GRE 2024 Quantitative Reasoning Practice Test 5 Question Paper with Solutions PDF

Step 1: Subtract 2 from both sides of the equation: \[ 3x + 2 - 2 = 11 - 2 \quad \Rightarrow \quad 3x = 9. \]
Step 2: Divide both sides by 3: \[ \frac{3x}{3} = \frac{9}{3} \quad \Rightarrow \quad x = 3. \]

Quick Tip: To solve linear equations, isolate the variable by performing inverse operations such as addition/subtraction and multiplication/division.

Step 1: Add the numbers: \[ 5 + 10 + 15 + 20 = 50. \]
Step 2: Divide the sum by the number of values (4): \[ \frac{50}{4} = 12.5. \]

Quick Tip: To find the arithmetic mean, add all the numbers together and divide by the total count of the numbers.

Step 1: Divide the total distance by the total time: \[ \frac{150}{2.5} = 60. \]
Thus, the average speed is \( 60 \) miles per hour.

Quick Tip: To calculate average speed, divide the total distance by the total time taken.

Step 1: Subtract \( 2y \) from both sides: \[ 2y - 7 - 2y = 3y + 4 - 2y \quad \Rightarrow \quad -7 = y + 4. \]
Step 2: Subtract 4 from both sides: \[ -7 - 4 = y + 4 - 4 \quad \Rightarrow \quad y = -11. \]

Quick Tip: When solving linear equations, isolate the variable by performing inverse operations like addition/subtraction or multiplication/division.

Step 1: Substitute \( 2 \) for \( x \) in the function: \[ f(2) = 2^2 - 3(2) + 2 = 4 - 6 + 2 = 0. \]
Thus, \( f(2) = 0 \).

Quick Tip: To evaluate a function at a specific value of \( x \), substitute the value of \( x \) into the expression and simplify.

Step 1: Use the distributive property: \[ (x + 3)(x - 2) = x(x - 2) + 3(x - 2) = x^2 - 2x + 3x - 6. \]
Step 2: Combine like terms: \[ x^2 - 2x + 3x - 6 = x^2 + x - 6. \]
Thus, the expanded form is \( x^2 + x - 6 \).

Quick Tip: To expand binomials, use the distributive property (also known as FOIL for two binomials): Multiply each term in the first binomial by each term in the second binomial.

Step 1: Take the square root of both sides: \[ x^2 = 16 \quad \Rightarrow \quad x = \pm 4. \]
Thus, the possible values of \( x \) are \( 4 \) or \( -4 \).

Quick Tip: When solving equations with squared terms, remember to take both the positive and negative square roots.

Step 1: Use the formula for the area of a triangle: \[ Area = \frac{1}{2} \times base \times height. \]
Step 2: Substitute the given values: \[ Area = \frac{1}{2} \times 8 \times 5 = 20 \, cm^2. \]
Thus, the area of the triangle is \( 20 \, cm^2 \).

Quick Tip: To find the area of a triangle, use the formula \( \frac{1}{2} \times base \times height \).

Step 1: Use the formula for the circumference of a circle: \[ C = 2\pi r. \]
Step 2: Substitute the given radius \( r = 7 \, cm \) and \( \pi \approx 3.14 \): \[ C = 2 \times 3.14 \times 7 = 43.96 \, cm. \]
Thus, the circumference of the circle is \( 43.96 \, cm \).

Quick Tip: To find the circumference of a circle, use the formula \( C = 2\pi r \), where \( r \) is the radius.

Step 1: Use the Pythagorean theorem: \[ a^2 + b^2 = c^2, \]
where \( a = 6 \, cm \), \( b = 8 \, cm \), and \( c \) is the length of the hypotenuse.

Step 2: Substitute the values: \[ 6^2 + 8^2 = c^2 \quad \Rightarrow \quad 36 + 64 = c^2 \quad \Rightarrow \quad 100 = c^2. \]

Step 3: Take the square root of both sides: \[ c = \sqrt{100} = 10 \, cm. \]
Thus, the length of the hypotenuse is \( 10 \, cm \).

Quick Tip: To find the length of the hypotenuse in a right triangle, use the Pythagorean theorem: \( a^2 + b^2 = c^2 \).

Step 1: Use the formula for the volume of a cylinder: \[ V = \pi r^2 h. \]
Step 2: Substitute the given values \( r = 3 \, cm \) and \( h = 5 \, cm \), and \( \pi \approx 3.14 \): \[ V = 3.14 \times 3^2 \times 5 = 3.14 \times 9 \times 5 = 141.3 \, cm^3. \]
Thus, the volume of the cylinder is \( 141.3 \, cm^3 \).

Quick Tip: To find the volume of a cylinder, use the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.

Step 1: Let the fifth number be \( x \). Then, the mean of the five numbers is given by: \[ \frac{7 + 9 + 12 + 5 + x}{5} = 8. \]
Step 2: Simplify the equation: \[ \frac{33 + x}{5} = 8. \]
Step 3: Multiply both sides by 5: \[ 33 + x = 40. \]
Step 4: Subtract 33 from both sides: \[ x = 7. \]
Thus, the fifth number is \( 7 \).

Quick Tip: To find a missing number when the mean is given, set up the equation for the mean, substitute the known values, and solve for the unknown number.

Step 1: Use the principle of inclusion and exclusion. The total number of people who like either coffee, tea, or both is: \[ 120 + 150 - 80 = 190. \]
Step 2: Subtract this from the total number of people surveyed: \[ 200 - 190 = 10. \]
Thus, 10 people do not like either coffee or tea.

Quick Tip: To solve problems involving sets, use the principle of inclusion and exclusion to avoid double-counting the people who like both coffee and tea.

Step 1: The median is the middle number in a sorted list. The given dataset is already sorted: \[ 5, 7, 9, 11, 13. \]
Step 2: The middle number is \( 9 \), which is the third number in the list.
Thus, the median is \( 9 \).

Quick Tip: To find the median of a dataset, first sort the numbers in increasing order. If there is an odd number of numbers, the median is the middle value.

Step 1: The total number of marbles is: \[ 4 + 5 + 6 = 15. \]
Step 2: The probability of picking a blue marble is: \[ \frac{5}{15} = \frac{1}{3}. \]
Thus, the probability of picking a blue marble is \( \frac{1}{3} \).

Quick Tip: To calculate probability, divide the number of favorable outcomes (blue marbles) by the total number of possible outcomes (total marbles).

Step 1: Distribute the \( 3 \) over the expression \( (x - 2) \): \[ 3(x - 2) = 3x - 6. \]
Step 2: Add the constant term \( 4 \) to the expression: \[ 3x - 6 + 4 = 3x - 2. \]
Thus, the simplified expression is \( 3x - 2 \).

Quick Tip: To simplify an expression, distribute the constant and combine like terms.

Step 1: Since \( x \) is directly proportional to \( y \), we can write the equation: \[ x = ky, \]
where \( k \) is the constant of proportionality.

Step 2: Use the given values \( x = 10 \) and \( y = 2 \) to find \( k \): \[ 10 = k \times 2 \quad \Rightarrow \quad k = 5. \]

Step 3: Now, when \( y = 8 \), substitute \( k = 5 \) into the equation: \[ x = 5 \times 8 = 40. \]
Thus, \( x = 40 \).

Quick Tip: For direct proportionality, use the formula \( x = ky \), and solve for \( k \) using known values. Then use this value of \( k \) to find the unknown \( x \).

Step 1: Start with the given equation: \[ 2x + 3 = 9. \]
Step 2: Subtract 3 from both sides: \[ 2x = 6. \]
Step 3: Divide both sides by 2: \[ x = 3. \]
Thus, the value of \( x \) is \( 3 \).

Quick Tip: To solve for \( x \) in a linear equation, isolate the variable by performing inverse operations (subtraction or division) on both sides of the equation.

Step 1: Use the Pythagorean theorem. Let the length of the other leg be \( x \). According to the Pythagorean theorem: \[ 5^2 + x^2 = 13^2. \]
Step 2: Simplify the equation: \[ 25 + x^2 = 169. \]
Step 3: Subtract 25 from both sides: \[ x^2 = 144. \]
Step 4: Take the square root of both sides: \[ x = 12. \]
Thus, the length of the other leg is \( 12 \, cm \).

Quick Tip: Use the Pythagorean theorem \( a^2 + b^2 = c^2 \) to find the missing side of a right triangle, where \( a \) and \( b \) are the legs and \( c \) is the hypotenuse.