An environmental scientist recorded the numbers of insects found in field samples. The mean number of insects per sample was . After the samples with the greatest numbers of insects were removed, the mean number of insects in the remaining samples was . The numbers of insects in the removed samples were consecutive integers. The smallest number of insects in any of the original samples was .
What was the range of the numbers of insects in the original samples?
For problems involving a full data set and a subset, change each mean into a total before doing anything else. Subtracting totals isolates the unknown subgroup. Then use the additional structure, such as consecutive integers, to identify the needed extreme value and calculate the range.
Hints
Convert means to totals
Multiply each mean by the number of samples in its data set to find a total number of insects.
Isolate the removed samples
Subtract the total for the remaining samples from the total for all samples.
Use the mean of consecutive integers
After finding the total for the four removed samples, divide by . Four consecutive integers with a half-integer mean have two middle integers on either side of that mean.
Desmos Guide
Calculate the removed total
Algebra is fastest for this question. For a Desmos verification, enter 18*52-14*45 to find the total number of insects in the four removed samples.
Find the removed mean
Enter (18*52-14*45)/4. Use the displayed mean to identify the four consecutive integer values centered on it.
Verify the range
After identifying the greatest removed value, enter that value minus 27 to verify the range.
Step-by-step Explanation
Find the totals
The total number of insects in all samples was . The total in the remaining samples was .
Find the removed values' total
The four removed samples had a total of insects. Their mean was therefore .
Use the consecutive-integer condition
Four consecutive integers with a mean of are , , , and . Thus, the greatest original value was . The least original value was , so the range was .