng ka mor'a litefo tse 6. P = R800\,000 x = R7\,853.15 i = (0.1025)/(12) ≈ 0.0085416667 k = 6 P_6 = 800\,000(1 + (0.1025)/(12))^6 - (7\,853.15((1 + 0.1025)/(12))^6 - 1)(0.1025)/(12) P_6 = 800\,000(1.0085416667)^6 - (7\,853.15((1.0085416667)^6 - 1))/(0.0085416667) P_6 = 800\,000(1.0523038) - (7\,853.15(0.0523038))/(0.0085416667) P_6 = 841\,843.04 - (410.9069)/(0.0085416667) P_6 = 841\,843.04 - 48\,106.90 P_6 = 793\,736.14 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. Step 1: Bala chelete e setseng ka mor'a khoeli ea bo9 (ka mor'a litefo tse 3 tse fosahetseng). Chelete e setseng ka mor'a tefo ea bo6 e ne e le P_6 = R793\,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: P_9 = P_6(1 + (0.1025)/(12))^3 P_9 = 793\,736.14(1.0085416667)^3 P_9 = 793\,736.14(1.025816) P_9 ≈ 814\,270.81 Chelete e setseng ka mor'a likhoeli tse 9 ke R814 270.81. Step 2: Fumana palo e setseng ea litefo. Nako ea pele ea kalimo e ne e le lilemo tse 20, e leng 20 × 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. Step 3: Bala tefo e ncha ea khoeli le khoeli. 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 = x[1-(1+i)^-n]i. Ho rarolla bakeng sa x: x_new = (P_9 · i)/(1 - (1+i)^-n_rem) x_new = (814\,270.81 × 0.1025)/(12)1 - (1 + (0.1025)/(12))^-231 x_new = (814\,270.81 × 0.0085416667)/(1 - (1.0085416667)^-231) x_new = (6956.887)/(1 - 0.144889) x_new = (6956.887)/(0.855111) x_new ≈ 8135.69 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. Step 1: Hlalosa menyetla bakeng sa sehlopha ka seng. E re P(T_1) e be monyetla oa hore Sehlopha sa 1 se tla matha ka tlas'a nako e tšoanelehang. P(T_1) = (4)/(9) E re P(T_2) e be monyetla oa hore Sehlopha sa 2 se tla matha ka tlas'a nako e tšoanelehang. P(T_2) = (3)/(7) Step 2: Bala menyetla ea hore sehlopha ka seng se ke ke sa matha ka tlas'a nako e tšoanelehang. P(T_1') = 1 - P(T_1) = 1 - (4)/(9) = (5)/(9) P(T_2') = 1 - P(T_2) = 1 - (3)/(7) = (4)/(7) Step 3: Bala monyetla oa hore hantle sehlopha se le seng sa lihlopha tsa relay 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. P(hantle sehlopha se le seng) = P(T_1 le T_2') + P(T_1' le T_2) P(hantle sehlopha se le seng) = (P(T_1) × P(T_2')) + (P(T_1') × P(T_2)) P(hantle sehlopha se le seng) = ((4)/(9) × (4)/(7)) + ((5)/(9) × (3)/(7)) P(hantle sehlopha se le seng) = (16)/(63) + (15)/(63) P(hantle sehlopha se le seng) = (16 + 15)/(63) P(hantle sehlopha se le seng) = (31)/(63) Monyetla oa hore sehlopha se le seng sa lihlopha tsa relay se tla matha ka tlas'a nako e tšoanelehang ke (31)/(63). That's 2 down. 3 left today — send the next one.

Accepted Answer · 2026-08-18T17:56:43.456Z

3.1.1: How many years ago did Sandile buy the car? Step 1: Ngola foromo ea ho theoha ha boleng. Foromo ea ho theoha ha boleng ka mokhoa oa ho fokotsa chelete ke: A = P(1 - i)^n 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. Step 2: Kenya boleng bo fanoeng ka har'a foromo. Re na le A = R79\,866.96, P = R180\,000, le i = 15\% = 0.15. 79\,866.96 = 180\,000(1 - 0.15)^n 79\,866.96 = 180\,000(0.85)^n Step 3: Rarolla bakeng sa n. Arola mahlakore ka bobeli ka 180\,000: (79\,866.96)/(180\,000) = (0.85)^n 0.443705333... = (0.85)^n Nka logarithm ea mahlakore ka bobeli ho rarolla bakeng sa n: (0.443705333...) = n (0.85) n = ((0.443705333...))/((0.85)) n ≈ 5 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? Step 1: Fumana nako ea letsete la Anil. 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. Step 2: Ngola foromo ea phaello e kopaneng. Foromo ea phaello e kopaneng ke: A = P(1 + (i)/(m))^n × m 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. Step 3: Kenya boleng le ho bala chelete e bokelletsoeng ea Anil. Re na le P = R49\,000, i = 10\% = 0.10, m = 4 (e kopanngoa kotara), le n = 5 lilemo. A = 49\,000(1 + (0.10)/(4))^5 × 4 A = 49\,000(1 + 0.025)^20 A = 49\,000(1.025)^20 A ≈ 49\,000(1.63861644) A ≈ 80\,292.21 Anil o bokelletse R80 292.21. Step 4: Bapisa chelete e bokelletsoeng ea Anil le boleng ba hona joale ba koloi ea Sandile. 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? Step 1: Bala sekhahla sa phaello sa khoeli le khoeli. 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: i = (0.1025)/(12) Step 2: Ngola foromo ea chelete e setseng ea kalimo. Chelete e setseng (P_k) ka mor'a litefo tse k e fanoa ke: P_k = P(1+i)^k - (x((1+i)^k - 1))/(i) 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. Step 3: Kenya boleng le ho bala chelete e setseng ka mor'a litefo tse 6. P = R800\,000 x = R7\,853.15 i = (0.1025)/(12) ≈ 0.0085416667 k = 6 P_6 = 800\,000(1 + (0.1025)/(12))^6 - (7\,853.15((1 + 0.1025)/(12))^6 - 1)(0.1025)/(12) P_6 = 800\,000(1.0085416667)^6 - (7\,853.15((1.0085416667)^6 - 1))/(0.0085416667) P_6 = 800\,000(1.0523038) - (7\,853.15(0.0523038))/(0.0085416667) P_6 = 841\,843.04 - (410.9069)/(0.0085416667) P_6 = 841\,843.04 - 48\,106.90 P_6 = 793\,736.14 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. Step 1: Bala chelete e setseng ka mor'a khoeli ea bo9 (ka mor'a litefo tse 3 tse fosahetseng). Chelete e setseng ka mor'a tefo ea bo6 e ne e le P_6 = R793\,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: P_9 = P_6(1 + (0.1025)/(12))^3 P_9 = 793\,736.14(1.0085416667)^3 P_9 = 793\,736.14(1.025816) P_9 ≈ 814\,270.81 Chelete e setseng ka mor'a likhoeli tse 9 ke R814 270.81. Step 2: Fumana palo e setseng ea litefo. Nako ea pele ea kalimo e ne e le lilemo tse 20, e leng 20 × 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. Step 3: Bala tefo e ncha ea khoeli le khoeli. 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 = x[1-(1+i)^-n]i. Ho rarolla bakeng sa x: x_new = (P_9 · i)/(1 - (1+i)^-n_rem) x_new = (814\,270.81 × 0.1025)/(12)1 - (1 + (0.1025)/(12))^-231 x_new = (814\,270.81 × 0.0085416667)/(1 - (1.0085416667)^-231) x_new = (6956.887)/(1 - 0.144889) x_new = (6956.887)/(0.855111) x_new ≈ 8135.69 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. Step 1: Hlalosa menyetla bakeng sa sehlopha ka seng. E re P(T_1) e be monyetla oa hore Sehlopha sa 1 se tla matha ka tlas'a nako e tšoanelehang. P(T_1) = (4)/(9) E re P(T_2) e be monyetla oa hore Sehlopha sa 2 se tla matha ka tlas'a nako e tšoanelehang. P(T_2) = (3)/(7) Step 2: Bala menyetla ea hore sehlopha ka seng se ke ke sa matha ka tlas'a nako e tšoanelehang. P(T_1') = 1 - P(T_1) = 1 - (4)/(9) = (5)/(9) P(T_2') = 1 - P(T_2) = 1 - (3)/(7) = (4)/(7) Step 3: Bala monyetla oa hore hantle sehlopha se le seng sa lihlopha tsa relay 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. P(hantle sehlopha se le seng) = P(T_1 le T_2') + P(T_1' le T_2) P(hantle sehlopha se le seng) = (P(T_1) × P(T_2')) + (P(T_1') × P(T_2)) P(hantle sehlopha se le seng) = ((4)/(9) × (4)/(7)) + ((5)/(9) × (3)/(7)) P(hantle sehlopha se le seng) = (16)/(63) + (15)/(63) P(hantle sehlopha se le seng) = (16 + 15)/(63) P(hantle sehlopha se le seng) = (31)/(63) Monyetla oa hore sehlopha se le seng sa lihlopha tsa relay se tla matha ka tlas'a nako e tšoanelehang ke (31)/(63). That's 2 down. 3 left today — send the next one.

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.

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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.

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