1. **State the problem:** A university academic council has 25 student representatives to be apportioned among 11 courses based on their student populations using Hamilton’s, Jefferson’s, Adam’s, and Webster’s methods. 2. **Given data:** | Course | Students | |--------|----------| | A | 1374 | | B | 2385 | | C | 5691 | | D | 4348 | | E | 1202 | | F | 245 | | G | 1263 | | H | 401 | | I | 471 | | J | 253 | | K | 790 | Total students $= 1374 + 2385 + 5691 + 4348 + 1202 + 245 + 1263 + 401 + 471 + 253 + 790 = 18,423$ 3. **Calculate the standard divisor (SD):** $$ SD = \frac{\text{Total students}}{\text{Total seats}} = \frac{18423}{25} = 736.92 $$ 4. **Calculate the standard quota (SQ) for each course:** $$ SQ_i = \frac{\text{Students in course } i}{SD} $$ | Course | Students | $SQ = \frac{Students}{736.92}$ | |--------|----------|-------------------------------| | A | 1374 | $\frac{1374}{736.92} \approx 1.865$ | | B | 2385 | $\frac{2385}{736.92} \approx 3.237$ | | C | 5691 | $\frac{5691}{736.92} \approx 7.722$ | | D | 4348 | $\frac{4348}{736.92} \approx 5.900$ | | E | 1202 | $\frac{1202}{736.92} \approx 1.631$ | | F | 245 | $\frac{245}{736.92} \approx 0.332$ | | G | 1263 | $\frac{1263}{736.92} \approx 1.714$ | | H | 401 | $\frac{401}{736.92} \approx 0.544$ | | I | 471 | $\frac{471}{736.92} \approx 0.639$ | | J | 253 | $\frac{253}{736.92} \approx 0.343$ | | K | 790 | $\frac{790}{736.92} \approx 1.072$ | 5. **Hamilton’s Method:** - Assign each course the lower quota (floor of SQ). - Calculate total assigned seats. - Distribute remaining seats to courses with largest fractional remainders. Lower quotas: | Course | SQ | Lower Quota | Fractional Part | |--------|-------|-------------|----------------| | A | 1.865 | 1 | 0.865 | | B | 3.237 | 3 | 0.237 | | C | 7.722 | 7 | 0.722 | | D | 5.900 | 5 | 0.900 | | E | 1.631 | 1 | 0.631 | | F | 0.332 | 0 | 0.332 | | G | 1.714 | 1 | 0.714 | | H | 0.544 | 0 | 0.544 | | I | 0.639 | 0 | 0.639 | | J | 0.343 | 0 | 0.343 | | K | 1.072 | 1 | 0.072 | Sum of lower quotas = $1+3+7+5+1+0+1+0+0+0+1=19$ Remaining seats = $25 - 19 = 6$ Assign 1 seat each to the 6 courses with largest fractional parts: D (0.900), A (0.865), C (0.722), G (0.714), E (0.631), I (0.639) Final Hamilton allocation: | Course | Seats | |--------|-------| | A | 2 | | B | 3 | | C | 8 | | D | 6 | | E | 2 | | F | 0 | | G | 2 | | H | 0 | | I | 1 | | J | 0 | | K | 1 | 6. **Jefferson’s Method:** - Use a divisor smaller than SD to lower quotas. - Trial and error to find divisor $d$ so that sum of lower quotas equals 25. Try $d=700$: Calculate $SQ_i = \frac{Students}{700}$ and take floor: | Course | $\frac{Students}{700}$ | Floor | |--------|-------------------------|-------| | A | 1.963 | 1 | | B | 3.407 | 3 | | C | 8.130 | 8 | | D | 6.211 | 6 | | E | 1.717 | 1 | | F | 0.350 | 0 | | G | 1.804 | 1 | | H | 0.573 | 0 | | I | 0.673 | 0 | | J | 0.361 | 0 | | K | 1.129 | 1 | Sum = 1+3+8+6+1+0+1+0+0+0+1 = 21 < 25 Try $d=680$: | Course | $\frac{Students}{680}$ | Floor | |--------|-------------------------|-------| | A | 2.021 | 2 | | B | 3.510 | 3 | | C | 8.365 | 8 | | D | 6.394 | 6 | | E | 1.768 | 1 | | F | 0.360 | 0 | | G | 1.858 | 1 | | H | 0.590 | 0 | | I | 0.693 | 0 | | J | 0.372 | 0 | | K | 1.162 | 1 | Sum = 2+3+8+6+1+0+1+0+0+0+1 = 22 < 25 Try $d=650$: | Course | $\frac{Students}{650}$ | Floor | |--------|-------------------------|-------| | A | 2.113 | 2 | | B | 3.669 | 3 | | C | 8.755 | 8 | | D | 6.689 | 6 | | E | 1.849 | 1 | | F | 0.377 | 0 | | G | 1.942 | 1 | | H | 0.616 | 0 | | I | 0.724 | 0 | | J | 0.389 | 0 | | K | 1.215 | 1 | Sum = 2+3+8+6+1+0+1+0+0+0+1 = 22 < 25 Try $d=600$: | Course | $\frac{Students}{600}$ | Floor | |--------|-------------------------|-------| | A | 2.29 | 2 | | B | 3.975 | 3 | | C | 9.485 | 9 | | D | 7.247 | 7 | | E | 2.003 | 2 | | F | 0.408 | 0 | | G | 2.105 | 2 | | H | 0.668 | 0 | | I | 0.785 | 0 | | J | 0.422 | 0 | | K | 1.317 | 1 | Sum = 2+3+9+7+2+0+2+0+0+0+1 = 26 > 25 Try $d=610$: | Course | $\frac{Students}{610}$ | Floor | |--------|-------------------------|-------| | A | 2.254 | 2 | | B | 3.911 | 3 | | C | 9.328 | 9 | | D | 7.131 | 7 | | E | 1.971 | 1 | | F | 0.402 | 0 | | G | 2.071 | 2 | | H | 0.657 | 0 | | I | 0.772 | 0 | | J | 0.415 | 0 | | K | 1.295 | 1 | Sum = 2+3+9+7+1+0+2+0+0+0+1 = 25 seats exactly. Jefferson’s allocation: | Course | Seats | |--------|-------| | A | 2 | | B | 3 | | C | 9 | | D | 7 | | E | 1 | | F | 0 | | G | 2 | | H | 0 | | I | 0 | | J | 0 | | K | 1 | 7. **Adam’s Method:** - Use a divisor larger than SD to increase quotas. - Round up quotas. - Trial and error to find divisor $d$ so that sum of upper quotas equals 25. Try $d=770$: Calculate $SQ_i = \frac{Students}{770}$ and round up: | Course | $\frac{Students}{770}$ | Ceiling | |--------|-------------------------|---------| | A | 1.784 | 2 | | B | 3.096 | 4 | | C | 7.389 | 8 | | D | 5.649 | 6 | | E | 1.561 | 2 | | F | 0.318 | 1 | | G | 1.641 | 2 | | H | 0.521 | 1 | | I | 0.612 | 1 | | J | 0.329 | 1 | | K | 1.026 | 2 | Sum = 2+4+8+6+2+1+2+1+1+1+2 = 30 > 25 Try $d=850$: | Course | $\frac{Students}{850}$ | Ceiling | |--------|-------------------------|---------| | A | 1.616 | 2 | | B | 2.806 | 3 | | C | 6.695 | 7 | | D | 5.115 | 6 | | E | 1.414 | 2 | | F | 0.288 | 1 | | G | 1.487 | 2 | | H | 0.471 | 1 | | I | 0.554 | 1 | | J | 0.298 | 1 | | K | 0.929 | 1 | Sum = 2+3+7+6+2+1+2+1+1+1+1 = 27 > 25 Try $d=900$: | Course | $\frac{Students}{900}$ | Ceiling | |--------|-------------------------|---------| | A | 1.527 | 2 | | B | 2.650 | 3 | | C | 6.323 | 7 | | D | 4.831 | 5 | | E | 1.335 | 2 | | F | 0.272 | 1 | | G | 1.403 | 2 | | H | 0.446 | 1 | | I | 0.523 | 1 | | J | 0.281 | 1 | | K | 0.878 | 1 | Sum = 2+3+7+5+2+1+2+1+1+1+1 = 26 > 25 Try $d=950$: | Course | $\frac{Students}{950}$ | Ceiling | |--------|-------------------------|---------| | A | 1.446 | 2 | | B | 2.511 | 3 | | C | 5.990 | 6 | | D | 4.574 | 5 | | E | 1.265 | 2 | | F | 0.258 | 1 | | G | 1.329 | 2 | | H | 0.422 | 1 | | I | 0.496 | 1 | | J | 0.266 | 1 | | K | 0.831 | 1 | Sum = 2+3+6+5+2+1+2+1+1+1+1 = 25 seats exactly. Adam’s allocation: | Course | Seats | |--------|-------| | A | 2 | | B | 3 | | C | 6 | | D | 5 | | E | 2 | | F | 1 | | G | 2 | | H | 1 | | I | 1 | | J | 1 | | K | 1 | 8. **Webster’s Method:** - Round quotas to nearest whole number. - Adjust divisor by trial and error to get total seats = 25. Try $d=740$: | Course | $\frac{Students}{740}$ | Rounded | |--------|-------------------------|---------| | A | 1.856 | 2 | | B | 3.222 | 3 | | C | 7.689 | 8 | | D | 5.873 | 6 | | E | 1.624 | 2 | | F | 0.331 | 0 | | G | 1.707 | 2 | | H | 0.542 | 1 | | I | 0.637 | 1 | | J | 0.342 | 0 | | K | 1.068 | 1 | Sum = 2+3+8+6+2+0+2+1+1+0+1 = 26 > 25 Try $d=750$: | Course | $\frac{Students}{750}$ | Rounded | |--------|-------------------------|---------| | A | 1.832 | 2 | | B | 3.180 | 3 | | C | 7.588 | 8 | | D | 5.797 | 6 | | E | 1.603 | 2 | | F | 0.327 | 0 | | G | 1.684 | 2 | | H | 0.535 | 1 | | I | 0.628 | 1 | | J | 0.337 | 0 | | K | 1.053 | 1 | Sum = 2+3+8+6+2+0+2+1+1+0+1 = 26 > 25 Try $d=770$ (already done in Adam’s method, sum 30), try $d=780$: | Course | $\frac{Students}{780}$ | Rounded | |--------|-------------------------|---------| | A | 1.762 | 2 | | B | 3.058 | 3 | | C | 7.298 | 7 | | D | 5.571 | 6 | | E | 1.541 | 2 | | F | 0.314 | 0 | | G | 1.620 | 2 | | H | 0.514 | 1 | | I | 0.604 | 1 | | J | 0.324 | 0 | | K | 1.013 | 1 | Sum = 2+3+7+6+2+0+2+1+1+0+1 = 25 seats exactly. Webster’s allocation: | Course | Seats | |--------|-------| | A | 2 | | B | 3 | | C | 7 | | D | 6 | | E | 2 | | F | 0 | | G | 2 | | H | 1 | | I | 1 | | J | 0 | | K | 1 | **Final summary:** | Course | Hamilton | Jefferson | Adam | Webster | |--------|----------|-----------|------|---------| | A | 2 | 2 | 2 | 2 | | B | 3 | 3 | 3 | 3 | | C | 8 | 9 | 6 | 7 | | D | 6 | 7 | 5 | 6 | | E | 2 | 1 | 2 | 2 | | F | 0 | 0 | 1 | 0 | | G | 2 | 2 | 2 | 2 | | H | 0 | 0 | 1 | 1 | | I | 1 | 0 | 1 | 1 | | J | 0 | 0 | 1 | 0 | | K | 1 | 1 | 1 | 1 | This completes the apportionment using the four methods for 25 representatives.