SAT

1. A sentence boundary needs a complete clause

Learning goal: Distinguish a complete sentence from a dependent clause before joining ideas.

A complete clause contains a subject and a finite verb and can stand as a sentence in the context of the exercise. A dependent clause may contain both but begins with a word such as although or because that makes it depend on another clause. Identifying the boundary helps you choose a punctuation form based on structure rather than the length of a pause.

Original example: the experiment ended. This is complete. Although the experiment ended is dependent because although introduces an unfinished contrast. Join it to a main clause: Although the experiment ended, the team kept analyzing data. The comma marks the introductory dependent clause; it is not a license to join any two full sentences with a comma alone.

For two independent clauses, a period is a safe separation. A semicolon can also join closely related independent clauses. A comma followed by a coordinating conjunction can work when the relationship fits. The original sentence the experiment ended, the team kept analyzing data has a comma splice. The fix follows the clause structure, not a preference for longer punctuation.

Practice set: identify the structure of because rain arrived, the game stopped, and the game stopped because rain arrived. The first is dependent alone, the second complete, and the third a complete sentence with a dependent reason clause. Now repair the team revised the plan, the deadline stayed fixed using either a period, a semicolon or a fitting conjunction. Explain which clauses are independent. Official exam reading/English scope includes sentence conventions; the examples here are original.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Repair: The trial ended, the researchers wrote the report.

1. Identify two independent clauses.
2. A comma alone is insufficient.
3. Use a period or semicolon.
4. The trial ended; the researchers wrote the report.

Mental model: Distinguish a complete sentence from a dependent clause before joining ideas.

Common trap: Treat punctuation as a pause rather than a clause relation.

2. Agreement follows the head of the subject

Learning goal: Match the verb with the actual subject, not the nearest noun.

Subject-verb agreement depends on the grammatical subject. A phrase between the subject and verb can contain a plural noun that distracts from a singular head. Remove the intervening phrase temporarily and read the core clause. The collection of maps is extensive reduces to the collection is extensive. Maps does not control the verb.

Original example: the results of the experiment are available. Results is plural, so are fits. In the result of the experiments is available, result is singular despite the plural experiments. The two sentences mean different things. Identify the noun phrase head before choosing a verb, and preserve the intended meaning while editing.

A compound subject joined by and is often plural in ordinary examples, as in the teacher and the student are ready. This lesson avoids special idiomatic exceptions and focuses on transparent cases. A pronoun must also have a clear referent, but agreement does not by itself solve ambiguity. The sentence can be grammatically matched yet still unclear about who acted.

Practice set: the list of requirements is short; the requirements on the list are clear. Then test the box containing several samples is sealed. Explain why samples does not make the verb plural. In a timed edit, underline the head and cross out modifiers mentally. These original exercises align with official language-conventions scope; they do not claim to cover every exception in English usage.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Choose verb: The collection of maps ___ useful.

1. Head subject is collection.
2. Of maps modifies collection.
3. Collection is singular.
4. Use is.

Mental model: Match the verb with the actual subject, not the nearest noun.

Common trap: Agree with the nearest noun instead of the subject head.

3. Transitions express the actual relationship

Learning goal: Choose a connector after naming contrast, consequence or continuation.

A transition guides the reader through a relationship between ideas. It is not decoration. First paraphrase the two statements without the options. Do they contrast, does the second follow as a result, or does it add another example? The connector should match that relation and not create a cause the text does not support.

Original pair: the device was inexpensive. Its replacement parts were costly. However fits the contrast between low purchase cost and high replacement cost. Therefore would imply that low purchase cost caused costly parts, which the two statements do not establish. Similarly, in addition merely adds a point without making the contrast explicit.

Context can change the relationship. The device used less energy, so operating costs fell describes a stated consequence. The device was compact; moreover, it was quiet adds a second favorable feature. These choices depend on the assertions, not on whether a transition sounds formal. Keep punctuation and clause boundaries separate from the logical relation.

Practice set: a proposal looked promising but a trial failed. Choose a contrast connector. A trial succeeded and funding continued because that result met a stated condition: choose a consequence connector. Two independent benefits are listed: choose an addition connector. For each, explain why the nearest tempting option fails. The invented examples teach expression of ideas in official exam scope, not a rule that every contrast must use the same word.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Low initial price; costly replacement parts. Choose relationship.

1. Compare the claims.
2. One cost is low; another high.
3. The relation is contrast.
4. However fits.

Mental model: Choose a connector after naming contrast, consequence or continuation.

Common trap: Choose a polished connector without checking logic.

4. A summary states the governing idea

Learning goal: Choose the claim that accounts for the passage, not an isolated detail.

A central-idea question asks which statement organizes the passage. A detail can be true and still fail as a summary. Read how examples connect to the author point. If every example illustrates trade-offs between speed and accuracy, a summary about one example technology is too narrow. An answer claiming all technologies are harmful may be too broad.

Original passage: a library adopted a faster search system. Staff found that unusual spelling sometimes hid older records, so they kept a manual index for difficult searches. The passage point is that faster digital retrieval can still benefit from an alternative route for exceptions. It is not that every digital system fails or that manual indexes always outperform search.

Summarize in one plain sentence before reading options. Compare each choice with that sentence, checking subject, claim and scope. A correct choice may use different words while preserving the relation. An attractive option may repeat a memorable noun but change the conclusion. Literal word overlap is not the same as support.

Practice set: classify three statements. The library uses a search system is a true detail. Combining tools can address exceptions is the governing idea. Digital search should be banned is unsupported. Now change the passage to say the manual index was never needed; the combined-tools conclusion would no longer follow. In this original exercise the evidence controls the summary. Use official practice separately for passage length and exam interface.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Library search is faster but manual index helps spelling exceptions. State main idea.

1. Find the recurring relationship.
2. Digital speed is a benefit.
3. Exceptions need another route.
4. Combining tools addresses different needs.

Mental model: Choose the claim that accounts for the passage, not an isolated detail.

Common trap: Pick a vivid detail as the whole-passage conclusion.

5. Evidence must answer the precise claim

Learning goal: Match a supporting observation to the conclusion it can actually bear.

An evidence question links a claim with information in a passage or table. Ask which exact part of the claim the observation supports. A number about attendance does not directly establish satisfaction, learning or revenue. Similar topics can hide a mismatched measure. State the target variable before evaluating the evidence.

Original claim: the extended opening hours attracted more evening visitors at one branch. A table showing evening visits at that branch before and after the change is relevant. A chart of all daytime visitors at a different branch is not direct evidence for the specified claim. The claim still may not be causal unless the design addresses other explanations.

An answer can be relevant but insufficiently specific. A statement that library membership rose concerns people with memberships, not necessarily actual evening visits. A survey that visitors like long hours reports opinion, not the measured attendance change. Prefer evidence whose location, time and variable match the claim rather than evidence with the most impressive number.

Practice set: a researcher claims an object became lighter after a process. Before/after mass measurements support the weight-change claim more directly than a color photograph. A claim that the process caused the change requires additional design evidence. Separate observation from causal interpretation. These original questions practice information-and-evidence reading; no real program effect or measured statistical pattern is claimed.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Claim: evening visits at branch A increased. What measure fits?

1. Match time: evening.
2. Match place: branch A.
3. Match variable: visits.
4. Compare before and after on that measure.

Mental model: Match a supporting observation to the conclusion it can actually bear.

Common trap: Match only the topic and ignore the measured variable.

6. Rhetorical notes need the requested goal

Learning goal: Select information based on purpose rather than include every available fact.

A writing task may provide notes and a specific goal. The best response serves that goal accurately, using relevant details and avoiding unsupported additions. A request to compare two methods differs from a request to introduce a person or explain a process. Read the goal before deciding which notes are essential.

Original notes: method A needs ten minutes and one tool; method B needs five minutes and three tools. Goal: emphasize the speed difference. A sentence stating B takes half as long as A serves it. A sentence listing only tool counts may be true but misses the purpose. A claim that B is always better adds a value judgment not established by the notes.

If the goal changes to explain the equipment trade-off, tool counts become central. If it asks to introduce both methods neutrally, a balanced description can include time and tools. Relevant omission is not inaccuracy: a concise sentence can leave out details that do not advance the requested purpose, provided it does not distort them.

Practice set: compare two routes, A twenty minutes and two transfers, B thirty minutes and no transfers. For a speed goal, A is ten minutes faster. For a convenience goal about transfers, B requires none. Do not say no transfers makes B universally best. The examples are original and align with writing/expression skills; exact task formats differ between exams, so use the course format lesson for the target test.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

A 10 minutes/1 tool; B 5 minutes/3 tools. Goal: emphasize speed.

1. Read the goal.
2. Choose time facts.
3. B takes half as long as A.
4. Do not claim universal superiority.

Mental model: Select information based on purpose rather than include every available fact.

Common trap: Include all notes without checking the writing purpose.

7. A modifier belongs beside its target

Learning goal: Make the opening phrase describe the subject that actually performed the action.

An introductory modifier can create an unintended meaning if it attaches to the wrong subject. In the original sentence walking into the lab, the report was on the table, the grammar appears to make the report do the walking. A clear revision names the walker: walking into the lab, Maya noticed the report on the table.

The fix is not simply to add a comma. The problem is the relationship between the opening phrase and the following subject. Ask who performed the action or had the described condition. A sentence can have correct punctuation but a misplaced descriptive phrase. Meaning and syntax need checking together.

Original pair: covered in dust, the researcher cleaned the shelf may describe the researcher as dusty. If the intended dusty object is the shelf, write the researcher cleaned the shelf, which was covered in dust, or covered in dust, the shelf needed cleaning. Different revisions can preserve different meanings. Choose the one that fits the surrounding context.

Practice set: after measuring the samples, the results were recorded by Jo. A clearer active version is after measuring the samples, Jo recorded the results. Next revise damaged during transit, Lin inspected the parcel so damaged plainly describes parcel. Explain the actor or target rather than calling the first sentence awkward. These are original convention exercises, not claims about actual laboratory work.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Revise: Walking into the lab, the report was visible.

1. Identify who walks.
2. A report cannot perform this action in the intended context.
3. Name the person as subject.
4. Walking into the lab, Maya saw the report.

Mental model: Make the opening phrase describe the subject that actually performed the action.

Common trap: Use punctuation to hide a meaning mismatch.

8. Mixed language practice reviews meaning and structure

Learning goal: Use a compact set to distinguish grammar, scope and writing purpose.

Language questions can require a structural correction or a meaning judgment. The two tasks overlap but are not identical. A sentence may be grammatical and still fail to support the passage claim. A summary may state a plausible idea but misattribute it. Read the target before choosing a revision.

Original set: choose the verb in the collection of maps is or are useful. Then summarize a library passage where digital search is faster but a manual index handles unusual spellings. Finally choose a transition between low initial cost and high replacement cost. Work the subject head, governing idea and relation separately.

The answers are is, complementary tools address exceptions, and a contrast connector such as however. The grammar answer follows singular collection, not plural maps. The summary preserves both a benefit and a limit. The transition contrasts two different cost observations rather than assert an unstated cause. Each answer needs a different reason.

For a new pass, use the results of the experiment, a passage about a manual tool that was never needed, and two independent benefits rather than contrasting costs. The verb becomes are, the earlier summary needs revision, and an addition connector may fit. Record which signal changed. These original items are practice, not an official test section, adaptive route or score prediction.

Source alignment: [Official English/Reading content scope; original practice]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

The collection of maps ___ useful.

1. Find head collection.
2. Ignore the intervening plural noun.
3. Use singular verb.
4. Is fits.

Mental model: Use a compact set to distinguish grammar, scope and writing purpose.

Common trap: Apply one memorized answer after the context changes.

9. Boundary cases reveal hidden assumptions

Learning goal: Test all permitted kinds of numbers, not one favourite integer.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

Boundary-case comparison asks whether one quantity is always greater, the quantities are equal, or the relationship cannot be determined from the information. Try varied values as an original editorial strategy. With real x, compare x² and x: a fraction between zero and one behaves differently from a negative number or an integer above one.

A single test can refute an always claim, but cannot prove it unless justified for the whole domain. A domain restriction may make an algebraic argument possible. Keep the exact restrictions visible: positive still permits fractions, whereas positive integer does not.

This boundary comparison is an editorial reasoning drill, not the format of a SAT question. Answer the numeric transfer separately.

Boundary checks test whether a conclusion holds for every permitted value. Do not assume positivity or integrality unless the question supplies it. Try zero, a fraction and a negative value when they are allowed. A comparison drill here is a teaching format, not an official SAT or ACT answer interface. To translate the idea into an exam-style calculation, solve the numeric transfer and check the domain before selecting a value.

Original practice: compare x and x squared when x equals one-half, two and minus one. The order changes, so an unrestricted comparison cannot have one universal direction. If x is at least two, x squared exceeds x; the extra restriction changes the result. Record the condition that made your inference possible rather than memorize the larger expression.

Worked example

x is real. Compare A=x² and B=x.

1. x=1/4 gives A=1/16 < B=1/4.
2. x=-1 gives A=1 > B=-1.
3. x=1 gives equality.
4. Since permitted cases give different relations, the relationship cannot be determined.

Mental model: Use boundary and sign cases before choosing a fixed relation.

Common trap: Positive does not mean positive integer.

10. Rates add; times do not

Learning goal: Model simultaneous work in units per time.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

These arithmetic foundations use ratios and algebra [S 2]. A worker completing a job in t hours has rate 1/t jobs per hour under a constant-rate assumption. Simultaneous rates add if the workers contribute independently and there is no leak or interference. The completion times themselves do not add or average.

The combined time should be shorter than either individual time for two positive filling rates. If a leak is supplied, subtract its rate rather than its time. These are explicitly simplified models, not descriptions of real pumps changing rate with pressure.

A work rate tells how much of a task is completed per unit time. If one worker completes a job in four hours, the ideal constant rate is one-quarter job per hour. Two workers operating independently at those specified constant rates add their completed fractions per hour. This model assumes they do not interfere and that the task can be shared. It is not a real productivity measurement.

Original follow-up: workers taking six and three hours have rates one-sixth and one-third, totaling one-half job per hour, so together they take two hours. Averaging the completion times gives four-and-a-half hours, which describes no relevant rate. If the first worker leaves after one hour, calculate the completed fraction before applying the remaining worker rate to the unfinished fraction.

Worked example

A fills in six hours, B in three. Together, constant rates and no leaks. Time?

1. A rate = 1/6 tank/hour.
2. B rate = 1/3 tank/hour.
3. Combined rate = 1/2 tank/hour.
4. One tank / (1/2 tank/hour) = 2 hours.

Mental model: Add compatible work rates.

Common trap: Never average completion times for simultaneous work.

11. Probability: count the event and the space

Learning goal: Specify equal likelihood and replacement assumptions.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

This course includes elementary probability and data analysis [S 2]. Count favourable outcomes over equally likely possible outcomes, or use a stated probabilistic model. Without replacement changes the second draw's denominator and composition. With replacement restores the original contents if mixing makes the draws independent under the model.

"At least one" can be easier to compute as one minus none. For a tiny sample space, enumerate outcomes to check the formula. These urns are toy models, not real-world frequencies. Hypothesis testing is outside this module's scope; this lesson teaches only the stated elementary probability model.

A probability is an event count divided by a total count only when the listed outcomes are equally likely. If outcomes have different weights, the count ratio can fail. Name the experiment, sample space and event before using a fraction. Drawing with replacement differs from drawing without replacement because the second selection conditions change.

Original practice: a bag contains three red and two blue tokens. One draw is red with probability three-fifths. Two draws without replacement are both red with probability three-fifths times two-fourths, or three-tenths. With replacement the probability is three-fifths squared, or nine-twenty-fifths. Explain the second denominator before multiplying. A correct multiplication with the wrong sample space still yields a wrong answer.

Worked example

A bag has three red and two blue tokens. Draw two without replacement. Probability both red?

1. First red probability = 3/5.
2. Given first red, remaining red probability = 2/4.
3. Multiply conditional probabilities: (3/5)(2/4)=3/10.
4. With replacement it would be (3/5)², a different model.

Mental model: Specify dependence before multiplying probabilities.

Common trap: A with-replacement shortcut fails without replacement.

12. A drawing is not a measurement

Learning goal: Use stated geometry and test scale-independent relations.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

This geometry lesson covers area, perimeter and scale. It does not cover all geometry or trigonometry in the official exam scope. A sketch can help organize relations, yet apparent size is not a measurement unless the problem specifies it. A rectangle's area uses both side lengths; perimeter uses their sum. Choose the requested quantity before inserting values.

Scaling all lengths by k scales area by k², not k. This follows from multiplying two length dimensions. A quick units check distinguishes square units from linear units. The exercises here provide exact side lengths so no conclusion depends on whether the pictured shape looks long or square.

A geometry diagram may illustrate relationships without being drawn to scale. Use marked lengths, angles and stated equations rather than visual guesses. A line that appears halfway across a shape does not establish a midpoint. If a triangle is declared right-angled, the Pythagorean relation applies; a merely similar-looking drawing does not supply that condition.

Original practice: a right triangle with perpendicular sides six and eight has hypotenuse ten because 36+64=100. A triangle with sides six and eight but no specified included angle does not have a fixed third side. Distinguish a given measurement from an inferred one, and label the formula conditions. After computing an area or length, check units: square units for area and linear units for length.

Worked example

A rectangle is 3 units by 8 units. Double both dimensions. New area?

1. Original area = 3×8 = 24 square units.
2. New sides = 6 and 16.
3. New area = 6×16 = 96 square units.
4. The factor is 2²=4, confirming 24×4=96.

Mental model: Use stated relations and the requested dimensions.

Common trap: Do not infer a right angle or equal length from appearance alone.

13. Data summaries do not tell the same story

Learning goal: Distinguish mean, median and weighted counts.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

This course includes descriptive data analysis [S 2]. The mean divides a total by its count; the median is the middle of a sorted list, or the average of the two middle values for an even count. A large extreme value can move the mean substantially while leaving the median unchanged. Neither summary is automatically the right one for every question.

All numbers in this activity are toy values. Describe exactly what a chosen summary captures rather than treating it as a complete distribution. A mean of four observations need not equal any observed value, and a median does not disclose the spread. Inferential conclusions about populations require more than these descriptive calculations and are outside this lesson.

A mean uses the total; a median uses ordered positions. An extreme observation can move the mean while leaving the median unchanged. Decide which statistic the task requests before making a claim about a typical value. A summary alone generally does not reconstruct every data point.

Original practice: values two, three, four, five, six have mean four and median four. Replace six with sixteen: mean becomes six but median stays four. The change affects the total and upper extreme, not the middle position. For an even number of observations, the median uses the two middle values in the stated elementary definition. Range measures maximum minus minimum; it does not tell how the interior observations are distributed.

Worked example

Toy values 2,3,3,12. Find mean and median.

1. Sum = 20, count = 4; mean = 5.
2. Sorted middle values are 3 and 3.
3. Median = (3+3)/2 = 3.
4. The high value raises the mean above the median; this is not a claim about a real sample.

Mental model: Choose the summary the question requests.

Common trap: A descriptive mean is not a population guarantee.

14. Percent changes use different bases

Learning goal: Translate sequential percent changes into multipliers before comparing endpoints.

Percent questions become simpler when every percentage is attached to a named base. A ten percent increase in a price means the new price is 1.10 times the old price. A ten percent decrease means the new price is 0.90 times the old price. The two changes do not cancel because the second percentage is calculated on a different amount. Avoid adding percentage changes unless the question explicitly defines them on a common base.

For a price that rises from 80 to 100, the increase is 20 divided by 80, or 25 percent. Reversing the journey gives a decrease of 20 divided by 100, or 20 percent. The dollar difference is the same; the percentage is not. Write the denominator before doing the arithmetic. This is useful when comparing discount claims, growth rates and changes in populations in original quantitative problems.

An index can represent a starting value without inventing a dollar amount. Set the initial value to 100, apply each multiplier and interpret the endpoint. If a value rises by 20 percent and then falls by 20 percent, the index becomes 100 times 1.20 times 0.80, or 96. The net decrease is four percent. An unknown original amount does not prevent a proportional answer.

Practice set: calculate the endpoint after a 50 percent increase and a one-third decrease; then after two successive ten percent increases. The first returns to the start, because 1.5 times two-thirds equals one. The second produces a 21 percent increase, not 20 percent. Before a timed set, record whether your mistake was a wrong base, a wrong multiplier or a rounding error. Keep exact fractions until the final requested precision.

Source alignment: [S 2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

An original price of 200 increases 25%, then decreases 20%. Find the final price.

1. Use the current price as each percentage base.
2. 200 × 1.25 = 250.
3. 250 × 0.80 = 200.
4. The reciprocal multipliers cancel here.

Mental model: Translate sequential percent changes into multipliers before comparing endpoints.

Common trap: Add successive percentages without checking their bases.

15. Linear equations can describe a whole family

Learning goal: Distinguish a unique solution from inconsistent equations or repeated information.

An equation expresses a condition, not necessarily a complete determination of every variable. With two unknown quantities, one equation usually leaves multiple possibilities. Substituting convenient values can reveal that freedom, but a general expression shows it more clearly. If x plus y equals ten, write y equals ten minus x. You can then ask whether the requested quantity stays fixed across the allowed values.

Two equations determine a unique solution when they provide independent information. If x plus y equals ten and x minus y equals two, adding them gives two x equals twelve. Thus x equals six and y equals four. Check both original equations, not only the simplified one. A sign error may satisfy the derived equation while failing the original pair.

If the second equation is twice the first, it gives no new information. x plus y equals ten and two x plus two y equals twenty describe the same line. By contrast, changing the second right-hand side to twenty-two creates a contradiction. The same left-hand side cannot equal twenty and twenty-two at once. Identify dependence before assuming every pair yields a unique intersection.

Practice set: compare three systems sharing x plus y equals eight. Pair it with x minus y equals two, with three x plus three y equals twenty-four, and with three x plus three y equals twenty-five. They have one solution, infinitely many solutions, and no solution respectively. State any domain restrictions: integer, positive or nonnegative constraints can reduce a family. On a boundary-case task, determine whether the requested expression is fixed even when the variables are not.

Source alignment: [S 2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

x + y = 10 and 2 x + 2 y = 20. Is x uniquely determined?

1. Check whether the second equation adds information.
2. Divide the second equation by two.
3. Both equations become x + y = 10.
4. x is not uniquely determined without more restrictions.

Mental model: Distinguish a unique solution from inconsistent equations or repeated information.

Common trap: Count equations instead of checking independence.

16. Weighted means need counts

Learning goal: Recover totals before combining averages from unequal-sized groups.

An average is a total divided by a count. When combining groups, the group averages alone are insufficient unless their sizes are known or equal. A group of two students averaging seventy and a group of eight averaging ninety do not combine to eighty. The larger group contributes more observations. Reconstruct each total, add the totals, and divide by the combined number of observations.

In this example the totals are two times seventy, or 140, and eight times ninety, or 720. Their sum is 860, spread across ten students, producing a mean of eighty-six. This weighted mean must lie between the two group means. That bound is a quick check, but it does not replace the calculation: many incorrect answers also lie between seventy and ninety.

Weighted means also appear in mixtures, portfolios and test-score problems. Define the weight in the units actually used: liters, dollars, students or observations. A statement that two solutions have different concentrations does not specify the resulting concentration without their quantities. If quantities are equal, the simple average works because the equal weighting is a fact, not a universal rule.

Practice set: combine three observations averaging ten with two observations averaging twenty-five. The total is thirty plus fifty, so the mean is sixteen. Then ask how large a second group averaging twenty must be to combine with four observations averaging ten into a mean of fifteen. Solve (40+20 n)/(4+n)=15 to obtain n=4. A missing value may be solved algebraically rather than guessed. Record whether the requested result is a total, a mean or a missing count.

Source alignment: [S 2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Two observations average 70 and eight average 90. Find the combined mean.

1. Recover each group total.
2. 2×70=140 and 8×90=720.
3. Add totals to get 860 and counts to get 10.
4. 860÷10=86.

Mental model: Recover totals before combining averages from unequal-sized groups.

Common trap: Average averages without checking equal weights.