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Solution ID:d0df86da

Problem-Solving and Data AnalysisStatistics (Center and Spread)Comparison of Weighted Means with Consecutive Integer Variables

Core Desmos Strategy

Pick small consecutive integers for a, b, and c in Desmos and directly evaluate the means and their difference.

Solved Result

Choice B (The mean of data set R is $\dfrac{2(11)}{14}$ greater than the mean of data set Q)

Extracted Problem (Vision OCR)

Complete Stem & Options

Data Set Q

Value	Frequency
a	110
b	30
c	0

Data Set R

Value	Frequency
a	0

Desmos Calculator Operations

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EvaluationAssign simple consecutive positive integer values for a, b, and c.

$a=1, b=2, c=3$

EvaluationCompute the mean of data set Q.

$Q=\dfrac{110a+30b}{110+30}$

$1.2142857$

EvaluationCompute the mean of data set R.

$R=\dfrac{3b+11c}{3+11}$

$2.7857143$

EvaluationFind the difference R - Q to see how much greater R is than Q.

$R-Q$

$1.5714286$

EvaluationEvaluate the expression in Choice B to confirm an exact match.

$\dfrac{2\cdot11}{14}$

$1.5714286$

Graph Inspection Tip:No graphing needed. The calculation in the Desmos scratchpad confirms R is 22/14 = 2(11)/14 greater than Q.

Analytical & Algebraic Solution

Step-by-step mathematical derivation with KaTeX proofs

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Step 1: Define the Relationship Between the Variables

We are given that $a, b,$ and $c$ are positive, consecutive integers such that $a < b < c$ . Therefore, we can express and in terms of the middle value :

Alternative Solving Methods

Alternative SAT test-taking techniques & secondary verification strategies

Algebraic Centering around b

algebraic

Express the consecutive integers in terms of the middle integer: a = b - 1 and c = b + 1. The mean of Q is (110(b - 1) + 30b) / 140 = (140b - 110) / 140 = b - 11/14. The mean of R is (3b + 11(b + 1)) / 14 = (14b + 11) / 14 = b + 11/14. Subtracting the two gives Mean(R) - Mean(Q) = (b + 11/14) - (b - 11/14) = 22/14 = 2(11)/14.

Deviation / Center of Mass Concept

shortcut

Notice both data sets are centered around b. Data set Q has 110 values at a distance of 1 below b, and 0 values above b, resulting in an average deviation of -110/140 = -11/14 relative to b. Data set R has 11 values at a distance of 1 above b, resulting in an average deviation of +11/14 relative to b. The net difference is (+11/14) - (-11/14) = +22/14 = 2(11)/14.

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Extracted Problem (Vision OCR)

Desmos Calculator Operations

Analytical & Algebraic Solution

Step 1: Define the Relationship Between the Variables

Alternative Solving Methods

Algebraic Centering around b

Deviation / Center of Mass Concept

Ask a Follow-Up Question

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Step 2: Calculate the Mean of Data Set Q

Step 3: Calculate the Mean of Data Set R

Step 4: Compare the Two Means

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