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.
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11. Musa sold a bag at sh. 160,000 making a profit of sh. 20,000. Find his cost price. Musa aliuza begi kwa sh. 160,000 akapata faida ya sh. 20,000. Tafuta bei yake ya kununulia. Step 1: Tambua bei ya kuuza na faida. Bei ya kuuza (Selling Price) = Sh. 160,000 Faida (Profit) = Sh. 20,000 Step 2: Tumia fomula ya bei ya kununulia. Bei ya kununulia (Cost Price) = Bei ya kuuza - Faida Bei\ ya\ kununulia = 160,000 - 20,000 Bei\ ya\ kununulia = 140,000 Bei ya kununulia ni Sh.\ 140,000. 12. Solve for p. 3p 5 7 Tafuta thamani ya p. Step 1: Tafuta namba ambayo ikizidishwa na 3, inatoa salio la 5 unapogawanya kwa 7. Tunaweza kujaribu namba tofauti za p kuanzia 0 hadi 6 (kwa sababu tunafanya kazi modulo 7). Ikiwa p=0, 3 × 0 = 0 5 7 Ikiwa p=1, 3 × 1 = 3 5 7 Ikiwa p=2, 3 × 2 = 6 5 7 Ikiwa p=3, 3 × 3 = 9 2 7 Ikiwa p=4, 3 × 4 = 12 5 7 (Hii ndiyo jibu sahihi) Step 2: Thibitisha jibu. 12 ÷ 7 = 1 na salio la 5. Hivyo, 3 × 4 5 7. Thamani ya p ni 4. 13. The probability of Sandra winning a computer game is 0.6. Find the probability of her losing the game. Uwezekano wa Sandra kushinda mchezo wa kompyuta ni 0.6. Tafuta uwezekano wa yeye kupoteza mchezo. Step 1: Tambua uwezekano wa kushinda. Uwezekano wa kushinda (P(W)) = 0.6 Step 2: Tumia fomula ya uwezekano wa kupoteza. Jumla ya uwezekano wote ni 1. Hivyo, uwezekano wa kupoteza (P(L)) ni 1 minus uwezekano wa kushinda. P(L) = 1 - P(W) P(L) = 1 - 0.6 P(L) = 0.4 Uwezekano wa Sandra kupoteza mchezo ni 0.4. 14. Using a ruler, a pencil and a pair of compasses only, construct an angle of 75^. Kwa kutumia rula, penseli na bikari pekee, chora pembe ya 75^. Step 1: Chora mstari wa msingi. Chora mstari ulionyooka AB. Step 2: Chora pembe ya 60^. Kutumia A kama kituo na radius yoyote, chora upinde unaokatiza AB kwenye C. Kutumia C kama kituo na radius ileile, chora upinde unaokatiza upinde wa kwanza kwenye D. Unganisha A na D. Pembe DAB = 60^. Step 3: Chora pembe ya 90^. Kutumia D kama kituo na radius ileile, chora upinde unaokatiza upinde wa kwanza kwenye E. Kutumia D na E kama vituo, chora pinde mbili zinazokatizana kwenye F. Unganisha A na F. Pembe FAB = 90^. Step 4: Gawanya pembe kati ya 60^ na 90^. Pembe kati ya mstari AD (60^) na AF (90^) ni 30^. Gawanya pembe hii ya 30^ mara mbili. Kutumia D na G (mahali ambapo AF inakatiza upinde wa kwanza) kama vituo, chora pinde mbili zinazokatizana kwenye H. Step 5: Chora pembe ya 75^. Unganisha A na H. Mstari AH utaunda pembe ya 60^ + 15^ = 75^ na mstari AB. Hivyo, HAB = 75^. 15. Solve for b: 3^b ÷ 3^3 = 27 Tafuta thamani ya b. Step 1: Tumia sheria za vipeo. Wakati wa kugawanya namba zenye 'base' sawa, tunatoa vipeo (exponents). 3^b-3 = 27 Step 2: Andika 27 kama kielelezo cha 3. Tunajua kuwa 3 × 3 × 3 = 27. Hivyo, 27 = 3^3. 3^b-3 = 3^3 Step 3: Linganisha vipeo. Kwa kuwa 'base' ni sawa, vipeo lazima viwe sawa. b-3 = 3 Step 4: Tafuta thamani ya b. Ongeza 3 pande zote mbili za mlinganyo. b = 3 + 3 b = 6 Thamani ya b ni 6. What's next?