The figures below (not drawn to scale) show two triangles, ABC and PQT, and their corresponding base and height lengths in terms of X. What is the ratio of the area of the triangle PQT to the area of triangle ABC?
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.
An Arithmetic Problem for TJ/SHSAT/SAT Prep (Easy). Target Time. 20-25 seconds.
A (= 5 x 7 x 7 x 9 x 13) , B = (5 x 5 x 3 x 3 x 3) and C (= 2 x 3 x 7) are three different integers. What is the ratio of the LCM: the GCF of A, B and C?
A Logic Problem for TJ/SAT/SHSAT/Blair Prep (Medium). Target Time. 25-45 seconds.
A 7th grade math student is designing a simple secret code based on the following rule. Each vowel is replaced by the next vowel in the English alphabet [ a becomes e] and each consonant is replaced by the consonant in the 2nd subsequent position [b is replaced by d and z is replaced by b]. Using this rule, what will the code words for coffee and pizza be? [Note. A typo has been corrected].
A Geometry Problem for SAT/TJ/AOS/SHSAT Prep (Medium). Target Time.25-45 seconds.
The figure [NOT DRAWN TO SCALE] shows a circle with radius 4 and three triangles -- a scalene, a right, and an equilateral triangle. What is the total area of the three sectors shaded blue?
An Algebra Word Problem for TJ/SHSAT/SAAT/Blair/AOS Prep (Medium). Target Time. 25-45 seconds.
A mouse has amassed cheese equal to three times its own weight. What % of the cheese must be eaten by the mouse so the mouse's weight just exceeds the weight of the leftover cheese. To solve this problem, please assume that the mouse is not excreting any waste.
A number problem for TJ/AOS/SAT/SHSAT Prep (Easy). Target Time. 20-35 seconds.
A four digit integer is made up of all odd digits. What is the difference between the greatest and the least possible such integer? [No digit can be repeated in each number]
A Geometry Word Prblem. (Easy). Tareget Time 20-35 seconds per part.
An angles measure is 60 deg more than 2/3 of its supplement. What are the measures of the angle and its supplement? The angle measure is how many deg greater than its supplement?
An Algebra Problem for TJ/SAT/AOS/Blair/SHSAT Prep (Medium). Target Time. 25-35 seconds.
The following three equations express the relationship between a, b, c and. d is what % of a?
2 a + b = 3 a
2 b + 3 c = 4 c
2 d + 3 b = c + 3 d
A Geometry for SAT/SHSAT/TJ/Blair Prep (Medium). Target Time. 25-60 seconds.
ABCDEFGH is a cube with side 6 (shown below). AR = AS = TF = HU = 2 units. What is the area of the polygon RSTU?
A Number line problem for TJ/AOS/Blair/SHSAT/SAT Prep (Easy). Target Time. 15-30 seconds.
17 students are standing in a line and are numbered 1 through 17. The teacher decided to reverse the numbering order, so number 1 became number 17 and number 17 became number 1. Which student's number remained unchanged?
A Rat & Cat % problem (Medium). Target Time. 30-70 seconds .
The weight of 11 rats equals the weight of 2 cats. 1) the average weight of 1 rat is what % less than the average weight of 1 cat? 2) the average weight of 1 cat is what % more than the average weight of 1 rat?
A Geometry Problem for TJ/AOS/Blair/SHSAT/SAT (Hard). Target Time. 20-50 seconds.
The figure below shows a polygon ABCDEF composed a rectangle, a right triangle and three equilateral triangles. What is the area of ABCDEF?
A % Problem for TJ/AOS/SHSAT/SAT/Blair Prep (Medium). Target Time. 15-45 seconds.
A classroom has 6 students aged 7 to 14 years and a teacher who is 50 years old. Of all the seven persons, whose % aging rate is the smallest?
An arithmetic Problem for TJ/AOS/SAT/SHSAT (medium). Target Time. 25-35 seconds.
Amy's mother makes lunch with a sandwich and a fruit. A different fruit is used per following schedule: Mon (Mango), Tue (Apple), Wed ( Blueberry), Thu (Strawberry), Friday (Grape), Saturday (Kiwi) and Sunday (Orange). If Today is Thursday, 297 days after today, which fruit will be in Amy's lunch bag?
An Average Problem for TJ/AOS/SSAT/SAT Prep (Easy). Target Time. 20-40 seconds.
The average weight of five teachers is 160 lbs.What should be the average weight of 15 students be so that the average weight of the entire group (5 teachers and 15 students) is 100 lbs? Note. There are many ways to solve this problem; the most efficient way should take 15-20 seconds.
An Algebra Proportion Problem for TJ/AOS/Blair/SSAT Prep (Medium). Target Time. 20-45 seconds.
What is the ratio of a:b if 2a +3 b = 3a + 2b -1 ?
An Profit and Loss Problem for TJ/Blair/AOS Prep (Medium). Target Time. 40-55 seconds.
James' stock portfolio grew 10% per years for the first two years but suffered a loss of 10% per year for the next two years. What is the ratio of the final value of his portfolio to the initial value of his portfolio?
A Variation Problem for TJ/AOS/Blair/SAT Problem (Medium). Target Time. 25-49 seconds.
T, F, U and N are positive numbers. If F, U and N are changed as describe in the figure below, U must be must be multiplied by what number to keep the equation valid?
A Proportion Problem for TJ/AOS/Blair/SAT Prep (Medium). Target Time. 35-45 seconds
The figure shows two linear line segments AB and PQ. with endpoints as marked. Point C on segment AB corresponds to 6 on segment PQ. What is the value of C?
A Ratio Problem for TJ/AOS/SSAT/SHSAT/SAT Prep (Easy). Target Time. 20-40 seconds.
Two touching gear wheels with sprockets (teeth) are in contact. The smaller wheel has 8 teeth whereas the larger wheel has 72 teeth. When the larger wheel makes 12 revolutions, how many revolutions will be completed by the smaller wheel?