Math Problem Statement

The data represent your hypothetical walking times for each of the months January – October (10 months). Calculate your mean and your median walking time for the 10 months. Mean: 172+182+187+198+204+220+230+245+268+290 / 10 = Median: 204 + 220 / 2 = How many minutes would you have to walk in November so that your mean for the 11 months would be 2 percentage points more than your mean for the 10 months? 172, 182, 187, 198, 204, 220, 230, 245, 268, 290

Solution

Let's break down the steps to solve this problem.

1. Mean Calculation for 10 Months:

The formula for calculating the mean is:

Mean=Sum of all valuesNumber of values\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}

So for the 10 months:

Sum=172+182+187+198+204+220+230+245+268+290=2196\text{Sum} = 172 + 182 + 187 + 198 + 204 + 220 + 230 + 245 + 268 + 290 = 2196

Thus, the mean for the 10 months is:

Mean=219610=219.6 minutes\text{Mean} = \frac{2196}{10} = 219.6 \text{ minutes}

2. Median Calculation for 10 Months:

To find the median, we need to list the numbers in order and find the middle value(s).

The sorted times are already provided in ascending order:
172, 182, 187, 198, 204, 220, 230, 245, 268, 290

Since there are 10 numbers (an even number of values), the median is the average of the 5th and 6th numbers:

Median=204+2202=212 minutes\text{Median} = \frac{204 + 220}{2} = 212 \text{ minutes}

3. Mean Increase for 11 Months:

Now, you want to walk a certain number of minutes in November so that the mean for the 11 months is 2 percentage points higher than the mean for the 10 months.

The current mean for 10 months is 219.6.
An increase of 2 percentage points would make the new mean for 11 months:

New mean=219.6×1.02=223.992\text{New mean} = 219.6 \times 1.02 = 223.992

Let the number of minutes you walk in November be xx. The total time walked over 11 months will be:

Total for 11 months=2196+x\text{Total for 11 months} = 2196 + x

The mean for the 11 months is:

New mean=2196+x11\text{New mean} = \frac{2196 + x}{11}

Set this equal to 223.992:

2196+x11=223.992\frac{2196 + x}{11} = 223.992

Now solve for xx:

2196+x=223.992×11=2463.9122196 + x = 223.992 \times 11 = 2463.912 x=2463.9122196=267.912x = 2463.912 - 2196 = 267.912

Thus, you would need to walk approximately 268 minutes in November to achieve the desired increase in the mean.


Summary:

  • Mean for 10 months: 219.6 minutes
  • Median for 10 months: 212 minutes
  • Walking time for November: Approximately 268 minutes to increase the mean by 2 percentage points.

Do you have any questions or want more details?

Here are five related questions to consider:

  1. How would the calculations change if the walking time for October was higher?
  2. What if you wanted the mean to increase by 3 percentage points instead of 2?
  3. Can you find the mode of the walking times if some months had the same value?
  4. What would be the effect on the median if you walked 300 minutes in November?
  5. How can you calculate the range of your walking times?

Tip: When calculating percentages, always remember to express them as decimals before applying them in equations.

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Math Problem Analysis

Mathematical Concepts

Statistics
Mean
Median
Percentage Increase

Formulas

Mean = (Sum of values) / (Number of values)
Median = (Middle value(s) of a sorted list)
Percentage increase = (Original value) * (1 + Percentage increase)

Theorems

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Suitable Grade Level

Grades 8-10