1. If \(a:b=3:2\) and \(b:c=3:2\), then \((a+b):(b+c) =\)_____.

2. 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

3. In trapezium ABCD, AD is parallel to BC. A straight line parallel to BC intersects AB and DC at points P and Q respectively. If \( AP : PB = 2 : 1 \), then what is the ratio \( DQ : QC \)?

(a) 1:1 (b) 1:2 (c) 1:4 (d) 2:1

4. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?

(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

5. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

6. In triangle ABC, \(\angle C = 90^\circ\) and AC : BC = 3 : 4, then what is the value of cosec A?

(a) \(\cfrac{3}{4}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{5}{4}\) (d) \(\cfrac{3}{5}\)

7. 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?

8. If A:B = 2:3 and B:C = 4:5, then what is the ratio A:B:C?

9. If A : B = 3 : 2 and B : C = 3 : 5, then what is the ratio A : B : C?

10. If A : B = 2 : 3, B : C = 5 : 8, and C : D = 6 : 7, then what is the ratio A : D?

(a) 2:7 (b) 7:2 (c) 5:8 (d) 5:14

11. If \(a : b = 2 : 3\) and \(b : c = 4 : 5\), then what is the value of \(a^2 : b^2 : bc\)?

(a) 8:18:21 (b) 16:36:45 (c) 16:20:36 (d) 8:15:18

12. If \(a : b = 2 : 1\) and \(x : y = 3 : 4\), then what is the value of \((2ax - by) : (2by - ax)\)?

(a) 2:1 (b) 3:1 (c) 4:1 (d) 5:1

13. If \(x + y = z\) and \(2x - z = y\), then what is the value of \(x : y : z\)?

(a) 3:2:1 (b) 1:2:3 (c) 4:5:6 (d) 2:1:3

14. If \(a : b = 2 : 3\) and \(b : c = 2 : 3\), then what is \((a + b) : (b + c)\)?

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

16. Here’s the English translation: *If the mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

17. “If \(\cfrac{a}{b} = \cfrac{3}{2}\) and \(\cfrac{b}{c} = \cfrac{3}{2}\), then what is \((a + b) : (b + c)\)?”