a teacher has 3 hours to grade all the papers...
a teacher has 3 hours to grade all the papers submitted by the 35 students in her class. she gets through the first 5 papers in 30 minutes. how much faster does she have to work to grade the remaining papers in the allotted time? 30% 25% 20%
Answer
# Explanation:
## Step1: Calculate initial grading - rate
The teacher grades 5 papers in 30 minutes. So the initial rate $r_1=\frac{5}{30}=\frac{1}{6}$ papers per minute.
## Step2: Calculate remaining time and papers
The teacher has 3 hours (3 * 60 = 180 minutes) in total. She has already spent 30 minutes, so the remaining time $t = 180 - 30=150$ minutes. The total number of papers is 35, and she has graded 5, so the remaining papers $n = 35 - 5 = 30$.
## Step3: Calculate required grading - rate
The required rate $r_2$ to grade the remaining 30 papers in 150 minutes is $r_2=\frac{30}{150}=\frac{1}{5}$ papers per minute.
## Step4: Calculate the percentage increase in speed
The formula for percentage increase in speed is $\text{Percentage increase}=\frac{r_2 - r_1}{r_1}\times100\%$. Substitute $r_1=\frac{1}{6}$ and $r_2=\frac{1}{5}$ into the formula:
\[
\begin{align*}
\frac{\frac{1}{5}-\frac{1}{6}}{\frac{1}{6}}\times100\%&=\frac{\frac{6 - 5}{30}}{\frac{1}{6}}\times100\%\\
&=\frac{\frac{1}{30}}{\frac{1}{6}}\times100\%\\
&=\frac{1}{30}\times6\times100\%\\
& = 20\%
\end{align*}
\]
# Answer:
20%