Week 3 · ACCURACY EARNS SPEED · 10–15 hrs

Algebra, statistics & Mock 1

Linear equations and algebraic manipulation

Quantitative

GMAT Focus tests linear equations less as "solve for $x$" than as translation and sufficiency. The topic rarely announces itself. It arrives as a word problem — rates, mixtures, ages, digits, profit — or as a Data Sufficiency stem where the variable you want will not isolate.

Two decisions recur. First, count independent equations against unknowns, then check for hidden constraints. "Positive integers", "distinct digits" or a bounded range can make one equation in two variables determinate; miss it and you call a statement insufficient when it is sufficient. Second, in DS, ask whether a unique numeric value is forced. $x=5-y$ is insufficient for $x$ — but may be sufficient for $x+y$. When the target is an expression such as $2x+3y$, do not solve for $x$ and $y$ separately; manipulate the given equations toward the expression.

The mechanical loss is arithmetic drift: solving for variables when the target is an expression, and dropping a sign when distributing a negative. Add or subtract equations first to see what cancels. If you clear a variable from a denominator, record the value it cannot take; that value will appear among the traps.

For $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$, a unique solution exists exactly when $a_1{b}_2-a_2{b}_1\mathrm{\ne }0$. When that determinant is zero, the system has either no solution or infinitely many, according to consistency.

Mean, median, mode and range

Quantitative

GMAT Focus rarely asks you to compute a mean or median from a clean list. It asks what happens when you add, remove, or replace a value, or whether a statement pins down the median. The arithmetic is trivial; the trap is distribution.

Two decisions separate answers. First, when a value changes, compare it to the current mean to predict the mean's direction, but locate it relative to the sorted set for the median. Adding a value equal to the mean leaves the mean unchanged; the median may still move if the new value's position shifts the middle. Second, in Data Sufficiency, mean and range are weak constraints: many sets share the same mean and range but have different medians. Ask whether the statements force one sorted arrangement or merely allow one.

Disguises: 'average' usually means arithmetic mean, but a question about 'typical' salary or home price is testing median's resistance to outliers. A set with duplicates makes mode and median position-counting traps. Removing a value can shrink the range only if it was the min or max; otherwise range is unchanged.

Worth memory: for $n$ sorted values, median position is $(n+1)\mathrm{/}2$ when $n$ is odd; for even $n$, average the two middle values. New mean after adding $x$ to $n$ values with old mean $M$: $(nM+x)\mathrm{/}(n+1)$.

Strong candidates lose points by computing the correct new mean and then assuming the median moved with it, or by reporting a mode for a set where every value appears once. The exam expects 'no mode', not a fabricated one.

Weighted averages

Quantitative

The exam almost never says "weighted average." It says overall average, combined mean, average price per unit, per-capita, or hands you a two-row table and asks for the mean of the whole column. Mixture problems — a 40% solution blended with a 70% one — are the same question in different clothing. The trigger to slow down is any setup that collapses two or more groups into a single figure without stating that their sizes are equal.

The mechanical failure that costs strong candidates points here is averaging the averages. If 30 values average 12 and 20 values average 17, the combined mean is $\frac{30(12)+20(17)}{50}=14$, not $14.5$. When the counts are equal both routes agree, so the habit goes unpunished until it does not.

Two decisions do most of the work:

• What is the weight — count, dollars, hours, distance? For average speed over equal distances, weight by time; that is why the answer is $\frac{2ab}{a+b}$, not $\frac{a+b}{2}$.
• Where does the answer sit relative to the midpoint? Always between the two group means, nearer the larger group.

Carry one formula: $\frac{w_1}{w_2}=\frac{\mathrm{\mid }{x}_2-\bar{x}\mathrm{\mid }}{\mathrm{\mid }\bar{x}-x_1\mathrm{\mid }}$ — the weights (counts) are inversely proportional to the distances from the combined mean. It answers ratio-of-group-size questions in one step, without simultaneous equations.

Critical Reasoning: strengthen and weaken

Verbal

Strengthen and weaken questions are assumption questions wearing a different verb. What is tested is not whether you can weigh evidence, but whether you can find the unstated link between the evidence given and the conclusion drawn, then judge which option moves that link.

Disguises: Which of the following, if true, would most seriously undermine the argument, provides the strongest support for, the argument is vulnerable to criticism on the grounds that, would most weaken the conclusion. The phrase if true is the tell — you are judging the effect of the option, not its plausibility.

Three decisions do the work:

1. Fix the conclusion. It is usually the prediction, recommendation, or causal claim, often carried by an implied therefore. Options that attack a premise, or a claim the author never made, are wrong.
2. State the gap before reading the options: what must hold for this evidence to support this conclusion?
3. Check direction and scope — the tempting option usually concerns a related quantity the argument never mentions.

The mechanical loss is treating strengthen as prove. The right answer only shifts likelihood; a modest effect on the assumption beats a striking fact about something else, and under time pressure strong candidates take the first true-and-relevant option and stop reading.

Use the negation test: if denying a candidate assumption would collapse the argument, that assumption carries the weight, and the correct answer is the option affirming or denying it — provided it stays inside the argument's terms.

Mock 1: taking a baseline that means something

Exam skills

Mock 1 is a measuring instrument, not a rehearsal for a good day. Its job is to record what you do by default when a section turns hostile and the clock is real. That is the only thing it can tell you that practice sets cannot, and the data is easy to corrupt.

Corruption has three usual sources: pausing the clock once ("just to finish this one"); taking it in slices across a day, which hides the fatigue that lands hardest in Data Insights, sat last; and best-effort mode, where you abandon your process to prove you can crack a hard question, then review a run that no longer resembles test day.

Three decisions carry the value:

• Finish every section, guessing the last questions rather than leaving blanks. A blank is a decision you would not make under the real clock.
• Sit it once, at the hour of your actual appointment, with the scratch materials you will be given.
• Log before you read explanations. Your memory of your reasoning overwrites the evidence of your behaviour.

The log is four cells: fast/slow x right/wrong. Fast-and-wrong is a content gap or a trap. Fast-and-right is ready. Slow-and-wrong is an abandonment rule you have not written yet. Slow-and-right is your speed backlog, and it is where most recoverable points sit — which is why reviewing only your errors is the standard way strong candidates waste a baseline.

Average pace $=$ section minutes $\mathrm{/}$ questions in section. It is an average, not a per-question cap: overspending on one question is fine only if you have banked the time elsewhere.

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