Question 7·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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An environmental scientist recorded the numbers of insects found in $18$ field samples. The mean number of insects per sample was $52$ . After the $4$ samples with the greatest numbers of insects were removed, the mean number of insects in the remaining $14$ samples was $45$ . The numbers of insects in the $4$ removed samples were consecutive integers. The smallest number of insects in any of the original $18$ samples was $27$ .

What was the range of the numbers of insects in the original $18$ samples?

Your answer

A mean is the total divided by the number of samples, so two means can tell you how much the removed samples added up to. Find both totals in Desmos and subtract. Then use the consecutive-integers condition to find the largest removed count. The range uses the original largest and smallest counts, not either mean.

Hints

Hint 1
A mean is one equal share of a group's total. Multiply each mean by its own number of samples to recover the total before removal and the total afterward.
Hint 2
The removed samples account for the difference between those two totals. Once you have their sum, write their counts as consecutive integers: start with a number and add $1$ each time.
Hint 3
The range is the largest count minus the smallest. Since the removed samples had the greatest counts, which of the four consecutive integers is the original largest?

Step-by-step

Recover the removed counts
Step 1
Find both sample totals
A mean is the total divided by the sample count, so mean times count gives the total. Type $52(18)$ and $45(14)$ in Desmos. It prints $936$ insects in the original samples and $630$ in the remaining samples.
Step 2
Recover the removed total
The original total includes the insects in the samples that stayed and those that were removed. So subtract the remaining total: type $52(18)-45(14)$ . Desmos prints $306$ , the sum of the four removed counts, not one sample's count.
Step 3
Find the smallest removed count
Consecutive integers step up by $1$ each time. If $x_1$ is the smallest removed count, the four counts are $x_1$ , $x_1+1$ , $x_1+2$ , and $x_1+3$ . Their sum is $306$ , so type $x_1+(x_1+1)+(x_1+2)+(x_1+3)\sim306$ . The $\sim$ tells Desmos to find $x_1$ ; under PARAMETERS, it reports $x_1=75$ .
Step 4
Identify the original maximum
The removed samples had the greatest counts, so the last of those four integers is the original maximum, or largest count. Type $x_1+3$ ; Desmos prints $78$ . Don't stop at $75$ : that's the smallest of the removed four.
Step 5
Find the original range
The range is maximum minus minimum. The original maximum is $78$ , and the given original minimum is $27$ . Type $x_1+3-27$ ; Desmos prints $51$ , the range of insect counts in the original samples. Grid in 51.

Question 7·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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