In writing all the numbers from 1 to 500, how many times digit 7 is used?

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"Outsmarting the Competition" - The Washington Post, 2001. (Hundreds of students accepted to TJ since 1994)

TJ test prep practice problems for our students and readers. The problems provide a very small sampling of what Optimal TJ Prep students work with.

In writing all the numbers from 1 to 500, how many times digit 7 is used?

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A cargo ship is transporting goats and dogs. The ratio of the number of goats to the number of dogs is 3:5. If the ratio of the average weight of a goat to the average weight of a dog is 3:2, goats account for what % (to the nearest whole number) of the weight of all the animals on the ship?

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A simulated spherical cancer cell has a radius of 7x10^-6 m and he diameter of an experimental nano particle is 3x10^-9 m. Assuming both to be spheres, what is the ratio of the volume of the cell to the volume of the nano particle?

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John is going on a 200 miles long trip and he must maintain a speed of 60 mph for the entire trip. If his average speed, for the first P miles, is x mph, what should be average speed, in terms of x and P for the rest of the trip ?

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A fair coin is tossed 4 times. What is the probability that an outcome other than no head or all four heads results?

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Melissa can paint her bedroom in six hours. If her younger brother joins her, together they can paint the room in 4 Hrs. Working alone, her brother will take how many hours topaint the same room?

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You are going from Ashville to Concertville to watch a concert. If you drive at 20 miles/Hr you arrive 8 Hrs late but if you drive at 50 mph, you will be only 2 Hrs late. How fast should you drive to be on time and what is distance between the two cities?

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In the figure below, multiple triangles are constructed using straight line segments including segment ABC. Angles are as marked. What are the measures of angles a, c and d?

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Wheel I has a diameter of 2' and revolves at 20 RPM. Wheel II has a diameter of 3' and revolves at 40 RPM. If both wheels revolve for 3 Hrs, what is the ratio of the distance traveled by wheel II to the distance traveled by wheel I?

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80 increased by X% is equal in value to 200 decreased by X%. Determine the exact value of X.

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The figure below shows a circle (O) with three triangles with vertex angles as shown. What is the total area of the triangles?

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The figure below shows scalene triangle ABC. Triangles PQR, STU and XYZ are created by connecting the midpoints of parent triangles respectively. What is the ray=tio of the total green area to total white area?

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The number line segment below ABCD is not drawn to scale. If the ratio AB:CD is 5:6 and the ratio BD:BC is 13:5, what is the ratio AB:BC?

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The figure (not drawn to scale) below shows a number line PT with line segment ratios as specified. What is the ratio of PQ:RT?

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On a trip from Startown to Destown and back, if you go at 60 mph, how fast should you drive on the return trip so that your round trip speed will be 70 mph?

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The figure below shows town A, B, C and D and paths connecting them. How may different ways are there to go from A to D and back (to A) while going through B exactly once?

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The future below shows a triangle ABC and aline segment DE such that DC= 2 AD and EC = 2 BE.

1. What is the ratio of the area of the triangle CDE to the area of triangle ABC?

2. What is the ratio of the area of triangle ABC to the area of trapezoid ABED ?

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How many integers between 19 and 155 are multiples of 6 and 8 but not both?

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The floor of a 15' x 18' room is to covered by tiles which are 5" x 3".

1. How many such tiles will be required?

2. If the length and the width of the tile is doubled, how many tiles will be required?

3. If the smaller tiles cost $3 each and the larger tiles cost $4 each, what will be the saving in $ by using the larger tiles?

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A large juice container measures 15 feet x 18 feet x 24 inches. A juice scoop measures 5"x3"x9". children are to be served one scoop juice during lunch. How may children can be fully served with one completely filled container?

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