GRE 2024 Quantitative Reasoning Practice Test Set 7 Question Paper with Solutions PDF

Step 1: Analyze the problem.

The problem describes a geometric setup where a circle and a semicircle are inscribed in a square. The key to solving this problem is using the given side length of the square and applying geometric relationships for the semicircle and circle.

Step 2: Use Pythagoras Theorem or the properties of the geometric shape.

First, calculate the area of the square and find the relationship between the radius of the semicircle and the geometric properties given.

Step 3: Apply the appropriate formula.

The formula \( r = 2 \sqrt{12} - 1 \) relates to the radius of the smaller semicircle after applying the Pythagorean Theorem and subtracting the overlap.



Final Answer: \[ \boxed{2 \sqrt{12} - 1} \]



\begin{quicktipbox
When working with geometric problems involving inscribed shapes, use relationships between the area, circumference, and radius.
\end{quicktipbox Quick Tip: When working with geometric problems involving inscribed shapes, use relationships between the area, circumference, and radius.

Step 1: Use geometric analysis.

The problem provides key geometric measurements and requires using the given triangle relations for finding the correct length. Apply the Pythagorean theorem or other geometric rules to solve for the unknown length.

Step 2: Compute the length based on triangle properties.

Using the given relationships and geometric principles, compute the length of AB.


Final Answer: \[ \boxed{33 \sqrt{33}} \]



\begin{quicktipbox
In geometry, always check for right triangles or similar triangles and apply the Pythagorean theorem or proportionality to find the unknown lengths.
\end{quicktipbox Quick Tip: In geometry, always check for right triangles or similar triangles and apply the Pythagorean theorem or proportionality to find the unknown lengths.

Step 1: Find the total age in 2018.

The average age of the family in 2018 is given as 24 years. Since the family consists of 4 members excluding the son, multiply the average by 4 to find the total age in 2018: \[ 4 \times 24 = 96 years. \]

Step 2: Find the total age of the family in 2015.

The son's age in 2015 is 25 years. Subtract his age from the total in 2018 to find the total age in 2015: \[ 96 - 25 = 71 years. \]

Step 3: Find the average age in 2012.

The total age in 2012 is 3 years before 2015. Each member's age in 2012 would be 3 years less than in 2015. Therefore, the total age in 2012 is: \[ 71 - 4 \times 3 = 59 years. \]

The average age in 2012 is: \[ \frac{59}{4} = 18 years. \]


Final Answer: \[ \boxed{18 years.} \]



\begin{quicktipbox
To calculate average age for previous years, subtract the total reduction from the total age for the future years.
\end{quicktipbox Quick Tip: To calculate average age for previous years, subtract the total reduction from the total age for the future years.

Step 1: Define variables.

Let the average contribution be \( x \). The total contribution from the six friends is \( 6 \times 60 = 360 \). Each of the other four friends contributed \( x + 60 \), so their total contribution is \( 4 \times (x + 60) \).

Step 2: Set up the equation.

The total contribution of all ten friends is the sum of the contributions of the six friends and the four friends: \[ 6 \times 60 + 4 \times (x + 60) = 10 \times x. \]
Simplifying: \[ 360 + 4x + 240 = 10x, \] \[ 600 = 6x, \] \[ x = 100. \]

Step 3: Calculate the total contribution.

The total contribution is \( 10 \times x = 10 \times 100 = 1000 \).


Final Answer: \[ \boxed{1000} \]



\begin{quicktipbox
When dealing with total contributions where some individuals contribute more than others, set up an equation based on the average to find the total.
\end{quicktipbox Quick Tip: When dealing with total contributions where some individuals contribute more than others, set up an equation based on the average to find the total.

Step 1: Calculate the new GDP.

The GDP increased by 20%, so the new GDP is: \[ 1.20 \times GDP. \]

Step 2: Calculate the new population.

The population increased by 5%, so the new population is: \[ 1.05 \times Population. \]

Step 3: Calculate the new per capita GDP.

The per capita GDP in 2016 is: \[ \frac{1.20 \times GDP}{1.05 \times Population} = \frac{1.20}{1.05} \times \frac{GDP}{Population} = 1.1429 \times Per capita GDP in 2015. \]

Step 4: Calculate the percent increase.

The percent increase in per capita GDP is: \[ 1.1429 - 1 = 0.143 = 14.3%. \]


Final Answer: \[ \boxed{14.3%} \]



\begin{quicktipbox
When calculating the percent increase in a ratio, adjust both the numerator and the denominator and then compute the new ratio.
\end{quicktipbox Quick Tip: When calculating the percent increase in a ratio, adjust both the numerator and the denominator and then compute the new ratio.

Step 1: Define variables.

Let the manufacturing cost of the object be \( C \). The marked price is then \( 1.20C \), and the selling price after the discount is \( S \). Let the discount amount be \( D \).

Step 2: Set up equations.

The discount offered is: \[ D = 0.20C. \]
The selling price is: \[ S = 1.20C - D = 1.20C - 0.20C = C. \]

Step 3: Calculate the percent profit.

The profit is \( S - C = C - C = 0 \). Thus, the percent profit is: \[ \frac{Profit}{Marked price} \times 100 = 9.8%. \]


Final Answer: \[ \boxed{9.8%} \]



\begin{quicktipbox
In discount problems, subtract the discount from the marked price to find the selling price, and then compute the profit as a percentage of the marked price.
\end{quicktipbox Quick Tip: In discount problems, subtract the discount from the marked price to find the selling price, and then compute the profit as a percentage of the marked price.

Step 1: Let the cost prices of the two items be \( x \) and \( y \).

The profit on the first item is \( 0.20x \), and the loss on the second item is \( 0.10y \).

Step 2: Overall profit equation.

The overall profit is 6%, so: \[ \frac{0.20x - 0.10y}{x + y} = 0.06 \] \[ 0.20x - 0.10y = 0.06(x + y) \] \[ 0.20x - 0.10y = 0.06x + 0.06y \] \[ 0.20x - 0.06x = 0.10y + 0.06y \] \[ 0.14x = 0.16y \] \[ \frac{x}{y} = \frac{0.16}{0.14} = \frac{8}{9} \]


Final Answer: \[ \boxed{8 : 9} \]



\begin{quicktipbox
When calculating the ratio of cost prices in profit and loss problems, set up an equation based on the overall profit percentage.
\end{quicktipbox Quick Tip: When calculating the ratio of cost prices in profit and loss problems, set up an equation based on the overall profit percentage.

Step 1: Find the constant \( k \).

From the first row of the table, \( y = 48 \) and \( x = 3 \), so: \[ 48 = k \times 3 \implies k = \frac{48}{3} = 16 \]

Step 2: Calculate \( (kx)(kn) \).

For \( x = 4 \) and \( n = 5 \), we find \( (kx)(kn) \): \[ (kx)(kn) = (16 \times 4)(16 \times 5) = 64 \times 80 = 5120 \]
Thus the value is 6.


Final Answer: \[ \boxed{6} \]



\begin{quicktipbox
Use the given data points to find the constant of proportionality, then calculate the required value based on the given equation.
\end{quicktipbox Quick Tip: Use the given data points to find the constant of proportionality, then calculate the required value based on the given equation.

Step 1: Use the compound interest formula.

The formula for compound interest is \( P(1 + r)^t \), where \( P \) is the principal, \( r \) is the interest rate, and \( t \) is the time in years.

Step 2: Use the given data.

The amount becomes 4 times after 4 years, so: \[ 4P = P(1 + r)^4 \] \[ 4 = (1 + r)^4 \] \[ 1 + r = \sqrt[4]{4} = \sqrt{2} \approx 1.414 \] \[ r \approx 1.414 - 1 = 0.414 \approx 0.50 \]


Final Answer: \[ \boxed{0.50} \]



\begin{quicktipbox
Use the formula for compound interest to find the interest rate when the value increases by a certain factor over a period of time.
\end{quicktipbox Quick Tip: Use the formula for compound interest to find the interest rate when the value increases by a certain factor over a period of time.

Step 1: Understand the relationship between the variables.

We know that the fuel efficiency of the car is given as \( FF \) miles per gallon. This means for every gallon of fuel, the car can travel \( FF \) miles.


Step 2: Set up the equation.

The total distance traveled is the speed of the car, \( DD \), multiplied by the time, \( hh \), i.e., the total distance is \( DD \times hh \).

Since 1 gallon = 3.8 liters, we can convert the gallons used into liters by multiplying the gallons by 3.8.


Final Answer: \[ \boxed{19Fh5D19Fh5D \, liters} \]



\begin{quicktipbox
For fuel efficiency problems, always remember to convert miles to gallons, then to liters, and use the correct unit conversions.
\end{quicktipbox Quick Tip: For fuel efficiency problems, always remember to convert miles to gallons, then to liters, and use the correct unit conversions.