7.2 Data Sufficiency: Algebra, Number Properties & Statistics

Now that we have firmly established the strategic AD/BCE framework and the core principles of Data Sufficiency, we must examine how these concepts apply across the specific mathematical domains tested on the exam. The Executive Assessment primarily evaluates your conceptual depth in Algebra, Number Properties, and Statistics through this sufficiency lens. Let's explore the specific rules, shortcuts, and traps within each area.

DS in Algebra: Equations, Unknowns, and the Linear Rule

A fundamental rule of linear algebra states that to solve for n unique, distinct variables, you typically need exactly n distinct, independent linear equations. The test makers know that you know this rule, and they will consistently try to manipulate it to see if you are paying attention.

The "n Equations, n Unknowns" Rule and Its Sneaky Exceptions If you see two variables in the question stem (for example, "What is the exact value of x + y?"), and Statement 1 gives you one equation while Statement 2 gives you another, your knee-jerk instinct might be to jump straight to choice C (both together are sufficient). However, you must pause and verify two critical conditions:

1. Are the equations truly independent? If Statement 1 tells you that x + y = 10, and Statement 2 tells you that 2x + 2y = 20, you must recognize that Statement 2 is merely Statement 1 multiplied by a scalar of 2. It provides absolutely zero new mathematical information. If you combine them, you still only have the equivalent of one equation for two variables, meaning they are still insufficient together (Choice E).
2. Do you actually need the individual values? Sometimes the question asks for the value of a combined expression, not the single variables themselves. If the question asks for the value of the expression (x + y), and Statement 1 is 3x + 3y = 15, you can simply divide the entire equation by 3 to arrive at x + y = 5. You have answered the question! Thus, Statement 1 ALONE is perfectly sufficient, even though you remain ignorant of the individual, isolated values of x or y.

The Quadratic Two-Solution Trap When dealing with variables raised to even exponents (like squares or to the fourth power) or quadratic equations, you must remember that solving them typically yields two possible roots—a positive and a negative.

• Imagine a Value Question that asks: "What is the value of x?"
• Statement 1 provides: x² = 25.
• Taking the square root tells us that x could be 5 OR x could be -5. Because there are two distinct possible values, Statement 1 does not provide a single, unique answer. Therefore, it is not sufficient.
• To achieve sufficiency and find a unique value, you would need additional information, such as a second statement declaring "x > 0", which would allow you to eliminate the negative root and confidently state that x is 5.

DS in Number Properties: Evens, Odds, Primes, and Divisibility

Number properties questions deal with the fundamental, theoretical characteristics of integers: evens and odds, prime factors, rules of divisibility, and positive/negative interactions. These are predominantly presented as Yes/No questions (e.g., "Is integer k divisible by 6?", "Is the product ab an odd integer?").

Advanced Re-framing Techniques The absolute secret to conquering number properties questions is re-framing the question stem into its simplest mathematical components before you even glance at the statements. Break the question down using prime factorization.

• Example Scenario: The question asks, "Is integer n divisible by 12?"
• The Re-frame Process: To be divisible by 12, the number n must be divisible by the prime factorization of 12. Since 12 = 3 × 4, and 4 = 2 × 2, the prime factorization of 12 is 2² × 3. Therefore, the real, simplified question you are trying to answer is: "Does the prime factorization of n contain at least one 3 and at least two 2s?"
• Once you have re-framed the question in this manner, evaluating the statements becomes a rapid exercise in checking off a mental checklist. You don't need to plug in massive numbers; you just look for the presence of those specific prime factors.

Another frequently tested domain is Even/Odd addition and multiplication rules. Memorizing the outcomes is critical, but applying them in reverse is where DS tests your skill. For instance, if a question asks "Is the product xy an even integer?", you should instantly re-frame this based on your knowledge that an even times anything is even. The real question becomes: "Is at least one of x or y an even integer?" If a statement allows you to prove that just one of the variables is even, you have achieved sufficiency, regardless of what the other variable happens to be.

DS in Statistics, Ratios, and Word Problems

Because the Executive Assessment excludes geometry, the Data Sufficiency format is most often applied to statistics, ratios, percentages, and word problems. The same sufficiency logic governs these problems, but the traps differ from algebra and number properties.

Statistics: Mean, Median, and Range A recurring DS trap is assuming that knowing one summary statistic determines the others.

• If a question asks "What is the median of set S?", a statement giving only the mean is insufficient — a symmetric set and a skewed set can share the same mean yet have different medians.
• If a question asks "What is the mean of set S?", re-state it as "What is (sum)/(count)?" A statement giving the sum alone is insufficient without the count, and a statement giving the count alone is insufficient without the sum; a statement giving both (or the average directly) is sufficient.
• For range questions, knowing both the maximum and minimum is sufficient; knowing only the maximum is not. Knowing the mean and the median together still does not determine the range.

Worked sufficiency check: Suppose the stem states that set S contains 5 distinct positive integers, and the question asks for the sum of S. Statement 1 gives "The mean of S is 10." Because sum = mean × count = 10 × 5 = 50, Statement 1 alone is sufficient (Choice A or D). A Statement 2 giving only "The median of S is 8" fixes the middle value but leaves the other four values free, so it cannot determine the sum — insufficient. Notice that the count came from the stem, not the statement; learning to read the stem for hidden constants is central to DS efficiency.

Ratios and Percentages Ratio DS questions test whether a ratio alone determines actual values.

• If the question asks "What is the total number of dogs and cats?", a statement giving only the ratio of dogs to cats (e.g., 3:5) is insufficient — infinitely many totals fit that ratio. A statement giving the total, or giving the count of one category, may be sufficient.
• If the question asks "What is the ratio of dogs to cats?", a statement giving both counts is sufficient; a statement giving only the difference between the counts is not.
• Percent-change traps: a 25% increase followed by a 20% decrease is not a 5% change — it returns to the original. In DS, test whether a statement gives the base (original) value or only the new value; without the base, a percent change cannot be evaluated for a specific result.

Word Problems and Overlapping Sets Overlapping-sets DS follows the formula Total = Group1 + Group2 − Both + Neither. With five quantities tied by one equation, you need enough statements to pin the specific variable the question asks for. A statement giving "Both" and another giving "Neither" is sufficient to solve for a missing group only if the total is also known; if the total is itself the question, neither statement alone may suffice.

The unifying skill is translation: convert the word problem into an equation, identify the exact variable the question asks for, and judge each statement by whether it uniquely determines that variable — not by whether it supplies "a lot" of information.