1. If 50% of A = 60% of B = \(\frac{4}{5}\) of C, then find the ratio A : B : C.

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

3. 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

4. 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

5. If \(4x = 5y = 6z\), what is the value of the ratio \(x : y : z\)?

(a) 12:10:15 (b) 10:12:15 (c) 15:12:10 (d) 15:10:12

6. 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?

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

8. If 75% of A equals 40% of B, find the ratio \(A : B\).

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

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

11. \(\triangle\)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.

12. \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6} \] This is a mathematical equation involving rational expressions. Here's the English translation of the statement: **"The sum of the following three fractions equals one-sixth:"** - The first fraction is: one divided by \((x - 1)(x - 2)\) - The second fraction is: one divided by \((x - 2)(x - 3)\) - The third fraction is: one divided by \((x - 3)(x - 4)\) And their total is equal to \(\frac{1}{6}\).

13. If the ratio of cost price to selling price is 10:11, Profit percentage = \( \frac{11 - 10}{10} \times 100 = 10\% \)

(a) 10% (b) 5% (c) 1% (d) 2%

14. If the ratio of cost price to selling price is 12:13, Profit percentage = \( \frac{13 - 12}{12} \times 100 = \frac{1}{12} \times 100 \approx 8.33\% \)

(a) \(7\cfrac{1}{3}\)% (b) \(7\cfrac{2}{3}\)% (c) 8% (d) \(8\cfrac{1}{3}\)%

15. In triangle ABC, if line DE is parallel to side BC and the ratio AD:DB = 3:2, What is the ratio of DE:BC?

16. 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

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

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

19. 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

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

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

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

23. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:p\), prove that \(\cfrac{(p+1)^2}{p} = \cfrac{b^2}{ac}\).

24. If 75% of \(A\) equals 40% of \(B\), then the ratio \(A:B\) is –?

25. A straight line parallel to the side BC of triangle ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, determine the ratio PB:QC.

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

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

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

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