During the baseball season so far this year, Pete has 55 hits in 198 at bats. How many consecutive hits does Pete need to raise his batting average to 0.350?
The correct answer and explanation is:
Pete currently has 55 hits in 198 at bats. His current batting average is calculated by dividing hits by at bats: Current average=55198≈0.2778\text{Current average} = \frac{55}{198} \approx 0.2778
He wants to raise his batting average to 0.350 by getting a certain number of consecutive hits without any additional outs. Let xx be the number of consecutive hits he needs.
After xx hits, his total hits will be 55+x55 + x and his total at bats will be 198+x198 + x. The new batting average will be: 55+x198+x=0.350\frac{55 + x}{198 + x} = 0.350
Multiply both sides by 198+x198 + x: 55+x=0.350(198+x)55 + x = 0.350 (198 + x)
Distribute on the right: 55+x=69.3+0.350×55 + x = 69.3 + 0.350x
Bring all terms involving xx to one side and constants to the other: x−0.350x=69.3−55x – 0.350x = 69.3 – 55 0.650x=14.30.650x = 14.3
Solve for xx: x=14.30.650≈22x = \frac{14.3}{0.650} \approx 22
Pete needs approximately 22 consecutive hits to raise his batting average to 0.350.
Explanation
A batting average is the ratio of hits to at bats. To increase a batting average, a player must increase the ratio by adding more hits than the number of at bats in a way that the new ratio exceeds the target average.
In this problem, Pete has 55 hits out of 198 at bats, giving a batting average below 0.350. By getting consecutive hits, both his hits and at bats increase equally. The question is how many additional hits are needed to make the ratio of total hits to total at bats equal to 0.350.
The approach is to set up an equation where the new average equals 0.350. The total hits are increased by the unknown xx, and the total at bats are also increased by xx. Solving the equation leads to the number of consecutive hits required.
The key insight is that since both numerator and denominator increase by xx, you solve for xx to reach the desired average. This type of problem is common in sports statistics to see how a streak of successful attempts can raise an average.
Thus, Pete needs to hit safely 22 times in a row to bring his batting average from about 0.278 to 0.350.