The Business of Politics

Algebra Level 2

There are 100 Senators in Congress,of whom r r are Republican.

It costs $1 million to successfully bribe a Republican, and $11 million to successfully bribe a Senator who is not a Republican. To break a filibuster, 60 Senators need to be successfully bribed.

Charles and David have $100 million in bribe money. To successfully break a filibuster, what is the minimum value of r r required?


The answer is 56.

Denton Young by Denton Young , 47, USA

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2 solutions

Hung Woei Neoh
Jun 5, 2016

They need to bribe at least 60 60 Senators. Let the number of Republicans be r r , and the number of non-Republicans be s s . This gives:

r + s ≥ 60 r+s \geq 60

They have a total of $ 100 \$100 million. Given the costs listed above, we have an equation

r + 11 s = 100 s = 100 − r 11 r+11s = 100\\ s=\dfrac{100-r}{11}

Substitute this into the inequality:

r + 100 − r 11 ≥ 60 11 r + 100 − r ≥ 660 10 r ≥ 560 r ≥ 56 r+\dfrac{100-r}{11} \geq 60\\ 11r+100-r \geq 660\\ 10r \geq 560\\ r \geq 56

Minimum number of r = 56 r=\boxed{56}

0 Helpful 0 Interesting 0 Brilliant 1 Confused
Denton Young
Jun 5, 2016

To break the filibuster, 60 successful bribes must happen. If r r < 60, then the amount in millions of bribe money used is r + 11 ( 60 − r ) r + 11(60 - r)

Set this equal to 100: 660 − 10 r = 100 660 - 10r = 100

10 r = 560 10r = 560

r = 56 r = 56

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Good approach. Since you're asking for the max/min, it would be better to use inequality signs throughout. In this case, where the algebra yields an integer, it doesn't matter. But otherwise, we have to figure out if we want the smaller or larger value.

Nice problem, quite hard, thank you dude

Guytou Charles - 5 years ago

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