This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is simplifying square roots by factoring out perfect squares. For √50 - √8, rewrite as √(252) - √(42) = 5√2 - 2√2 = 3√2. This simplification combines like terms under the radicals. The correct answer of 3√2 satisfies the expression because it is the exact simplified form. A representative incorrect option like 5√2 might result from adding instead of subtracting the terms. Another error, such as choosing √58, could stem from combining the numbers without factoring.

This is a Multiple Choice, Select One GRE Quant problem requiring solving a linear equation. The fundamental principle involves isolating the variable by combining like terms and using inverse operations. Starting with 2x - 3 = 5x + 12, we subtract 2x from both sides to get -3 = 3x + 12, then subtract 12 from both sides to get -15 = 3x. Dividing both sides by 3 yields x = -5. The Dorrect answer D (x = -5) satisfies the original equation: 2(-5) - 3 = -10 - 3 = -13, and 5(-5) + 12 = -25 + 12 = -13. A common mistake would be to incorrectly combine terms, such as getting x = 3 by making a sign error when moving terms across the equals sign.

This is a Multiple Choice, Select One GRE Quant problem involving percent discount calculations. The core principle is that if an item is 25% off, the customer pays 75% of the original price. Since the sale price is $72 and this represents 75% of the original price, we can set up the equation: 0.75 × (original price) = $72. Solving for the original price: original price = $72 ÷ 0.75 = $96. The Borrect answer B ($96) satisfies the requirement because 0.75 × $96 = $72. A common error would be to add 25% of $72 to $72 (giving $90), which incorrectly treats the sale price as the base for the percentage calculation rather than recognizing that the discount is based on the original price.

This is a Multiple Choice, Select One GRE Quant problem involving ratios and linear equations. The principle is that if x:y = 3:5, then x = 3k and y = 5k for some constant k. Given x + y = 64, we substitute to get 3k + 5k = 64, which gives 8k = 64, so k = 8. Therefore, x = 3×8 = 24 and y = 5×8 = 40. The Aorrect answer A (24) satisfies both conditions: x:y = 24:40 = 3:5 and x + y = 24 + 40 = 64. A common error would be to incorrectly set up the ratio relationship or make an arithmetic mistake when solving for k.

This is a Multiple Choice, Select One GRE Quantitative problem that tests function evaluation. The core principle involves substituting a given value into a function and performing the arithmetic operations. For f(x) = x² - 4x + 1, we substitute x = 5: f(5) = 5² - 4(5) + 1 = 25 - 20 + 1 = 6. The correct answer is 6 because it results from correctly applying the order of operations to evaluate the function at x = 5. Answer choice C (10) might result from an error such as calculating 5² - 4(5) + 5 or misapplying the function formula.

This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is the relationship between least common multiple (LCM), greatest common divisor (GCD), and the product of two numbers, where LCM(a, b) * GCD(a, b) = a * b. Given LCM(a, b) = 60 and GCD(a, b) = 6, the product ab = 60 * 6 = 360. The correct answer of 360 satisfies the relationship because it directly follows from multiplying the given LCM and GCD. A representative incorrect option like 120 might result from dividing instead of multiplying the values. Another error, such as choosing 180, could stem from misapplying prime factorizations or assuming incorrect factors.

This is a Multiple Choice, Select One GRE Quantitative problem that tests percentage increase calculations. The core principle involves calculating 12% of the initial value and adding it to the original amount. A 12% increase means the new value is 112% of the original: $2,000 × 1.12 = $2,240. Alternatively, 12% of $2,000 = 0.12 × $2,000 = $240, so the final value is $2,000 + $240 = $2,240. The correct answer is $2,240 because it accurately reflects a 12% increase from the initial investment. Answer choice A ($2,120) incorrectly calculates only 6% increase instead of 12%, giving $2,000 × 1.06 = $2,120.

This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is setting up and solving a system of linear equations based on given totals. Let n represent the number of notebooks and p the number of pens; then n + p = 14 and 3n + 2p = 34. Substituting p = 14 - n into the second equation gives 3n + 2(14 - n) = 34, which simplifies to n + 28 = 34, so n = 6. The correct answer of 6 notebooks satisfies the equations because 6 notebooks cost $18 and 8 pens cost $16, totaling $34 for 14 items. A representative incorrect option like 4 might result from mistakenly assuming equal costs or miscalculating the substitution. Another error, such as choosing 8, could stem from swapping the prices in the equations.

This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is solving a proportion by cross-multiplying or inverting. Given 5/x = 2/7, cross-multiplying gives 2x = 35, so x = 35/2. Alternatively, x = 5 * (7/2) = 35/2. The correct answer of 35/2 satisfies the equation because 5/(35/2) = 5 * 2/35 = 10/35 = 2/7. A representative incorrect option like 14/5 might result from inverting incorrectly. Another error, such as choosing 7/10, could stem from swapping numerator and denominator.

This is a Multiple Choice, Select One GRE Quantitative problem that tests fraction addition with different denominators. The core principle requires finding a common denominator before adding fractions. To add 3/4 + 5/6, we first find the least common denominator, which is 12. Converting: 3/4 = 9/12 and 5/6 = 10/12. Therefore, 3/4 + 5/6 = 9/12 + 10/12 = 19/12. The correct answer is 19/12 because it represents the sum in its simplest form as an improper fraction. Answer choice A (19/24) incorrectly assumes 24 as the common denominator without properly converting the fractions, yielding 18/24 + 20/24 = 38/24, which doesn't equal 19/24.