Q14. - Algebra 2

Question

In an election, a total of 4000 votes were cast for three candidates, A, B and C. Candidate C received 800 votes. If candidate B received more votes than candidate C and candidate A received more votes than candidate B, what is the least number of votes that candidate A could have received?

Step-by-Step Solution

Answer

The least number of votes that candidate A could have received are 1601.

Step-1 – Define the variables

Three candidates A, B and C stand in an election. Total number of votes casted are 4000.

Candidate C received 800 votes.

Let candidate A received x votes and candidate received y votes.

Step-2 – Frame the equations

Total number of votes casted are 4000.

$\begin{array}{c}x+y+800=4000\\ x+y=4000-800\\ x+y=3200\end{array}$

According to the question candidate B received more votes than candidate C.

Mathematically, this is expressed below.

$y>800$

Also, candidate A received more votes than candidate B.

Mathematically, this is expressed below.

$x>y$

Step-3 – Solve the equations

The equations obtained from the word problem are provided below.

$\begin{array}{c}x+y=3200\\ y>800\\ x>y\end{array}$

Suppose both candidates A and B receive equal votes. So each gets 1600 votes. But inequalities are not satisfied in this case.

Let candidate B received 1599 votes.

So, the value of y is 1599.

Now, substitute y as 1599 in the equation $x+y=3200$ and solve for x.

$\begin{array}{c}x+1599=3200\\ x=3200-1599\\ x=1601\end{array}$

Inequalities and all conditions are satisfied.

Therefore, the least number of votes that candidate A could have received are 1601.