Question

Donald is on a school baseball team and has 12 base hits in 98 at bats so far this year. He wants to raise his average to .210. How many of the next consecutive at bats need to be base hits to raise his batting average to .210? Write and solve an equation to describe this situation for the number of consecutive base hits.

Answer 1

The average is equal to the quotient between the number of base hits and the total times at bats.

`\bar{X}=\frac{nº\text{base hits}}{nº\text{bats}}`

You have to determine the number of bats (and base hits) he needs to raise his average to 0.210.

Let "x" represent the number of times at bats, and base hits needed to reach this average.

You can express the calculation needed as follows:

`0.210=(12+x)/(98+x)`

From this expression, you can determine the value of x:

-Multiply both sides by (98+x)

`\begin{gathered} 0.210(98+x)=(98+x)(12+x)/(98+x) \\ 0.210(98+x)=12+x \end{gathered}`

-Distribute the multiplication on the parentheses term:

`\begin{gathered} 0.210\cdot98+0.210\cdot x=12+x \\ 20.58+0.210x=12+x \end{gathered}`

-Pass x to the left side of the equation by applying the opposite operation to both sides of the equal sign:

`\begin{gathered} 20.58+0.210x-x=12+x-x \\ 20.58-0.79x=12 \end{gathered}`

-Repeat to pass 20.58 to the right side of the equation

`\begin{gathered} 20.58-20.58-0.79x=12-20.58 \\ -0.79x=-8.58 \end{gathered}`

-Divide both sides by -0.79

`\begin{gathered} (-0.79x)/(-0.79)=(-8.58)/(-0.79) \\ x=10.86\cong11 \end{gathered}`

He needs to have approximately 11 consecutive base hits to raise his average to 0.210