In the figure below two triangles are shown. In triangle ABC, a new triangle is formed by connecting midpoints of sides AB, BC and CA resulting in formation of 4 smaller triangles. Find the ratio of the blue region to the white region. In triangle PQR, the process used in triangle ABC is repeated successively three times. What is the ratio of the area of the green region to the white region in triangle PQR.
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 SSAT/TJ/AOS/Blair Problem. Target Time 25-35 seconds.
In the figure (not drawn to scale) below two triangles ABC and PCQ are shown with some measurements as specified. Find the length of CQ.
A Geometry Problem for TJ/AOS/Blair Prep (Easy). Target Time. 25-45 seconds.
In the figure below, BC and DE are line segments drawn parallel to the base FG of the triangle AFG. Some measurements are as specified in the figure. Find the lengths of line segments FG, BC, AB and BD.
A Geometry Problem for TJ/SHSAT/Blair/AOS Prep (Hard). Target Time. 30-50 seconds.
In the figure below 4 congruent circles are drawn with centers at A, B, C and D..Circle A is ts tangent to circle B and circle D, and circle C is tangent to circles B and D. What fraction of the quadrilateral ABCD is shaded purple? (The region between three circles is shaded purple.)
An Arithmetic Ratio Problem for TJ/Blair/SSAT/HSPT Prep (Medium). Target Time 20-30 seconds.
A and B are positive integers. If the sum of A and B is 17, what is the maximum possible ratio of A:B What is the minimum possible ratio of A:B? Would a unique answer be possible if the restriction that "A and B are positive integers is removed?"
An Arithmetic Problem for TJ/HSPT/SSAT/Blair Prep (Medium). Target Time. 25-50 seconds.
A and B are positive integers such that 1 <= B < A <= 10. What is the maximum possible value of A/ (A+B)? What is the minimum possible value of A/(A-B)?
An Aithmetic Problem for TJ/AOS/SSAT Prep (Easy). Target Time. 15-35 seconds.
Hareesh has a collection of fossils of different types. If the ratio of trilobites, bivalves and cephalopods is 2:3:7 and if his collection of three types of fossils contains 960 fossils, how many bivalves are in his collection?
A Difficult Geometry Problem for TJ/AOS/Blair Prep. Target Time. 25-120 seconds.
In the figure below, a right triangle ABC is shown.a circle is drawn with diameter AC and half of the circle is shaded blue. Two semicircles are drawn with diameters a and c. The region between the circle and the semicircles are shaded green. What is the area of the green regions? What is the ratio of the green regions to the blue region?
An Updated Version of a Geometry Question for TJ/Blair/AOS/SHSAT Prep (Medium). Target Time. 25-40 seconds.
In the figure below, a pentagon ABCDE is shown. What is the measure of angle R?
A Geometry Question for TJ/AOS/Blair Test Prep Problem (Medium). Target Time. 25-40 seconds.
The figure below shows a polygon ABCDEF. Sides AF, FE, BC and CD are 6, 8, 3, 4 respectively. Angles F and C are right angles. What is the perimeter of the polygon?
An Arithmetic Problem for TJ/SSSAT/AOS/Blair Prep (Medium). Target Time 20-30 seconds.
The ratio of Bob's age to Jill's age is 32:36. If Jill is 45 years old, how old is Bob?
ALiteral Equation Question for TJ/AOS/Blair Test Prep (Medium). Target Time. 2035 seconds.
6 M - 8 + 2 N = 2 M P. Solve for M.
A Percent Question for TJ/Blair/AOS/SHSAT Prep (Medium). Target Time. 20-45 seconds.
A% of B is what % of 20 B?
Algebra Question for TJ/AOS/Blair Test Prep (Easy). Target 15-50 seconds.
Please simplify the following Expressions:
- 52 (X - Y) + 52 (Y - X)
- 12 (U - V) + 6 (S - T) - 8 (V - U) - 14 (T - S)
- 6 a (b -c) + 9 b (a - c)
A Mathematics Problem for TJ/AOS/Blair Test Prep (Easy). Target Time. 20-35 seconds.
A quadrilateral is drawn by connecting 4 points A (0,4), B (8,8), C (4,0) and D(0,0). What is the area of ABCD?
A Math Problem for TJ/SSAT/AOS Test Prep Problems (Medium). Target Time. 20-45 seconds.
On a model, the scale is 1 mm = 100 meters. What will be the volume of a pile of sand which can be represented by a model cube with volume of 216 cubic mm?
A Geometry Question for TJ/SHSAT/AOS/Blair Test Prep (Medium). Target Time 25-40 seconds.
In a far away galaxy, far beyond the reach of our most powerful rockets and weapons exist three planets -- Helios, Feros and Plumbos. Assume the densities of the three planets are in the ratios of 1:40:60 and their diameters are 2, 4 and 8 million meters. Recall that Mass (commonly referred to as weight) = volume x density. What is the ratio of the volume of the largest plant to the volume of the smallest planet? What is the ratio of the mass of the largest planet to the mass of the smallest planet?
A Geometry Problem for TJ/AOS/Blair Problem (Medium). Target Time. 25-45 seconds.
A regular hexagon ABCDEF is shown in the figure below. On sides AB, CD and EF a semicircle is drawn as shown. What is the ratio of the total area of the semicircles to the area of the hexagon?
A Geometry Question for TJ/HSPT/Blair Test Prep (Medium). Target Time. 30
Three circles are shown in the figure below. The area of white region with the largest circle is how many times the area of the blue region?
A Simple Equation Question for TJ/AOS/SAT Prep (Easy). Target Time. 15-35 seconds.
Find solution to the equation 2X - 2(2 - 2X) = 6X -4.