7.3 Ratio and Percentage in a Population/Group Split

Example 43: In a school, the ratio of boys to girls is 5 : 4. If 40% of the boys and 25% of the girls receive a scholarship, what percentage of the total students receive a scholarship? Let boys = 5x, girls = 4x, total = 9x. Scholarship boys = 0.40 × 5x = 2x Scholarship girls = 0.25 × 4x = 1x Total scholarship students = 3x Percentage = (3x/9x) × 100 = 33.33%.

Example 44: The ratio of the salaries of A and B was 4 : 5 last year. This year, A’s salary increased by 25% and B’s salary increased by 12%. If A’s new salary is ₹15,000, find B’s new salary. Let last year’s salaries = 4x, 5x. A’s new salary = 4x × 1.25 = 5x (since 1.25 = 5/4, this simplifies neatly). Given A’s new salary = 15,000, we get 5x = 15,000 → x = 3,000. B’s last year salary = 5x = 5 × 3,000 = 15,000. B’s new salary = 15,000 × 1.12 = ₹16,800.

Example 45: Two numbers are such that the ratio between them is 4 : 5. If each number is increased by 40, the ratio becomes 5 : 6. Find the numbers. Let numbers = 4x, 5x. (4x+40)/(5x+40) = 5/6. Cross-multiply: 6(4x+40) = 5(5x+40) → 24x + 240 = 25x + 200 → x = 40. Numbers = 4×40 = 160 and 5×40 = 200. (This is the “ratio + constant added” variant, closely related to the percentage-based variant above — both are solved the same way: assign ax, bx to the original ratio, apply the stated change, and cross-multiply the new ratio.)

Example 45A (chained percentage-to-ratio, additional): A’s salary is 40% more than B’s salary, and B’s salary is 20% less than C’s salary. If C’s salary is ₹25,000, find A’s salary. B’s salary = 25,000 × (100−20)/100 = 25,000 × 0.8 = ₹20,000. A’s salary = 20,000 × (100+40)/100 = 20,000 × 1.4 = ₹28,000.

Example 45B (reverse-solve original numbers from a ratio-after-addition, additional): Two numbers are in the ratio 5 : 7. If 15 is added to each number, the ratio becomes 3 : 4. Find the numbers. Let numbers = 5x, 7x. (5x+15)/(7x+15) = 3/4. Cross-multiply: 4(5x+15) = 3(7x+15) → 20x + 60 = 21x + 45 → x = 15. Numbers = 5×15 = 75 and 7×15 = 105. Check: (75+15)/(105+15) = 90/120 = 3/4. ✔