A) Solve; 1) P + 8 = 21 ii) y + y + 4 = 16 b) Given that g = 1/2 and n = 24, p = 2/3. What is pgn?

$$P = 13$$

Jibu ni $\boxed{P=13}$.

31. a) ii) Tatua: $y+y+4=16$.

$$2y + 4 - 4 = 16 - 4$$ $$2y = 12$$

$$\frac{2y}{2} = \frac{12}{2}$$ $$y = 6$$

Jibu ni $\boxed{y=6}$.

31. b) Ukipewa $g = \frac{1}{2}$ na $n = 24$, $p = \frac{2}{3}$. Je, $pgn$ ni nini?

Unaweza kurahisisha kabla ya kuzidisha.

$$pgn = \frac{2 \times 1 \times 24}{3 \times 2}$$ $$pgn = \frac{48}{6}$$

$$pgn = 8$$

Jibu ni $\boxed{8}$.

32. a) Tafuta idadi kubwa zaidi ya mipira inayotolewa kwa wachezaji 30 au 45 kwa usawa bila baki.

Hii inahitaji kutafuta Kigezo Kikubwa Zaidi cha Pamoja (HCF) cha 30 na 45.

Vigawo vikuu vya kawaida ni 3 na 5.

$$\text{HCF}(30, 45) = 3 \times 5 = 15$$

Idadi kubwa zaidi ya mipira ni $\boxed{\text{15}}$.

32. b) Andika 729 kwa vipeo vya 3.

$$729 \div 3 = 243$$ $$243 \div 3 = 81$$ $$81 \div 3 = 27$$ $$27 \div 3 = 9$$ $$9 \div 3 = 3$$ $$3 \div 3 = 1$$

3 ilitumika mara 6.

$$729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6$$

Jibu ni $\boxed{3^6}$.