Mistake Master
Right cell, right base, right type
A table hands you every number you need and several you do not. The arithmetic is light. The points slip away when a value comes from the wrong cell or an unread scale, when a fraction stands on the wrong base, and when a count answers a share question. Read the labels, find the of-phrase, then match the type of answer to the ask.
§1
What this topic is about
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A table or graph packs many numbers into one display, and every question about it turns on three reads: pulling the value from the right place (both labels, plus any scale), building fractions on the right base, and reporting the right type of answer, a count or a share. The SAT writes a wrong answer for each.
§2
Both labels, then the scale
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A two-way table pins every number under two labels at once, and a graph's axis label is part of every value on it. One label alone points at a whole row or column; the scale ignored turns thousands into units.
- A cell needs BOTH its row and its column confirmed; totals live at the edges.
- Read the axis title before any bar: "in thousands" multiplies every reading by $1{,}000$.
- In a pictograph, the key converts symbols to things, half symbols included.
Worked example. A survey of $200$ students: of $100$ juniors, $60$ chose science, $40$ art; of $100$ seniors, $70$ science, $30$ art. How many seniors chose art?
The senior row, the art column: $30$.
The juniors-art cell holds $40$, the senior total $100$; both are one label short of the question.
§3
The of-phrase names the base
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Fraction-of questions hide their denominator in an "of" phrase. "What fraction OF THE JUNIORS chose science" divides by juniors; "of all students" divides by everyone; "of the science choosers" divides by that column. Same numerator, three different answers.
- Find the of-phrase first, and put that group's total in the denominator.
- No restriction stated means the grand total.
- "What fraction of X are Y" and "what fraction of Y are X" are different questions.
Worked example. Same survey: what fraction of the juniors chose science?
Of the juniors: $\dfrac{60}{100} = \dfrac{3}{5}$.
Dividing by all $200$ gives $\dfrac{3}{10}$, and by the $130$ science choosers gives $\dfrac{6}{13}$; each answers a question with a different of-phrase.
§4
Complements and combinations
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Many table questions ask about everyone EXCEPT a group, or about two groups together. Build these from the pieces, and keep the total in view.
- NOT a group means the total minus that group, all of it.
- Two categories "or" each other by adding their counts (when nothing overlaps).
- A difference question subtracts totals of the same kind: column total minus column total.
Worked example. Of $80$ students, $45$ bus, $25$ walk, $10$ bike. What fraction of the class walks or bikes?
$25 + 10 = 35$ of the $80$: $\dfrac{35}{80} = \dfrac{7}{16}$.
$\dfrac{25}{80}$ counts walkers only, and $35$ alone is a count, not a fraction.
§5
Answer as a count or as a share
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The last read is the answer's type. "How many" wants a count. "What fraction" or "what percent" wants a share. The table hands you both, and the choices will contain both.
Confirm both labels and the scale, let the of-phrase set every denominator, build complements from the total, and match the answer's type, count or share, to the question's ask.
§6
Three patterns that cost real points
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Three patterns recur on table and graph questions. They are the same ones the diagnostic routes on.
The value comes from the wrong place.
A neighboring cell, a row total standing in for a cell, or a bar read without its axis scale. The number is real; its address is wrong.
Fix. Check the row label, the column label, and the axis title before trusting any value. Two labels pin a cell; one label pins only a line of the table.
The fraction stands on the wrong base.
The numerator is right and the denominator came from the grand total, or the column, when the question restricted to a row.
Fix. The of-phrase is the denominator's name. Find it before dividing, and when none appears, the base is everyone.
A count answers a fraction question, or the reverse.
The work is right and the answer arrives as the wrong type: $35$ for a fraction question, $\dfrac{7}{16}$ for a how-many question.
Fix. Reread the ask. "How many" is a count; "what fraction of" is a comparison. Match the type before gridding.
Ten quick checks across the patterns: pinning cells with both labels, applying scales and keys, choosing the base the of-phrase names, complements, and count-versus-share. Pick or type your answer, then check. Progress is saved.