Molly paid an interest of sh. 150,000 after borrowing sh. 450,000 at 20% per annum. Calculate the time she took to pay the money.

Haya master, twende na maswali haya.

16. Molly paid an interest of sh. 150,000 after borrowing sh. 450,000 at 20% per annum. Calculate the time she took to pay the money.

• Hatua ya 1: Andika fomula ya riba rahisi na utambue vigezo. Fomula ya riba rahisi ni $I = PRT$, ambapo: $I$ = Riba (Interest) = sh. 150,000 $P$ = Kiasi kikuu (Principal) = sh. 450,000 $R$ = Kiwango cha riba (Rate) = 20% kwa mwaka = 0.20 $T$ = Muda (Time) = ?

• Hatua ya 2: Badilisha vigezo kwenye fomula na utatue kwa T.

  $$150,000 = 450,000 \times 0.20 \times T$$ $$150,000 = 90,000 \times T$$

  Gawanya pande zote mbili kwa 90,000:

  $$T = \frac{150,000}{90,000}$$ $$T = \frac{15}{9}$$

  Rahisisha sehemu:

  $$T = \frac{5}{3} \text{ miaka}$$ $$T = 1\frac{2}{3} \text{ miaka}$$

  Muda aliochukua kulipa pesa ni: $\boxed{\textbf{1}\frac{\textbf{2}}{\textbf{3}} \textbf{ miaka}}$

17. Peter is twice as old as James. The difference of their ages is 18 years. Find James' age.

• Hatua ya 1: Unda milinganyo kulingana na taarifa. Acha umri wa Peter uwe $P$ na umri wa James uwe $J$. "Peter ana umri mara mbili ya James":

  $$P = 2J \quad \text{(Mlinganyo 1)}$$

  "Tofauti ya umri wao ni miaka 18":

  $$P - J = 18 \quad \text{(Mlinganyo 2)}$$

• Hatua ya 2: Badilisha Mlinganyo 1 kwenye Mlinganyo 2 na utatue kwa J. Badilisha $P$ na $2J$ katika Mlinganyo 2:

  $$2J - J = 18$$ $$J = 18$$

  Umri wa James ni: $\boxed{\textbf{Miaka 18}}$

18. A motorist drove at 120km/ hour from town P to town Q for 80 minutes. Calculate the distance he covered.

• Hatua ya 1: Badilisha muda kutoka dakika kwenda masaa. Kasi imetolewa kwa km/saa, kwa hivyo muda lazima uwe katika masaa.

  $$80 \text{ dakika} = \frac{80}{60} \text{ masaa} = \frac{4}{3} \text{ masaa}$$

• Hatua ya 2: Tumia fomula ya umbali. Umbali = Kasi $\times$ Muda

  $$\text{Umbali} = 120 \text{ km/saa} \times \frac{4}{3} \text{ masaa}$$ $$\text{Umbali} = (120 \div 3) \times 4 \text{ km}$$ $$\text{Umbali} = 40 \times 4 \text{ km}$$ $$\text{Umbali} = 160 \text{ km}$$

  Umbali aliosafiri ni: $\boxed{\textbf{160 km}}$

19. A circular tank has a circumference of 1320cm. Find its diameter. (take $\pi$ as $\frac{22}{7}$)

• Hatua ya 1: Andika fomula ya mduara na utambue vigezo. Fomula ya mduara (circumference) ni $C = \pi d$, ambapo: $C$ = Mduara = 1320 cm $\pi = \frac{22}{7}$ $d$ = Kipenyo (diameter) = ?

• Hatua ya 2: Badilisha vigezo kwenye fomula na utatue kwa d.

  $$1320 = \frac{22}{7} \times d$$

  Ili kutafuta $d$, zidisha pande zote mbili kwa $\frac{7}{22}$:

  $$d = 1320 \times \frac{7}{22}$$ $$d = \frac{1320}{22} \times 7$$ $$d = 60 \times 7$$ $$d = 420 \text{ cm}$$

  Kipenyo cha tanki ni: $\boxed{\textbf{420 cm}}$

20. The prime factors of 72 is $2^3 \times p^2$. Work out the value of p.

• Hatua ya 1: Tafuta vigawo vikuu vya 72. Anza kugawanya 72 kwa nambari kuu ndogo zaidi:

  $$72 \div 2 = 36$$ $$36 \div 2 = 18$$ $$18 \div 2 = 9$$ $$9 \div 3 = 3$$ $$3 \div 3 = 1$$

  Kwa hiyo, vigawo vikuu vya 72 ni $2 \times 2 \times 2 \times 3 \times 3$.

• Hatua ya 2: Andika vigawo vikuu katika mfumo wa vipeo na ulinganishe.

  $$72 = 2^3 \times 3^2$$

  Tunapewa kuwa $72 = 2^3 \times p^2$. Kwa kulinganisha milinganyo miwili, tunaona kwamba $p^2 = 3^2$. Hii inamaanisha $p = 3$. Thamani ya $p$ ni: $\boxed{\textbf{3}}$