1.3 Problem Solving Strategies & Elimination Techniques

Translate First: Name the Target

Every Quant item is a short reading task wrapped around arithmetic or algebra. Testers who dive into numbers before naming the target produce the most common GMAT miss: they compute a real quantity that is not the quantity asked. Write a three-word label on the scratch pad — "target: senior count," "target: s − L," "target: final price" — then translate.

Worked translation. "The product of 3 more than n and 3 less than n is 40, and n is a positive integer. What is n?" Translation: (n + 3)(n − 3) = 40 → n² − 9 = 40 → n² = 49 → n = 7 or n = −7. The extra words "positive integer" discard −7. Target is n, so 7. If the last sentence had asked for n², the answer would be 49 — and 7 would be sitting in the list as a trap.

Worked translation (order). "5 less than twice a number is 17." That is 2n − 5 = 17, so 2n = 22, n = 11. The distractor setup 5 − 2n = 17 produces a negative and is how "less than" gets reversed.

Read the last sentence twice after you have a number. If the stem defined x and y and asked for 2x + 1, submitting x is a complete-the-wrong-problem miss. GMAC writes that miss into the answer list on purpose.

Backsolving: Test a Mid-Range Number First

Backsolving means inserting a numeric answer into the story and checking whether every condition holds. Use it when:

• the question asks for a single concrete quantity ("how many," "what is the cost"),
• the answers are sorted numbers,
• and writing the algebra would produce a system, a quadratic, or a messy rational equation.

Start with a mid-range value in the ordered list. One trial tells you whether the true quantity is smaller or larger, which knocks out an entire half of the remaining answers. A second trial usually finishes the item. You rarely need to test every option.

Worked example. A firm splits a $13,600 bonus among 14 analysts. Seniors receive $1,200 each and juniors receive $800 each. How many seniors received a bonus? Possible values: 4, 6, 8, 10.

Let S be seniors, so juniors = 14 − S. Test the mid-range value 8: 8 seniors and 6 juniors produce 8(1,200) + 6(800) = 9,600 + 4,800 = $14,400, which is $800 too high. Seniors are the expensive group, so 8 seniors is too many. Eliminate 8 and 10. Test 6: 6 seniors and 8 juniors produce 6(1,200) + 8(800) = 7,200 + 6,400 = $13,600. Match. The count of seniors is 6.

If the first mid-range test had been too low, you would have moved to a larger remaining value instead. Either way, the first test is a directional cut, not a guess.

Backsolving is a poor fit when the answers are algebraic expressions, or when plugging a number would require more arithmetic than solving the linear equation you already have. In those cases, use picking numbers or straight algebra.

Picking Numbers (Smart Substitution)

Picking numbers is the mirror image of backsolving. The answers are expressions or percents, and the stem never gives a starting dollar, a starting population, or a concrete x. You assign legal numbers, compute a numerical target, then evaluate each expression until one matches.

Worked example (percents, no base). A merchant buys at wholesale w, marks up p percent to list, then sells at d percent off list. What is the gross profit per unit?

Pick w = 100, p = 50, d = 20. List = 150. Sale = 150 − 30 = 120. Profit = 120 − 100 = 20. Any correct expression in w, p, and d must equal 20 at these values. The trap expression that adds p and −d as if they used the same base will equal 30, not 20.

Worked example (variables in answers). If a = 2b and the answers are expressions in b, pick b = 3 so a = 6, compute the asked quantity, and test the expressions. If two expressions both hit the same number, pick a second pair (b = 5) to break the tie. That is why 0 and 1 are dangerous first picks: too many expressions agree at those values.

After you pick numbers, evaluate every remaining option. Stopping at the first match is how a poorly chosen 0 or 1 lets two expressions survive.

Elimination, Distractors, and Extreme Values

Incorrect Quant answers are not random. They are intermediate results and algebra errors GMAC can predict:

1. Partial calculation. The question needs three steps; step 1's output is an answer. Example: you found 27 small cakes when the question asked how many more small than large.
2. Wrong variable. The stem defines x and y and asks for y; x is in the list.
3. Opposite sign or reciprocal. −k instead of k, 3/4 instead of 4/3, after an inequality flip or a division.
4. Added percents. +20% then −20% offered as "no change."

When the item is a "must be true" number-property statement, test a short extreme set rather than a single friendly integer:

• 0
• 1 and −1
• a large integer and a large negative
• a fraction strictly between 0 and 1 (1/2 or 1/4)

Fractions between 0 and 1 reverse several school reflexes. If x = 1/2, then x² = 1/4 smaller than x, √x ≈ 0.707 larger than x, and 1/x = 2 larger than x. An option that claims "squaring always increases a positive number" dies immediately.

When Algebra Is Faster

Do not become a strategy zealot. Direct algebra wins when the equation is already on the page.

• Linear in one variable: 3x + 5 = 20 → x = 5 is faster than testing four answers.
• Asked for an expression, not x: If 3x + 5 = 20, what is 6x + 10? Double the entire equation: 6x + 10 = 40. Solving for x first is extra work and creates a "submit 5" trap if 5 is listed.
• Clean consecutive-integer sums: n + (n+1) + (n+2) = 3n + 3 = 48 → n = 15 is faster than backsolving.
• Ugly answer numbers that are not ordered around the unknown: picking 7.3 as a test value is slower than solving.

A 15-second scan decides the tool: answers are neat sorted numbers → consider backsolving; answers contain variables or percents → consider picking numbers; stem is already an equation → algebra; stem is "must be true" → extreme values. Then commit. Switching methods at the 90-second mark is how items become 4-minute blanks.

Bookmark Strategy That Still Answers Every Item (3-Edit Budget)

Review & Edit does not let you skip hard items during the section. You must submit an answer for all 21 questions before the review screen exists, and the screen never appears if time expires. Build a first-pass protocol that respects that gate:

1. Answer every item as you meet it. If the method is unclear at about 1 minute 45 seconds, eliminate, guess from what remains, bookmark, and go.
2. Cap bookmarks at two or three items — the same size as the three-edit budget. Bookmarking ten questions is a promise you cannot keep.
3. Bookmark only items where extra time could change the answer: two remaining values, or arithmetic you may have signed wrong. Do not bookmark items you already know you guessed blindly unless time is plentiful.
4. Reach item 21 with a couple of minutes left whenever you can. That is the only way the review screen opens.
5. On review, recompute a bookmarked item from scratch. If the new value disagrees, spend one of the three edits. If it agrees, leave the original answer and keep the remaining edits.
6. Do not spend the last 40 seconds opening random non-bookmarked items to "just check." Random edits convert correct adaptive answers into wrong ones and exhaust the budget.

The 3-edit budget is a correction tool, not a second exam. The scoring engine still saw your first-pass answers when it chose later items; changing an early miss can still raise the final ability estimate, which is why the feature exists. It cannot rescue three unanswered items at the end. Finish the string first, then edit.

First-Pass Routine (Use on Every Problem Solving Item)

1. Box the target from the last sentence.
2. Glance at the answers: numbers vs. expressions; wide vs. tight spread.
3. Choose algebra, backsolving, picking numbers, or estimation.
4. Compute, then match the target, not an intermediate.
5. If time is gone and you are not sure, select, bookmark if it is one of your two or three, and start the next item.