ABC is an equilateral triangle with sides of length 6. BCD is a 30-60-90 triangle with angle CBD = 30 deg and angle BCD = 60 deg.Squares are constructed on sides AB, AC, CD and BD. What is the sum of the areas of the four squares ?
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TJ Prep Problems
TJ Prep practice problems for our students and readers. The problems provide a very small sampling of what Optimal TJ Prep students work with.
A Geometry Problem for TJ Exam Prep. (Hard). Target Time. 20-35 seconds.
The figure below shows square ABCD and an inscribed circle. The midpoints of the sides of ABCD are connected to make a smaller square. The region between the circle and the smaller square is shaded yellow. What is the ratio of the yellow area to the remaining area (white) inside ABCD ?
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A Geometry Problem for TJHSST (TJ) Exam Prep. (Medium). 15-25 seconds/part.
The figure below shows a quadrilateral PQRS with angles x, z, m, y, w and n as marked.
1. What is w + n in terms of x and z ?
2. What is w and n in terms of m and y ?
A Word Problem for TJHSST (TJ) Exam Prep. (Medium). Target Time. 15-25 seconds per part.
In 2 years Bob will 16 times as old as his grandson will be then, and in 10 years Bob will be 12 times as old as his grandson will be then.
1. How old are Bob and his grandson today?
2. How old was Bob when his grandson was born?
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An Algebra Problem for TJHSST Admission Test Prep. (Easy). Target Time. 15-25 seconds
If u is+9 is increased by 2 t, the result is 7 times as large as u, what is u in terms of t?
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A Word Problem for TJ Admission Exam/SAT Prep (Hard). Target Time. 35-50 seconds.
A village is located at the center of a dense forest. The population of Coyotes is distributed as follows:50% of the coyotes are located with two miles of the village center, 40% coyotes are located at a distance of 2 to 4 miles from the village center, and the remaining 10% of the coyotes are located between 4 to 5 miles from the city center. What is the ratio of the population density of the coyotes in the area less than 2 miles of the village center to the population density between 4 and 5 miles from the city center?
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A Geometry Problem for TJ Admission Test Prep. (Hard). 35-50 seconds.
The figure below is not drawn to scale and shows two triangles PQO and QTR. If the area of PQO is equal to the area of QTR, what is the height of triangle QTR ?
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An Algebra Problem for TJ/SHSAT/HSPT Prep. (Medium). Target Time. 15-40 seconds.
Please simplify the following algebraic expressions:
1. (X^2 - Y^2) (X + Y)^-1 (X - Y)^-1
2. (X + Y) (X^2 - XY + Y^2)
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A Geometry Problem for TJ/HSPT/SSAT/SHSAT/SAT Prep. (Hard). Target Time. 25-40 seconds.
A revolving wheel with a diameter of 1/4 m will make how many revolutions if it travels 6280 m without sliding ? (Assume Pi = 3.14)
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A Basic Geometry Problem for TJ Test/HSPT/SHSAT/SAT Prep. (Hard). Target Time. 35-50 seconds.
A toy train playground maker can choose from one of the two types of stakes for its fencing. Each stake is constructed from a strong alloy metal wire. Stake I combines a rectangle and a triangle whereas stake II combines a rectangle and a semi circle; dimension in units as shown. In making a fence that is 3600 units long, which stake will use less wire and by how many units ?
A Word Problem for TJ/HSPT/SSAT/SHSAT Prep. (Medium). Target Time. 30-35 seconds.
Three goats and 5 cows cost $1180.00 and 4 goats and 2 cows cost $640.00. What is the price of 3 Goats and three cows ?
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Al Algebra/Proportion Problem for TJ/HSPT/SSAT/SHSAT Prep. (Medium). Target Time. 20-45 seconds/part.
The weight 5 dogs and 3 cats is the same as the weight of 3 dogs and 8 cats.
1. The weight of a cat is what % less than the weight of a dog?
2. The weight of a dog is what % more than the weight of a cat?
3. What is the radio of the weight of 5 cats to the weight of 3 dogs?
An Algebra Problem for TJ/HSPT/SHSAT Prep (Medium). Target Time. 20-30 seconds.
M. N and O are positive numbers. If MN = 3, NO= 4 and OM = 12, what is the product of MNO?
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A Geometry Problem for TJ Test/HSPT/SSAT/SAT Prep. (Hard). Target Time. 20-35 seconds per part.
The figure below shows a triangle ABC. DE is parallel to AB such that AD is 1/2 of CD.
1. What is the ratio of the perimeter of CDE to the perimeter of CAB?
2. What is the ratio of the area of CDE to the area of CAB
3. What is the ratio of the area of CDE to the area of ADEB?
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A Proportion Problem for TJ Test/HSPT/SSAT Prep. (Medium). Target Time. 15=35 seconds.
AB and CD are two line segments. The ratio of CD to AB is 5:3. If CD is increased by 20% and AB is decreased by 20%, what is the ratio of new CD to new AB?
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A Literal Equation Problem for TJ Test/SHSAT/HSPT Prep. (Medium). Tsarget Time. 15-35 seconds.
Solve for b in in terms of a, in the equation ( a + b is nonzero):
a / (a + b) = 1 / (a + b)
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A Proportion Problem for TJ Test/HSPT/SSAT Prep. (Hard). Target Time. 10-40 seconds.
How many pieces, each T yard long, can be cut from a P' long wire?
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A Percent Problem for TJ Test/HSPT/SSAT/SHSAT Prep (Medium). Target Time. 20-30 seconds.
In the equation 3 X + 2 Y = 60, Y is what % greater than X ?
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A Speed-Distance Problem for TJ Test/HSPT/SSAT/HSPT Prep. (Medium). Target Time. 25-40 seconds.
Mohan is driving from Motown to Yotown via Transitville. The distance from Motown to Transitville is half of the distance from Transitville to Yotown. If Mohan's driving speed from Transitville to Yotown is 25% greater than his driving speed from Motown to Transitville, what fraction of Mohan's total driving time was spent traveling from Motown to Transitville?
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A Geometry Problem for TJ/HSPT/SAT Prep. (Medium). Target Time. 15-35 seconds.
What is the length of the side of a cube whose volume and surface area are equal in magnitude?
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