Question

In the 75th Congress (1937−1939) there were in the Senate 75 Democrats, 17 Republicans, and 4 members of other parties. Suppose that a bill passed the Senate with 32 more votes in favor than against, with three times as many Democrats voting in favor as non-Democrats voting in favor, and 34 more Democrats voting in favor than Republicans voting in favor. If every member voted either for the bill or against it, how many Democrats, how many Republicans, and how many members of other parties voted in favor of the bill?

Answer 1

Answer:

48 Democrats

14 Republicans

2 members of other parties

Step-by-step explanation:

This can be solved by a system of equations.

I am going to say that:

x is the number of democrats that voted for the bill

ax is the number of democrats that voted against the bill

y is the number of republicans that voted for the bill

ay is the number of republicans that voted against the bill

z is the number of members of other parties that voted for the bill

az is the number of members of other parties that voted against the bill.

Now we use the informations the exercise gives us so we can build the system.

There were 75 democrats, so:

$x + ax = 75$

17 republicans, so:

$y + ay = 17$

4 members of other parties, so:

$z + az = 4$

A bill passed the Senate with 32 more votes in favor than against, so

$x + y + z = 32 + ax + ay + az$

Three times as many Democrats voting in favor as non-Democrats voting in favor

$x = 3(y + z)$

And 34 more Democrats voting in favor than Republicans voting in favor:

$x = y + 34$

The easiest equation to replace is this:

$x + y + z = 32 + ax + ay + az$

The first step is writing everything just with x, y and z, so:

$x + ax = 75$

$ax = 75 - x$

-------------

$y + ay = 17$

$ay = 17 - y$

------------

$z + az = 4$

$az = 4 - z$

So we have:

$x + y + z = 32 + ax + ay + az$

$x + y + z = 32 + 75 - x + 17 - y + 4 - z$

$2x + 2y + 2x = 128$

Dividing by 2

$x + y + z = 64$

We also have that

$x = y + 34$

So

$y + 34 + y + z = 64$

$2y + z = 30$

And

$x = 3(y + z)$

$3z = x - 3y$

$3z = y + 34 - 3y$

$3z = 34 - 2y$

$z = \frac{34 - 2y}{3}$7

So:

$2y + z = 30$

$2y +  \frac{34 - 2y}{3} = 30$

$6y + 34 - 2y = 90$

$4y = 56$

$y = 14$

14 republicans voted in favor of the bill.

$2y + z = 30$

$2(14) + z = 30$

$z = 2$

2 members of other parties voted in favor of the bill.

$x = y + 34$

$x = 14 + 34$

$x = 48$

48 democrats voted in favor of the bill