Question

1. the table shows the party affiliation of the 140 senators of the 115th congress. the ratio of republicans to democrats is 3:5. how many senators are democrats? independent?

party affiliation	number of senators
democrat
independent

Explanation:

Step1: Solve for number of Democrats

Let the number of Republicans be $5x$ and Democrats be $6x$. We know Republicans = 60, so:
$5x = 60$
$x = \frac{60}{5} = 12$
Number of Democrats: $6x = 6\times12 = 72$

Step2: Solve for number of Independents

Total senators = 140. Subtract Republicans and Democrats:
$140 - 60 - 72 = 8$

Step3: Solve Part B (band trumpet players)

First band: ratio trumpet:flute = $4:7$. Let trumpet players be $t_1$.
$\frac{t_1}{21} = \frac{4}{7}$
$t_1 = \frac{4\times21}{7} = 12$
Second band: ratio trumpet:flute = $5:6$. Let trumpet players be $t_2$.
$\frac{t_2}{21} = \frac{5}{6}$
$t_2 = \frac{5\times21}{6} = 17.5$ (18 if rounding, but 17.5 mathematically)
The second band has more trumpet players.

Step4: Solve Part B (science club)

First club: ratio chemistry:physics = $2:3$. Let chemistry students be $c_1$.
$\frac{c_1}{15} = \frac{2}{3}$
$c_1 = \frac{2\times15}{3} = 10$
Second club: ratio chemistry:physics = $4:5$. Let chemistry students be $c_2$.
$\frac{c_2}{15} = \frac{4}{5}$
$c_2 = \frac{4\times15}{5} = 12$
The second science club has more chemistry students.

Answer:

1. Senate problem:

• Number of Democrats: 72
• Number of Independents: 8

1. Band problem: The second band (ratio 5:6) has more trumpet players.
2. Science club problem: The second science club (ratio 4:5) has more chemistry students.