Directions : The charts given below represents data of number of students of…

Concept

In a pie-chart problem, each sector's percentage of its chart's grand total gives an absolute count. When some sectors are unknown, use the constraint that all sectors sum to 100% (i.e. the parts add up to the given total), and translate any worded note (a given count, or a stated difference between two quantities) into a small equation. Solve for the unknowns first, then read off whatever combination the question asks for.

Given data

• Students chart total = 2500; known sectors: P = 20%, S = 25% (so Q + R = 55%).

• Boys chart total = 1200; known sectors: Q = 20%, R = 30% (so P + S = 50%).

• Note 1: girls in Q = 10.

• Note 2: boys in P − girls in P = 100, with boys > girls.

Application — step by step

1. Boys from the boys-chart: boys in Q = 20% of 1200 = 240; boys in R = 30% of 1200 = 360.

2. Students of known sectors: P = 20% of 2500 = 500; S = 25% of 2500 = 625.

3. Use Note 1 to fix Q's total: girls in Q = 10, and boys in Q = 240, so total students in Q = 240 + 10 = 250 (which is 10% of 2500).

4. Now R's total students = 2500 − (P + Q + S) = 2500 − (500 + 250 + 625) = 1125.

5. Use Note 2 for P: boys + girls = 500 and boys − girls = 100. Adding the two equations: 2 × boys = 600, so boys in P = 300 (and girls in P = 200).

6. Boys in S = total boys − boys in (P + Q + R) = 1200 − (300 + 240 + 360) = 300.

7. Girls in R = students in R − boys in R = 1125 − 360 = 765; girls in S = students in S − boys in S = 625 − 300 = 325.

Putting the asked combination together

Required ratio = (boys in Q + R) : (girls in R + S) = 600 : 1090. Divide both terms by their HCF, 10, to get 60 : 109.

Cross-check

• Total students used: 500 + 250 + 1125 + 625 = 2500 ✔

• Total boys used: 300 + 240 + 360 + 300 = 1200 ✔

• So the consistent, fully-determined answer is 60 : 109.