QUANTITATIVE REASONING · GRAPHICAL + DATA REASONING

Name the evidence.
Respect the limit.

Read complete graph and table contracts, preserve units and denominators, and prove uniqueness across every value the prompt allows.

Start the lessons ↓Build a balanced set →
2guided lessons
6retrieval prompts
8practice questions
5choices per item

The evidence audit

Inventory. Align. Test. Conclude.

  1. 01Inventory

    Name axes, units, source values, variables, groups, and the stated domain.

  2. 02Align

    Match scales and denominators before calculating or comparing.

  3. 03Test

    Check invariants or enumerate every allowed sufficiency candidate.

  4. 04Conclude

    State only the quantity or uniqueness claim supported by the model.

The two lesson labels follow the official Mathematical Problems scope. DAT TRAIN’s sequence, examples, and questions are original study tools and do not imply unpublished topic weights.

Two connected topic lessons

Make every denominator and boundary visible.

Each model exposes the complete source values, exact axis or table contract, and evidence limit required to support its conclusion.

01

LESSON 1 · 16 MIN

QR-MAT-GRA-01QR-MAT-GRA-02

Graphical analysis

Read axes, units, scale, points, slope, intercept, and domain, then test whether a table, equation, and graph encode the same relation.

ESSENTIAL QUESTIONWhich marks are observations, which values are modeled between them, and where does the stated domain stop?
Semantic graph contract

Axes, points, connection rule, and domain travel together.

Distance traveled over four hoursA straight-segment distance-time graph through zero hours and zero miles, two hours and one hundred miles, and four hours and two hundred miles. The stated domain is zero through four hours.. The complete values follow in a data table.
Measured points with stated straight segmentscircle marker · solid line
Observed distance values are connected by stated straight segments only from 0 through 4 hours; 3 hours is interpolation and 5 hours is outside the domain.
Open the complete data table
Distance traveled over four hours: source values
SeriesPointTime (h)Distance (mi)
Measured points with stated straight segmentsStart00
Measured points with stated straight segmentsTwo-hour observation2100
Measured points with stated straight segmentsFour-hour observation4200
Collected volume over five minutesA straight-segment volume-time graph through zero minutes and zero milliliters, two minutes and twenty-four milliliters, and five minutes and sixty milliliters.. The complete values follow in a data table.
Measured points with stated straight segmentscircle marker · solid line
The exact change ratio is 12 milliliters per minute across every supplied straight segment.
Open the complete data table
Collected volume over five minutes: source values
SeriesPointTime (min)Volume (mL)
Measured points with stated straight segmentsStart00
Measured points with stated straight segmentsTwo-minute observation224
Measured points with stated straight segmentsFive-minute observation560
Equivalent linear representationsA straight-segment graph through x equals zero and y equals four, x equals two and y equals ten, and x equals four and y equals sixteen, with domain zero through four.. The complete values follow in a data table.
Measured points with stated straight segmentscircle marker · solid line
The table, graph, and equation y = 3x + 4 agree only when the stated domain 0 through 4 is preserved.
Open the complete data table
Equivalent linear representations: source values
SeriesPointInput (x-units)Output (y-units)
Measured points with stated straight segmentsIntercept04
Measured points with stated straight segmentsMiddle observation210
Measured points with stated straight segmentsDomain endpoint416
Complete axis, unit, connection, and domain ledger
Graphx-axis contracty-axis contractConnection ruleStated domain
Distance traveled over four hoursTime: 0–4 h; tick 1 hDistance: 0–200 mi; tick 50 miStraight segments between adjacent source points0 ≤ time ≤ 4 h
Collected volume over five minutesTime: 0–5 min; tick 1 minVolume: 0–60 mL; tick 12 mLStraight segments between adjacent source points0 ≤ time ≤ 5 min
Equivalent linear representationsInput: 0–4 x-units; tick 1 x-unitsOutput: 0–16 y-units; tick 4 y-unitsStraight segments between adjacent source points0 ≤ input ≤ 4 x-units
VALIDATED QUANTITATIVE MODEL · COMPLETE SOURCE TABLE + TEXT EQUIVALENT INCLUDED
01

Read the contract

Name both variables, units, axis ranges, tick intervals, series encoding, source points, and stated domain before calculating.

  • Tick spacing—not visual distance alone—sets scale.
  • The source table is the complete nonvisual equivalent.
02

Separate point types

Observed points come from the source data; interpolation uses an explicit between-point model; extrapolation leaves the stated domain.

  • A connected segment authorizes only the stated in-domain interpretation.
  • Never continue a trend past an endpoint without an explicit model.
03

Match invariants

Equivalent equations, tables, and graphs must agree on intercept, slope or shape, values, and allowed inputs.

  • Slope carries output-units per input-unit.
  • A matching equation with a different domain is a different model.

Worked example

A straight-segment graph contains (0 h, 0 mi), (2 h, 100 mi), and (4 h, 200 mi). What does it show at 3 h, and what may it claim at 5 h?

  1. 1

    The axes identify time in hours and distance in miles; the stated domain is 0 through 4 hours.

  2. 2

    Three hours lies between the 2-hour and 4-hour observations, so the stated straight segment gives 150 miles.

  3. 3

    Five hours lies beyond the stated domain, so 250 miles would be an unsupported extrapolation unless another model is supplied.

ConclusionThe graph supports 150 miles at 3 hours by interpolation and supports no value at 5 hours under the given contract.

Calculator decision ledger

Model mentally
Read axis labels, units, source points, connection rule, and domain before keying values.
Estimate
Predict direction, intercept, and approximate change per input unit from the complete table.
Calculator
Use division only when the exact change ratio is not immediate from the supplied values.
Audit
Reattach slope units and classify the input as observed, interpolated, or beyond the domain.

Close the notes first

Retrieve the model.

01What must be read before comparing two drawn slopes?
Both axis scales and units

Drawing angle changes when an axis scale changes; slope is an exact change ratio with units.

02How does interpolation differ from extrapolation?
Interpolation stays between modeled points inside the domain; extrapolation goes beyond it.

The evidence boundary is set by the source points, connection rule, and stated domain.

03Why can two equations that share points still represent different models?
They may differ in rule, shape, behavior, or allowed domain.

A few shared values do not establish agreement over every permitted input.

02

LESSON 2 · 17 MIN

QR-MAT-DAI-01QR-MAT-DAI-02

Data analysis, interpretation, and sufficiency

Identify variables, groups, totals, rates, and denominators, then test every allowed candidate to decide whether supplied information determines one answer.

ESSENTIAL QUESTIONWhat is the denominator—and has every alternative allowed by the stated domain been eliminated?
Denominator + finite-domain ledger

Keep each denominator. Eliminate every surviving alternative.

Unequal-denominator count comparison
GroupSuccessful countTotal countSuccess rate
Group A305060%
Group B408050%
Pooled-percentage source values
CohortSuccessesTotalWithin-cohort rate
Cohort 1183060%
Cohort 2246040%
Combined429046 2/3%
Exhaustive information-sufficiency survivor ledger
Stated domainStatement I survivorsStatement II survivorsCombined survivorsConclusion
Integers 1–10{3, 6, 9}{6, 8, 10}{6}Both together; neither alone
Integers 1–12{7}{7}{7}Each statement alone

The sufficiency routine evaluates the complete stated finite domain. It does not add positivity, integrality, or any other condition unless the prompt supplies it.

VALIDATED QUANTITATIVE MODEL · COMPLETE SOURCE TABLE + TEXT EQUIVALENT INCLUDED
01

Inventory the data

Name the population, variable, group, unit, numerator, denominator, total, trend, and anomaly before choosing a summary.

  • A larger rate can coexist with a smaller count.
  • Association and displayed change do not establish causation.
02

Align denominators

Compare counts, rates, percentages, and percentage-point differences only after naming the question each quantity answers.

  • Pool counts and totals before computing a combined percentage.
  • Do not average subgroup percentages when group sizes differ.
03

Exhaust the domain

For sufficiency, test statement I alone, statement II alone, and then their intersection across the complete stated domain.

  • One compatible value is not proof of uniqueness.
  • Do not add unstated positivity, integrality, or other restrictions.

Worked example

For integer x from 1 through 10, is x determined? I: x is a multiple of 3. II: x is even and greater than 4.

  1. 1

    Statement I leaves 3, 6, and 9, so it is not sufficient alone.

  2. 2

    Statement II leaves 6, 8, and 10, so it is not sufficient alone.

  3. 3

    The intersection contains only 6, so the statements together determine x.

ConclusionBoth statements together are sufficient, but neither statement alone is sufficient.

Calculator decision ledger

Model mentally
Write each numerator beside its own denominator and list the complete allowed domain for sufficiency.
Estimate
Compare rough counts and pooled rates before exact division to catch denominator mistakes.
Calculator
Use multiplication or division for awkward rates only after the comparison base is fixed.
Audit
Rebuild pooled totals and inspect every statement survivor before declaring a unique answer.

Close the notes first

Retrieve the model.

01Why can 60% of one group be fewer people than 50% of another?
The group denominators can differ.

A count equals the rate multiplied by its own total.

02How is a pooled percentage computed?
Add all relevant counts, add all relevant totals, then divide the two sums.

Pooling preserves each group’s denominator weight.

03What makes a sufficiency statement sufficient?
Every allowed candidate that survives it gives the same requested answer.

The task is to prove uniqueness across the stated domain, not find one example.

Randomized retrieval bank

Now reason without the lesson label.

All four objectives appear in the bank. Complete graph and table values remain in each prompt while question and answer order shuffle.

8 PRACTICE QUESTIONS

Retrieve before you review.

Question order and all five answer options are shuffled when you begin. The correct answer stays attached to the same underlying choice.

Scope and score notice

Use results for study guidance.

The ADA names these topics but does not publish their individual question quotas. DAT TRAIN does not invent them.

These original questions are draft and uncalibrated. Accuracy can direct review; it cannot predict an official DAT score.