# Chapter 101, module 1: Quantitative reasoning practice

Original supplementary practice, not an official ACER or AAMC question set. Paired with the Japanese file of the same module ID. This exercise is separate from the approximately ten-minute narration text. Suggested practice time: eight minutes, followed by explanation review. This timing is a study suggestion, not an exam specification.

## Passage

A laboratory is testing a colour-producing reaction. Its stock reagent has a concentration of 2 millimoles per litre. A technician transfers 5 millilitres of stock to a vessel and adds solvent until the total volume reaches 20 millilitres. Assume that mixing causes no chemical reaction or solute loss.

The technician then measures a response in arbitrary units. Three independently prepared control samples give readings of 18, 20, and 22. Three independently prepared treated samples give readings of 27, 30, and 33. All six samples come from the same experimental system. Controls are measured in the morning and treated samples in the afternoon. The instrument is not recalibrated between sessions. The investigator concludes that treatment caused a 50 percent increase and that the result will apply to other biological systems.

## Questions

1. What is the diluted reagent concentration? Show a calculation with units.
2. What amount of reagent, in micromoles, was transferred from the stock?
3. Calculate the control and treated means. Is the investigator's percentage calculation correct?
4. If the mean response later falls from 30 back to 20, what is the percentage decrease? Why is it not the same percentage as the original increase?
5. Identify one design feature that weakens the causal conclusion. Give one improvement addressing that feature.
6. Do three samples from one system establish that the effect generalises to other systems? Explain what additional evidence would address that claim.
7. A second instrument consistently reads every sample 5 units too high. Is this primarily a precision problem or an accuracy problem? Would subtracting the control mean from the treated mean remove that fixed additive offset?

## Worked answers

1. Concentration after dilution is 2 millimoles per litre multiplied by 5/20, giving 0.5 millimoles per litre. The volume ratio is dimensionless because both volumes are in millilitres. The denominator is final solution volume, not added solvent volume.

2. Five millilitres is 0.005 litres. Multiplying by 2 millimoles per litre gives 0.010 millimoles, or 10 micromoles. Dividing 10 micromoles by the final 0.020 litres gives 500 micromoles per litre, which agrees with 0.5 millimoles per litre.

3. The means are 20 and 30 units. The absolute difference is 10 units. Relative to the control mean, the increase is 10/20, or 50 percent. The arithmetic is correct; the causal and generalisation claims require separate justification.

4. The decrease is 10/30, or about 33.3 percent. Percentage change uses the starting value as its denominator. Reversing a change changes that reference value.

5. Treatment is confounded with measurement session: instrument drift or another morning–afternoon difference could contribute to the result. Randomising or interleaving measurement order and including controls in each session would help separate treatment from session effects. Calibration checks would also help, but calibration alone does not remove every possible session difference.

6. No. These samples provide replication within one system, not evidence across all systems. Independent experiments across relevant biological systems and conditions would test generalisability. More measurements of the same sample would mainly assess measurement repeatability, not biological diversity.

7. A fixed offset is a systematic accuracy problem; repeatability can remain excellent. If exactly the same additive offset affects both groups, it cancels in the difference of means. It does not cancel in a ratio: 35/25 is 1.4, whereas 30/20 is 1.5. Therefore even a shared calibration error can alter a relative-effect estimate.

## Retrieval link and review status

Connect this exercise to cards IMF-101-01-0001 through IMF-101-01-0008. The passage adds design and calibration reasoning beyond those cards; it is not evidence of complete module coverage. Numerical answers have been checked. Independent editorial review, bilingual review, and learner difficulty calibration remain pending.