Question 55·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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A restaurant tracked the number of customers served during 14 lunch shifts. The mean number of customers served per shift was 52, and the least number of customers served in any of the 14 shifts was 41.

After data from 1 additional lunch shift is included, the mean number of customers served per shift becomes 50.2, and the range for all 15 shifts is 45.

If the additional shift had the least number of customers served of any of the 15 shifts, which choice is the range of the original 14 shifts?

Adding one value changes a data set’s mean, so rebuild each total using mean times count. Subtract the totals to find the added value. Then use the range, largest value minus smallest value, to find the maximum. The key trap is treating the new minimum as though it belonged to the original data set.

Hints

Hint 1
The mean is the total divided by the number of shifts. Reverse that relationship: multiply each mean by its shift count to find the two totals.
Hint 2
Only one shift was added, so the difference between the totals is that shift’s customer count. Subtract the totals, not the means.
Hint 3
The range is the maximum minus the minimum. If the added shift is the new minimum, the maximum still belongs to the original shifts. How can the 15-shift range help you find it?

Step-by-step

Recover the totals, then track the endpoints
Step 1
Find the original total
The mean is the total divided by the count, so total equals mean times count. Type $a=14\cdot52$ in Desmos, using $a$ for the original total. Desmos shows $a=728$ customers.
Step 2
Find the total after the addition
There are now 15 shifts, with a mean of $50.2$ . Add $b=15\cdot50.2$ in Desmos, using $b$ for the new total. It shows $b=753$ customers.
Step 3
Find the added shift’s count
The only difference between the two totals is the added shift. Type $b-a$ ; Desmos shows $25$ . So that shift served $25$ customers. Don’t subtract the means: each mean is spread across a different number of shifts.
Step 4
Find the maximum shared by both sets
The added shift is the new minimum, $25$ . A range measures from the minimum up to the maximum, so the maximum is $45$ above $25$ . Type $m=25+45$ , using $m$ for that maximum; Desmos shows $m=70$ . Adding a new minimum changes the minimum, but leaves the original maximum in place.
Step 5
Use the original minimum
The original shifts still have their minimum of $41$ and maximum of $70$ . Type $m-41$ ; Desmos shows $29$ . So the range of the original $14$ shifts is $29$ customers. Choice B.

Question 55·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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