Chapter 3.2, Problem 66PPE

To determine

To calculate: The possible numbers of additional senators who must vote in favor of the treaty in order to ratify it.

Answer to Problem 66PPE

The possible numbers of additional senators who must vote in favor of the treaty in order to ratify it are any natural numbers from 19 to 52.

Explanation of Solution

Given information: The U.S. Senate is composed of 2 senators from each of the 50 states. In order for a treaty to be ratified, at least two thirds of the senators present must approve the treaty. All senators are present and 48 of them have voted in favor of a treaty.

Calculation:

Let the possible numbers of additional senators who must vote in favor of the treaty in order to ratify it be $\text{'}x\text{'}$

The total numbers of senators $=100$

The least numbers of senators needed to ratify the treaty $=\frac{2}{3}\times 100=\mathrm{66.666...}\approx 67$

The numbers of senators already voted in favor of treaty $=48$

The least numbers of senators needed in addition to 48 senators to vote in favor of treaty $=67-48=19$

Also, maximum number of senators who can vote in favor of the treaty cannot exceed 100.

Therefore, the number of the rest senators other than 48 who van vote in favor $=100-48=52$

Hence, $19\le x\le 52$