Problem: With 400 members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the re-vote was 1112 of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time?
Answer Choices:
A. 75
B. 60
C. 50
D. 45
E. 20
Solution:
Let y1,n1, respectively, represent the numbers voting for, against the bill originally, and let y2,n2, respectively, represent the numbers voting for, against the bill on the re-vote.
y1+n1∴2(y1+n1)∴4n1∴y2∴n1−n2=400,y2+n2=400,y2=1112n1,y2−n2=2(n1−y1)=2(y2+n2),2(−y1+n1)=y2−n2=3y2+n2=3⋅1112n1+n2∴n1=811n2=1112⋅811n2=23n2, so that23n2+n2=400,n2=160,n1=220=60,or sincey2=240andy1=180,y2−y1=60