Mistake Master
The bigger group pulls the average
When unequal groups combine, their averages never meet in the middle. The arithmetic is totals: each amount times its own average or concentration, added, divided by everything. The points slip away when the weights get skipped or swapped, when concentrations get added, and when the total gets reported in place of the part. Work through totals and name what was asked.
§1
What this topic is about
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When groups of different sizes combine, their averages do not meet in the middle: the bigger group drags the result its way. Every mixture and weighted-average question runs on one idea: work through the totals. Multiply each average or concentration by its own amount, add, and divide by the combined amount. The three failures are skipping the weights, gluing a percent to the wrong batch, and reporting the wrong member of the mixture family.
§2
Combine averages through their totals
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An average is a total in disguise: group size times average. To combine groups, recover the totals, add them, and divide by the combined size. The unweighted shortcut, averaging the averages, only works when the groups match exactly.
- Combined average $= \dfrac{n_1 a_1 + n_2 a_2}{n_1 + n_2}$.
- The result always lands between the two averages, closer to the bigger group's.
- Equal groups are the only time $\dfrac{a_1 + a_2}{2}$ is right.
Worked example. A class of $20$ averages $80$, and a class of $30$ averages $90$. What is the average across all $50$ students?
$20 \cdot 80 = 1600$ points and $30 \cdot 90 = 2700$ points: $4300$ in all.
$$\dfrac{4300}{50} = 86.$$ The unweighted $85$ ignores that the $90$-class has ten more students; the true average leans toward it.
§3
Weighted grades and prices work the same way
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Weights are group sizes wearing percent signs. A grade that is $30\%$ homework and $70\%$ tests multiplies each average by its weight and adds. Mixture prices do the same with pounds and dollars.
- Multiply each average by ITS OWN weight; the weights must total $1$ (or the full amount).
- Swapping the weights is the second-most-common slip after ignoring them.
- A per-unit price of a blend is total cost over total amount, never the average of the two prices.
Worked example. Homework ($30\%$ weight) averages $90$; tests ($70\%$) average $80$. What is the grade?
$0.3 \cdot 90 = 27$ and $0.7 \cdot 80 = 56$.
$$27 + 56 = 83.$$ The even split gives $85$, and swapped weights give $87$; both ignore which piece the syllabus says matters more.
§4
Mixtures: track the actual substance
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When solutions combine, the thing to follow is the pure substance. Each batch contributes its volume times its concentration; concentrations themselves never add and never average evenly unless the volumes match.
- Pure amount $=$ volume $\times$ concentration, batch by batch.
- Mixture concentration $= \dfrac{\text{total pure}}{\text{total volume}}$, always between the two inputs.
- For a target concentration, set pure-in equal to pure-out: $v_1 p_1 + v_2 p_2 = (v_1 + v_2)\, p_{\text{target}}$.
Worked example. $6$ liters of $20\%$ juice mix with $4$ liters of $50\%$ juice. What percent of the mix is juice?
$6 \cdot 0.2 = 1.2$ liters and $4 \cdot 0.5 = 2$ liters: $3.2$ liters of juice in $10$ liters.
$$\dfrac{3.2}{10} = 32\%.$$ The even average $35\%$ ignores the volumes, and $70\%$ adds concentrations, a move that would let two weak solutions make a strong one.
§5
The mixture family: report the member asked
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One mixture computation produces the total volume, the pure amount, the concentration, and each batch's contribution. All of them are correct numbers; the question names one.
Worked example. The $10$-liter mix above: how many liters of PURE juice does it hold?
Total $10$ liters; pure juice $3.2$ liters; concentration $32\%$.
The question says liters of pure juice: $$3.2.$$ Gridding $10$ or $32$ hands in a different family member.
Recover totals before averaging, keep every weight glued to its own group, track the pure substance through a mixture, and report the family member the question names.
§6
Three patterns that cost real points
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Three patterns recur on mixture and weighted-average questions. They are the same ones the diagnostic routes on.
The averages get averaged.
Two class averages meet at their midpoint even though one class is half again as large, or the biggest group's average stands in for everyone. Size never entered the computation.
Fix. Go through totals: size times average for each group, added, over the combined size. Check that the result leans toward the larger group.
The mixture is built on the wrong quantity.
Concentrations get added, weights swap groups, or the pure amount gets divided by one batch instead of the whole. The bookkeeping connected numbers that never belonged together.
Fix. Write each batch as volume times concentration before combining anything. Every percent stays attached to its own batch, and the final divide uses the whole mixture.
The wrong family member gets reported.
The total volume goes in when the pure amount was asked, or the concentration when the question wanted liters, or the average when it wanted the total.
Fix. After the mixture math, reread the final line: total, pure amount, or percent? Match your number to that exact name before committing.
Ten quick checks across the patterns: weighting averages by group size, weighted grades and blend prices, tracking the pure substance through a mixture, and reporting the family member asked. Pick or type your answer, then check. Progress is saved.