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The U.S. Senate has 100 members. After a certain election, there were 12 more Democrats than Republicans, with no other parties represented. How many members of each party were there in the Senate?

◻ Democrats

◻ Republicans

The U.S. Senate has $100$ members. After a certain election, there were $12$ more Democrats than Republicans, with no other parties represented. How many members of each party were there in the Senate? $\newline$ $\square$ Democrats $\newline$ $\square$ Republicans

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Find a value using two-variable equations: word problems

Full solution

Q. The U.S. Senate has

100

members. After a certain election, there were

12

more Democrats than Republicans, with no other parties represented. How many members of each party were there in the Senate?

\newline

\square

Democrats

\newline

\square

Republicans

Define Variables: Let's call the number of Democrats $D$ and the number of Republicans $R$ . We know that $D = R + 12$ because there are $12$ more Democrats than Republicans.
Total Senate Members: We also know that the total number of Senate members is $100$ . So, $D + R = 100$ .
Substitute and Solve: Now we have two equations: $D = R + 12$ and $D + R = 100$ . We can substitute the first equation into the second one to find $R$ .
Combine Like Terms: Substituting $D$ from the first equation into the second, we get $(R + 12) + R = 100$ .
Isolate Term with R: Combining like terms, we get $2R + 12 = 100$ .
Solve for R: Subtract $12$ from both sides to isolate the term with $R$ , we get $2R = 100 - 12$ , which is $2R = 88$ .
Find D: Divide both sides by $2$ to solve for R, we get $R = \frac{88}{2}$ , which is $R = 44$ .
Calculate D: Now that we have $R$ , we can find $D$ by plugging $R$ back into the first equation: $D = R + 12$ . So, $D = 44 + 12$ .
Check Total: Calculating $D$ gives us $D = 56$ . So, there are $56$ Democrats.
Check Total: Calculating $D$ gives us $D = 56$ . So, there are $56$ Democrats. We've found that there are $44$ Republicans and $56$ Democrats. Let's check if the total adds up to $100$ : $44 + 56 = 100$ .

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