1. If \(a \propto b\), \(b \propto \cfrac{1}{c}\), and \(c \propto d\), then what will be the proportional relationship between \(a\) and \(d\)?

2. If \(a \propto \frac{1}{b}\) and \(b \propto \frac{1}{c}\), then which of the following is correct?

(a) \(a \propto c\) (b) \(a \propto \cfrac{1}{c}\) (c) \(a^2 \propto c\) (d) \(a \propto c^2\)

3. If \(a \propto (b + c)\), \(b \propto (c + a)\), and \(c \propto (a + b)\), and \(k, l, m\) are non-zero distinct constants respectively, then what is the value of \[ \frac{k}{k+1} + \frac{l}{l+1} + \frac{m}{m+1} \, ? \]

(a) 1 (b) 3 (c) 5 (d) 7

4. If \(a \propto b+c\), \(b \propto c+a\), and \(c \propto a+b\), and \(l, m, n\) are three distinct constants, prove that \(\cfrac{l}{l+1}+\cfrac{m}{m+1}+\cfrac{n}{n+1}=1\).

5. If \(a \propto b\), and \(b \propto c\), then show that \(a^3 b^3 + b^3 c^3 + c^3 a^3 \propto abc(a^3 + b^3 + c^3)\).

6. If \(a \propto b\) and \(b \propto c\), then show that \[ a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3) \]

7. If \(a \propto b^3\) and \(c \propto \sqrt{a}\), then which of the following is correct?

(a) \(b \propto c^2\) (b) \(b \propto c^3\) (c) \(b^3 \propto c^3\) (d) \(b^3 \propto c^2\)

8. If \(a \propto b\) and \(b \propto c\), then prove that \(a + b + c \propto 5abc\).

9. If \(a \propto b\) and when \(a = 2\), \(b = 14\), then what is the value of \(b\) when \(a = 5\)?

10. If \(x \propto yz\), \(y \propto ab^2\), and \(z \propto \cfrac{b}{a}\), then \(x \propto\) _____ (express in terms of \(a\) and \(b\)).

11. \(a \propto b^2\) and \(1+b \propto 6\), and if \(a = 1\) when \(c = 9\) and \(b = 5\), determine the value of \(c\).

12. If \(a\propto b\) and \(b\propto c\), then show that \((pa+qb+rc) \propto (l\sqrt{bc}+m\sqrt{ca}+n\sqrt{ab})\).

13. If \(a \propto b\) and \(b \propto c\), then prove that \(a^3 + b^3 + c^3 \propto 3abc\)

14. If \(a \propto b\) and \(b \propto c\), then show that \[a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3)\]

15. If \(a \propto b\) and \(b \propto c\), then prove that \(a^3 + b^3 + c^3 \propto 3abc\).

16. Given that \(a \propto b\) and \(b \propto c\), show that \(a^3 + b^3 + c^3 \propto 5abc\).

17. \(A \propto B\); when \(A = 2\), then \(B = 14\). Find the value of \(B\) when \(A = 5\).

18. Given: \(a \propto b\) and \(b \propto c\) Show that: \(a^3 + b^3 + c^3 \propto abc\)

19. If \(a\propto b\) and \(b\propto c\), prove that: \[ a^2+b^2+c^2 \propto ab+bc+ca \]

20. If \(b\propto a^3\) and \(a\) increases in the ratio \(2:3\), determine the ratio in which \(b\) will increase.

21. If \(a \propto b\) and \(b \propto c\), prove that \(a^3 + b^3 + c^3 \propto 3abc\). Let me know if you'd like this proven step by step. I'm here for it.

22. If \(a \propto b\) and \(b \propto c\), prove that \(a^3 + b^3 + c^3 \propto 3abc\).

23. If \(a^2 + b^2\propto ab\), then prove that \(a + b\propto a - b\).

24. If \(a^2 + b^2 \propto ab\), then prove that \(a + b \propto a - b\).

25. If \(a^2 + b^2 \propto 5ab\), then **show that** \(a + b \propto a - b\).