Rani goes to school from her house in 30 minutes. Raja takes 45 minutes in covering the same distance. Find the ratio between time taken by Rani and Raja.

### Concept & Simple Ratios A ratio compares two quantities of the same kind. To find the ratio between two times, simply place them in a fraction and simplify to its lowest terms. $$Ratio = \frac{\text{Time}_1}{\text{Time}_2}$$ ### Step-by-Step Solution 1. Time taken by Rani = 30 minutes. 2. Time taken by Raja = 45 minutes. 3. Form the ratio of Rani's time to Raja's time: $$Ratio = \frac{30}{45}$$ 4. Simplify the ratio by dividing the numerator and denominator by their greatest common divisor (15): $$Ratio = \frac{30 \div 15}{45 \div 15} = \frac{2}{3}$$ 5. Expressed as a ratio, this is 2 : 3. ### Exam Strategy & Shortcut Both numbers end in 5 or 0, making them multiples of 5, but quickly scanning for larger multiples reveals they are both part of the 15 times table ($15 \times 2 = 30$, $15 \times 3 = 45$). This gives the simplified ratio of 2:3 instantly. ### Common Pitfall Inverting the ratio by accident. The question asks for "Rani and Raja" in that specific order. Writing 3:2 (Raja to Rani) would be incorrect. ### Final Answer Therefore, the correct answer is **2 : 3**.