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Chapter 2 • Theory & Derivations

Unit 2 — Igneous classification and textures

Describe what can be observed before interpreting rock origins.

§2.1 Central question

Describe what can be observed before interpreting rock origins.

§2.2 Unit outcomes and perspective

Create a defensible rock name from mineral mode, chemistry and texture; document cases where a formal scheme cannot be applied. The central approach is to separate measurement, interpretation, and uncertainty before assigning any historical meaning to a rock. Petrography identifies minerals, textures and overprinting relationships; geochemistry expresses chemical patterns; field mapping constrains geometry and chronology; quantitative models test whether a proposed history is physically plausible. No one method can normally reconstruct the entire history of a complex terrain. The unit therefore uses deliberately simplified calculations alongside explicit alternatives and checks.

§2.3 Prior knowledge and the logic of the lesson sequence

Recall that composition is not the same as mineral assemblage: two rocks can have similar bulk chemistry yet different minerals after differing thermal or pressure histories. Revisit the distinction between an intensive variable such as temperature or pressure and an extensive quantity such as mass or volume. When sketching diagrams, annotate axis scales and units and include the assumptions that make a boundary meaningful. For any result, ask whether equilibrium, a closed reservoir or ideal geometry was used. These assumptions should be stated before applying equations to natural rocks.

§2.4 Evidence ladder for this unit

Begin with repeatable descriptions of a specimen or synthetic dataset. Next ask which process could account for those descriptions under plausible Earth conditions. Finally test a process model against a competing explanation. Even a good process model does not automatically identify a unique geographic origin. A strong field interpretation shows where each independent line of evidence enters, what could overturn it and which uncertainty is irreducible without new sampling. Use contextual evidence from the surrounding stratigraphy and structures to prevent a microscope-scale feature from being overinterpreted.

§2.5 Phaneritic, aphanitic, glassy and porphyritic textures

Mechanism: what physical or chemical change is being proposed?

Cooling and nucleation histories influence grain size, but grain size also reflects supersaturation, chemistry and residence time. To reason quantitatively about phaneritic, aphanitic, glassy and porphyritic textures, first distinguish the part of the system that is open to transport from the part treated as a closed parcel. Track changes in the mineral or melt assemblage, not just the name assigned to the whole rock. A rock records its latest preserved state together with some relics, so the working question is not simply “What rock is this?” but “What sequence of processes could have produced these specific observations?”

Evidence: what would a geologist actually measure?

A coarse crystal population in fine groundmass documents at least two growth or residence regimes. Organize the field notebook as a sequence of observation, measurement conditions and tentative interpretation. Geological evidence can come from hand specimens, thin sections, mapped contacts, electron-microprobe analyses, isotopes or dated mineral domains. The methods are complementary rather than interchangeable. Do not describe a map-scale contact as a microscopic mineral reaction, or a crystal core analysis as a whole-rock average.

Competing interpretations and a discriminating test

A porphyritic texture can arise during storage, recharge or decompression rather than from a single two-stage cooling event. The alternative is worth considering because two very different histories can converge on similar final textures or compositions. Measure grain-size distributions and compare crystal zoning with the groundmass. A test must be designed before the final preferred interpretation is chosen: otherwise the analyst risks using every possible feature as evidence for the same story. Identify the response variable, a comparison sample and the result each proposed process would predict.

Use two samples at contrasting spatial scales where possible: a local thin section and a regional outcrop pattern. Agreement across scales is more informative than repeated estimates of a single proxy.

Worked quantitative example 6: Mean temperature fall

Synthetic teaching inputs: 360 °c cooling over 240 h; not an instantaneous cooling rate. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is (1120-760)/240. Substituting the stated values gives 1.5 °C/h. Check whether the result has appropriate units and physical order of magnitude before using it.

Why this inference can fail

Annealing, deformation and alteration may erase or modify the original magmatic fabric. The scale of observation controls what the measured record can resolve. Two samples showing similar assemblages are not necessarily coeval; two mineral compositions agreeing within analytical precision are not necessarily in equilibrium. Evidence of replacement can survive after bulk chemistry has changed. Use two samples at contrasting spatial scales where possible: a local thin section and a regional outcrop pattern. Agreement across scales is more informative than repeated estimates of a single proxy. No degree of numerical polishing eliminates a missing source reservoir or undocumented sample context.

Guided field or laboratory investigation

  1. Define the most diagnostic feature for phaneritic, aphanitic, glassy and porphyritic textures and record it without a causal verb.
  2. Sketch the predicted field or thin-section expression of the principal mechanism: cooling and nucleation histories influence grain size, but grain size also reflects supersaturation, chemistry and residence time.
  3. Describe an observation that would be expected under the rival: a porphyritic texture can arise during storage, recharge or decompression rather than from a single two-stage cooling event.
  4. Propose the sampling or measurement comparison: measure grain-size distributions and compare crystal zoning with the groundmass.
  5. Recompute the quantitative case after changing its largest uncertain input by a plausible amount; describe which conclusion survives.
  6. Produce a short evidence matrix that ranks the explanations as supported, consistent-but-not-unique, contradicted or untested.

Interpretive outcome: Infer the minimum number of events warranted by the texture, not an invented detailed story. A strong answer states both the observation that supports a conclusion and the observation still needed to reject an alternative.

Review questions

  • Which measured variable provides the strongest test of phaneritic, aphanitic, glassy and porphyritic textures, and which confounder is most threatening?
  • Why might the competing hypothesis produce an apparently similar rock, and what data would discriminate?
  • Which parameter in the worked numerical example is least securely known in a real field setting?
  • Can a preserved texture be assigned to one instant in time, or might it be a record of overprinting?
  • What conclusion would you withdraw if a second laboratory contradicted the key measurement?

§2.6 QAPF modal rock naming

Mechanism: what physical or chemical change is being proposed?

Plutonic felsic rock names use relative proportions of quartz, alkali feldspar, plagioclase and feldspathoids in applicable IUGS fields. To reason quantitatively about qapf modal rock naming, first distinguish the part of the system that is open to transport from the part treated as a closed parcel. Track changes in the mineral or melt assemblage, not just the name assigned to the whole rock. A rock records its latest preserved state together with some relics, so the working question is not simply “What rock is this?” but “What sequence of processes could have produced these specific observations?”

Evidence: what would a geologist actually measure?

Point counts in a representative thin section can provide modal Q, A and P fractions. Organize the field notebook as a sequence of observation, measurement conditions and tentative interpretation. Geological evidence can come from hand specimens, thin sections, mapped contacts, electron-microprobe analyses, isotopes or dated mineral domains. The methods are complementary rather than interchangeable. Do not describe a map-scale contact as a microscopic mineral reaction, or a crystal core analysis as a whole-rock average.

Competing interpretations and a discriminating test

A visually pink coarse rock is not necessarily granite; colour and trade names are unreliable classifiers. The alternative is worth considering because two very different histories can converge on similar final textures or compositions. Normalize Q+A+P appropriately and compare the estimate with a defined classification diagram. A test must be designed before the final preferred interpretation is chosen: otherwise the analyst risks using every possible feature as evidence for the same story. Identify the response variable, a comparison sample and the result each proposed process would predict.

An independent mineralogical check can be more valuable than simply increasing precision of an already biased major-element measurement. Plan discrimination first, then precision.

Worked quantitative example 7: Normalized modal quartz Q

Synthetic teaching inputs: Q=24, a=29, p=47 points; exclude mafic phases in qapf normalization. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*24/(24+29+47). Substituting the stated values gives 24 Q/(Q+A+P) %. Check whether the result has appropriate units and physical order of magnitude before using it.

Why this inference can fail

QAPF fields are not intended for every ultramafic or glass-rich specimen. The scale of observation controls what the measured record can resolve. Two samples showing similar assemblages are not necessarily coeval; two mineral compositions agreeing within analytical precision are not necessarily in equilibrium. Evidence of replacement can survive after bulk chemistry has changed. An independent mineralogical check can be more valuable than simply increasing precision of an already biased major-element measurement. Plan discrimination first, then precision. No degree of numerical polishing eliminates a missing source reservoir or undocumented sample context.

Guided field or laboratory investigation

  1. Define the most diagnostic feature for qapf modal rock naming and record it without a causal verb.
  2. Sketch the predicted field or thin-section expression of the principal mechanism: plutonic felsic rock names use relative proportions of quartz, alkali feldspar, plagioclase and feldspathoids in applicable IUGS fields.
  3. Describe an observation that would be expected under the rival: a visually pink coarse rock is not necessarily granite; colour and trade names are unreliable classifiers.
  4. Propose the sampling or measurement comparison: normalize Q+A+P appropriately and compare the estimate with a defined classification diagram.
  5. Recompute the quantitative case after changing its largest uncertain input by a plausible amount; describe which conclusion survives.
  6. Produce a short evidence matrix that ranks the explanations as supported, consistent-but-not-unique, contradicted or untested.

Interpretive outcome: Report the analysed mode, uncertainty and the classification method used. A strong answer states both the observation that supports a conclusion and the observation still needed to reject an alternative.

Review questions

  • Which measured variable provides the strongest test of qapf modal rock naming, and which confounder is most threatening?
  • Why might the competing hypothesis produce an apparently similar rock, and what data would discriminate?
  • Which parameter in the worked numerical example is least securely known in a real field setting?
  • Can a preserved texture be assigned to one instant in time, or might it be a record of overprinting?
  • What conclusion would you withdraw if a second laboratory contradicted the key measurement?

§2.7 TAS and altered volcanic rocks

Mechanism: what physical or chemical change is being proposed?

Total alkalis versus silica is a chemical naming tool commonly used where reliable modal mineralogy is unavailable. To reason quantitatively about tas and altered volcanic rocks, first distinguish the part of the system that is open to transport from the part treated as a closed parcel. Track changes in the mineral or melt assemblage, not just the name assigned to the whole rock. A rock records its latest preserved state together with some relics, so the working question is not simply “What rock is this?” but “What sequence of processes could have produced these specific observations?”

Evidence: what would a geologist actually measure?

Fresh whole-rock major-element measurements supply SiO2 and combined Na2O+K2O for a plot. Organize the field notebook as a sequence of observation, measurement conditions and tentative interpretation. Geological evidence can come from hand specimens, thin sections, mapped contacts, electron-microprobe analyses, isotopes or dated mineral domains. The methods are complementary rather than interchangeable. Do not describe a map-scale contact as a microscopic mineral reaction, or a crystal core analysis as a whole-rock average.

Competing interpretations and a discriminating test

Secondary sodium or potassium mobility can shift a volcanic sample into an inappropriate TAS field. The alternative is worth considering because two very different histories can converge on similar final textures or compositions. Inspect loss on ignition, alteration petrography and immobile-element consistency before classification. A test must be designed before the final preferred interpretation is chosen: otherwise the analyst risks using every possible feature as evidence for the same story. Identify the response variable, a comparison sample and the result each proposed process would predict.

Document negative evidence as carefully as positive evidence. A predicted reaction product that is absent from an adequately sampled rock deserves an explanation rather than being silently ignored.

Worked quantitative example 8: TAS combined total alkalis

Synthetic teaching inputs: Major-element na₂o=3.6 wt % and k₂o=1.9 wt % on comparable basis. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 3.6+1.9. Substituting the stated values gives 5.5 wt % Na₂O+K₂O. Check whether the result has appropriate units and physical order of magnitude before using it.

Why this inference can fail

Analytical totals and volatile corrections matter; TAS is not a petrogenetic explanation. The scale of observation controls what the measured record can resolve. Two samples showing similar assemblages are not necessarily coeval; two mineral compositions agreeing within analytical precision are not necessarily in equilibrium. Evidence of replacement can survive after bulk chemistry has changed. Document negative evidence as carefully as positive evidence. A predicted reaction product that is absent from an adequately sampled rock deserves an explanation rather than being silently ignored. No degree of numerical polishing eliminates a missing source reservoir or undocumented sample context.

Guided field or laboratory investigation

  1. Define the most diagnostic feature for tas and altered volcanic rocks and record it without a causal verb.
  2. Sketch the predicted field or thin-section expression of the principal mechanism: total alkalis versus silica is a chemical naming tool commonly used where reliable modal mineralogy is unavailable.
  3. Describe an observation that would be expected under the rival: secondary sodium or potassium mobility can shift a volcanic sample into an inappropriate TAS field.
  4. Propose the sampling or measurement comparison: inspect loss on ignition, alteration petrography and immobile-element consistency before classification.
  5. Recompute the quantitative case after changing its largest uncertain input by a plausible amount; describe which conclusion survives.
  6. Produce a short evidence matrix that ranks the explanations as supported, consistent-but-not-unique, contradicted or untested.

Interpretive outcome: Use provisional chemical labels when alteration makes the formal class uncertain. A strong answer states both the observation that supports a conclusion and the observation still needed to reject an alternative.

Review questions

  • Which measured variable provides the strongest test of tas and altered volcanic rocks, and which confounder is most threatening?
  • Why might the competing hypothesis produce an apparently similar rock, and what data would discriminate?
  • Which parameter in the worked numerical example is least securely known in a real field setting?
  • Can a preserved texture be assigned to one instant in time, or might it be a record of overprinting?
  • What conclusion would you withdraw if a second laboratory contradicted the key measurement?

§2.8 Vesicles, amygdales and pyroclastic fabrics

Mechanism: what physical or chemical change is being proposed?

Gas bubbles, later mineral-filled cavities and broken clasts record distinct physical histories. To reason quantitatively about vesicles, amygdales and pyroclastic fabrics, first distinguish the part of the system that is open to transport from the part treated as a closed parcel. Track changes in the mineral or melt assemblage, not just the name assigned to the whole rock. A rock records its latest preserved state together with some relics, so the working question is not simply “What rock is this?” but “What sequence of processes could have produced these specific observations?”

Evidence: what would a geologist actually measure?

Vesicle shape, flattened pumice fragments, welding textures and clast sorting provide process evidence. Organize the field notebook as a sequence of observation, measurement conditions and tentative interpretation. Geological evidence can come from hand specimens, thin sections, mapped contacts, electron-microprobe analyses, isotopes or dated mineral domains. The methods are complementary rather than interchangeable. Do not describe a map-scale contact as a microscopic mineral reaction, or a crystal core analysis as a whole-rock average.

Competing interpretations and a discriminating test

Rounded cavities may be primary vesicles, solution features or later voids. The alternative is worth considering because two very different histories can converge on similar final textures or compositions. Combine cross-cutting relationships with microscopy and field-scale bedding or flow structures. A test must be designed before the final preferred interpretation is chosen: otherwise the analyst risks using every possible feature as evidence for the same story. Identify the response variable, a comparison sample and the result each proposed process would predict.

Do not infer time from spatial distance without independent constraints; a cooling front, reaction isograd and intrusive contact are different physical objects.

Worked quantitative example 9: Vesicular pore fraction

Synthetic teaching inputs: 19 ml cavity volume in 95 ml bulk specimen; no closed-pore undercount. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*19/95. Substituting the stated values gives 20 vol %. Check whether the result has appropriate units and physical order of magnitude before using it.

Why this inference can fail

Compaction, hydrothermal filling and weathering can substantially change original porosity. The scale of observation controls what the measured record can resolve. Two samples showing similar assemblages are not necessarily coeval; two mineral compositions agreeing within analytical precision are not necessarily in equilibrium. Evidence of replacement can survive after bulk chemistry has changed. Do not infer time from spatial distance without independent constraints; a cooling front, reaction isograd and intrusive contact are different physical objects. No degree of numerical polishing eliminates a missing source reservoir or undocumented sample context.

Guided field or laboratory investigation

  1. Define the most diagnostic feature for vesicles, amygdales and pyroclastic fabrics and record it without a causal verb.
  2. Sketch the predicted field or thin-section expression of the principal mechanism: gas bubbles, later mineral-filled cavities and broken clasts record distinct physical histories.
  3. Describe an observation that would be expected under the rival: rounded cavities may be primary vesicles, solution features or later voids.
  4. Propose the sampling or measurement comparison: combine cross-cutting relationships with microscopy and field-scale bedding or flow structures.
  5. Recompute the quantitative case after changing its largest uncertain input by a plausible amount; describe which conclusion survives.
  6. Produce a short evidence matrix that ranks the explanations as supported, consistent-but-not-unique, contradicted or untested.

Interpretive outcome: Recognize when a texture implies gas exsolution and when it only documents an empty space. A strong answer states both the observation that supports a conclusion and the observation still needed to reject an alternative.

Review questions

  • Which measured variable provides the strongest test of vesicles, amygdales and pyroclastic fabrics, and which confounder is most threatening?
  • Why might the competing hypothesis produce an apparently similar rock, and what data would discriminate?
  • Which parameter in the worked numerical example is least securely known in a real field setting?
  • Can a preserved texture be assigned to one instant in time, or might it be a record of overprinting?
  • What conclusion would you withdraw if a second laboratory contradicted the key measurement?

§2.9 Crystal size distributions and residence histories

Mechanism: what physical or chemical change is being proposed?

Population shapes in crystal-size plots reflect nucleation, growth, mixing and recycling rather than just cooling rate. To reason quantitatively about crystal size distributions and residence histories, first distinguish the part of the system that is open to transport from the part treated as a closed parcel. Track changes in the mineral or melt assemblage, not just the name assigned to the whole rock. A rock records its latest preserved state together with some relics, so the working question is not simply “What rock is this?” but “What sequence of processes could have produced these specific observations?”

Evidence: what would a geologist actually measure?

Counts by size class and zoning traverses test whether the population is single-stage. Organize the field notebook as a sequence of observation, measurement conditions and tentative interpretation. Geological evidence can come from hand specimens, thin sections, mapped contacts, electron-microprobe analyses, isotopes or dated mineral domains. The methods are complementary rather than interchangeable. Do not describe a map-scale contact as a microscopic mineral reaction, or a crystal core analysis as a whole-rock average.

Competing interpretations and a discriminating test

An apparent kink can arise from sampling bias or intersection effects in two-dimensional sections. The alternative is worth considering because two very different histories can converge on similar final textures or compositions. Compare replicated grain counts and uncertainty envelopes, then test alternative population models. A test must be designed before the final preferred interpretation is chosen: otherwise the analyst risks using every possible feature as evidence for the same story. Identify the response variable, a comparison sample and the result each proposed process would predict.

Change one assumption at a time when constructing an alternative model. This approach demonstrates whether the conclusion is stable, or whether a single unmeasured parameter controls the interpretation.

Worked quantitative example 10: Crystal population occurrence

Synthetic teaching inputs: 42 size-class grains in 125 counted intersections; not a volume mode. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*42/125. Substituting the stated values gives 33.6 %. Check whether the result has appropriate units and physical order of magnitude before using it.

Why this inference can fail

Small crystals may be missed and large ones rare; stereological conversion is uncertain. The scale of observation controls what the measured record can resolve. Two samples showing similar assemblages are not necessarily coeval; two mineral compositions agreeing within analytical precision are not necessarily in equilibrium. Evidence of replacement can survive after bulk chemistry has changed. Change one assumption at a time when constructing an alternative model. This approach demonstrates whether the conclusion is stable, or whether a single unmeasured parameter controls the interpretation. No degree of numerical polishing eliminates a missing source reservoir or undocumented sample context.

Guided field or laboratory investigation

  1. Define the most diagnostic feature for crystal size distributions and residence histories and record it without a causal verb.
  2. Sketch the predicted field or thin-section expression of the principal mechanism: population shapes in crystal-size plots reflect nucleation, growth, mixing and recycling rather than just cooling rate.
  3. Describe an observation that would be expected under the rival: an apparent kink can arise from sampling bias or intersection effects in two-dimensional sections.
  4. Propose the sampling or measurement comparison: compare replicated grain counts and uncertainty envelopes, then test alternative population models.
  5. Recompute the quantitative case after changing its largest uncertain input by a plausible amount; describe which conclusion survives.
  6. Produce a short evidence matrix that ranks the explanations as supported, consistent-but-not-unique, contradicted or untested.

Interpretive outcome: Make a process argument conditioned on measurement resolution. A strong answer states both the observation that supports a conclusion and the observation still needed to reject an alternative.

Review questions

  • Which measured variable provides the strongest test of crystal size distributions and residence histories, and which confounder is most threatening?
  • Why might the competing hypothesis produce an apparently similar rock, and what data would discriminate?
  • Which parameter in the worked numerical example is least securely known in a real field setting?
  • Can a preserved texture be assigned to one instant in time, or might it be a record of overprinting?
  • What conclusion would you withdraw if a second laboratory contradicted the key measurement?

§2.10 Unit 2 summary and assessment

The five lessons in this unit share a logic of observation → explanation → discrimination → calculation → qualified conclusion. Submit the five worked calculations with their units, three alternative-hypothesis comparisons and an annotated conceptual diagram. The diagram should distinguish arrows that represent measured directions from arrows that indicate proposed process pathways. Your final statement should name the observation that would most efficiently falsify your preferred account. A final confidence label is required, but the label is not a substitute for the written justification.

THREE WORKED QUESTIONS

Step-by-Step Petrology Solutions

Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.

Solved Problem Example 2.1: Mean temperature fall

Synthetic teaching inputs: 360 °c cooling over 240 h; not an instantaneous cooling rate. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is (1120-760)/240. Substituting the stated values gives 1.5 °C/h. Check whether the result has appropriate units and physical order of magnitude before using it.

Step

Synthetic teaching inputs: 360 °c cooling over 240 h; not an instantaneous cooling rate. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is (1120-760)/240. Substituting the stated values gives 1.5 °C/h. Check whether the result has appropriate units and physical order of magnitude before using it.

Solved Problem Example 2.2: Normalized modal quartz Q

Synthetic teaching inputs: Q=24, a=29, p=47 points; exclude mafic phases in qapf normalization. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*24/(24+29+47). Substituting the stated values gives 24 Q/(Q+A+P) %. Check whether the result has appropriate units and physical order of magnitude before using it.

Step

Synthetic teaching inputs: Q=24, a=29, p=47 points; exclude mafic phases in qapf normalization. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*24/(24+29+47). Substituting the stated values gives 24 Q/(Q+A+P) %. Check whether the result has appropriate units and physical order of magnitude before using it.

Solved Problem Example 2.3: TAS combined total alkalis

Synthetic teaching inputs: Major-element na₂o=3.6 wt % and k₂o=1.9 wt % on comparable basis. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 3.6+1.9. Substituting the stated values gives 5.5 wt % Na₂O+K₂O. Check whether the result has appropriate units and physical order of magnitude before using it.

Step

Synthetic teaching inputs: Major-element na₂o=3.6 wt % and k₂o=1.9 wt % on comparable basis. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 3.6+1.9. Substituting the stated values gives 5.5 wt % Na₂O+K₂O. Check whether the result has appropriate units and physical order of magnitude before using it.

Solved Problem Example 2.4: Vesicular pore fraction

Synthetic teaching inputs: 19 ml cavity volume in 95 ml bulk specimen; no closed-pore undercount. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*19/95. Substituting the stated values gives 20 vol %. Check whether the result has appropriate units and physical order of magnitude before using it.

Step

Synthetic teaching inputs: 19 ml cavity volume in 95 ml bulk specimen; no closed-pore undercount. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*19/95. Substituting the stated values gives 20 vol %. Check whether the result has appropriate units and physical order of magnitude before using it.

Solved Problem Example 2.5: Crystal population occurrence

Synthetic teaching inputs: 42 size-class grains in 125 counted intersections; not a volume mode. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*42/125. Substituting the stated values gives 33.6 %. Check whether the result has appropriate units and physical order of magnitude before using it.

Step

Synthetic teaching inputs: 42 size-class grains in 125 counted intersections; not a volume mode. Treat the listed inputs as constructed exercise values, not measurements from a real locality. The mathematical relation in this intentionally simplified calculation is 100*42/125. Substituting the stated values gives 33.6 %. Check whether the result has appropriate units and physical order of magnitude before using it.