1. 75% of A = 50% of B; Find the ratio A : B.

2. If \(\frac{2}{3}\) of A = 75% of B = 0.6 of C, then find the ratio A : B : C.

3. If \(a : b = 3 : 4\) and \(x : y = 5 : 7\), then find the value of the ratio \((3ax - by) : (4by - 7ax)\).

4. If \(a : b = 3 : 2\) and \(b : c = 3 : 2\), then find the value of the ratio \((a + b) : (b + c)\).

5. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

6. In triangle \( \triangle ABC \), AD is a median. Point E divides AD in the ratio 1:2. The extended line BE intersects AC at point F. If \( AC = 10 \) cm, find the length of \( AF \).

(a) 5 cm (b) 4 cm (c) 2 cm (d) None of the above

7. If \(a : b : c = 2 : 3 : 5\), then find the value of \(\frac{2a + 3b - 3c}{c}\).

(a) \(=-\cfrac{2}{5}\) (b) \(=-\cfrac{3}{5}\) (c) \(=\cfrac{2}{5}\) (d) \(=\cfrac{3}{5}\)

8. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

9. If \(a : b = 3 : 4\) and \(x : y = 5 : 7\), then find the value of \((3ax - by) : (4by - 7ax)\).

10. If \(a + b : \sqrt{ab} = 4 : 1\), then what is the ratio of \(a : b\)?

11. If \((3x - 2y) : (x + 3y) = 5 : 6\), then what is the value of the ratio \(x : y\)?

12. If \(a : 2 = b : 5 = c : 4\), then 50% of \(a\) = 20% of \(b\) = ______% of \(c\).

13. \(\triangle\)ABC ~ \(\triangle\)DEF; BC and EF are corresponding sides. If BE : EF = 1 : 3, then the ratio of the areas of \(\triangle\)ABC and \(\triangle\)DEF will be 1 : 27.

14. AOB is the diameter of a circle, and point C lies on the circumference. Given: AC = 3 cm and BC = 4 cm Then, find the length of AB.

15. If the ratio of the cost price to the selling price is 10:11, then the percentage profit is: Profit = Selling Price – Cost Price = 11 – 10 = 1 (in ratio units) Now, Percentage Profit = \(\frac{\text{Profit}}{\text{Cost Price}} \times 100 = \frac{1}{10} \times 100 = 10\%\) So, the translated sentence is: If the ratio of cost price to selling price is 10:11, then the profit percentage is 10%.

(a) 9 (b) 11 (c) \(10\frac{1}{9}\) (d) 10

16. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

17. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

18. If the similar triangles \(\triangle\)ABC and \(\triangle\)PQR have AB:PQ = 3:5, then the area ratio of \(\triangle\)ABC to \(\triangle\)PQR is—

(a) 9.25 (b) 25.9 (c) 3.5 (d) 5.3

19. In \(\triangle\)ABC, if DE \(\parallel\) BC and AD:DB = 3:2, then what is the ratio of DE:BC?

20. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1:r\), then show that \((r+1)^2ac = b^2r\).

21. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

22. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:2\), then prove that \(2b^2=9ac\).

23. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

24. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

25. If \(m + \cfrac{1}{m} = \sqrt{3}\), then find the simplest value of: (a) \(m^2 + \cfrac{1}{m^2}\), and (b) \(m^3 + \cfrac{1}{m^3}\).

26. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

27. If \(A : B = 6 : 7\) and \(B : C = 8 : 7\), then find the value of \(A : C\).

28. If \(A : B = 2 : 3\), \(B : C = 4 : 5\), and \(C : D = 6 : 7\), then find the value of \(A : D\).

29. If \(a : b = c : d = e : f\), then prove that each of these ratios is equal to \(\cfrac{5a - 7c - 13e}{5b - 7d - 13f}\)

30. If \(a\) is a positive number and \[ a : \cfrac{27}{64} = \cfrac{3}{4} : a, \] then find the value of \(a\).

(a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)