A ratio is not two numbers; it is a count of equal parts. Writing 3:5 as "3k and 5k" turns almost every question in this topic into one linear equation, and it is the single habit that makes the topic easy.
The questions that look hardest - a ratio that changes when something is added, or three ratios that have to be combined - are the ones where this helps most.
Try each one before opening the answer. A solution you have read is not a solution you can produce under time pressure, which is the only kind that counts.
Q1 Two numbers are in the ratio 3 : 5. Adding 10 to each makes the ratio 5 : 7. What are the numbers?
15 and 25.
Write them as 3k and 5k. Then (3k + 10)/(5k + 10) = 5/7, so 7(3k + 10) = 5(5k + 10), giving 21k + 70 = 25k + 50 and k = 5. The numbers are 15 and 25, and adding 10 gives 25 : 35, which is 5 : 7.
Q2 A sum of 7800 is divided among A, B and C in the ratio 1/2 : 1/3 : 1/4. How much does A get?
3600.
Fractional ratios have to be cleared first. Multiply every term by the LCM of the denominators, 12, to get 6 : 4 : 3. The parts total 13, so A gets 6/13 × 7800 = 3600. Splitting in the ratio 1 : 2 : 3 - reading the fractions backwards - is the common mistake.
Q3 A invests 20,000 for 6 months and B invests 15,000 for 12 months. They make a profit of 12,000. What is B’s share?
7200.
Profit divides by capital × time, not by capital. A: 20,000 × 6 = 120,000. B: 15,000 × 12 = 180,000. That is 2 : 3, so the parts total 5 and B gets 3/5 × 12,000 = 7200. Note that B put in less money and still takes the larger share - going by capital alone would have given B 3/7 of the profit, about 5143, and put the two of them the wrong way round.