Word Problems(Geometric Problems)

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✔A rectangular plot of ground is 120 yards long and 80 yards wide. To double the area of the plot, while retaining the rectangular shape, strips of equal width will be added at one end and along one side. Find the width of the strip.
✔Find the length of a side of an equilateral triangle whose altitude is 3 feet shorter than a side.
✔Find the side of a square whose diagonal is 5 feet longer than a side.
✔Find the dimensions of a rectangle where the altitude is four sevenths of the base and the perimeter is 330feet
✔ A rope 75 feet long is cut into two pieces where one is 11 feet longer than the other. Find their lengths

Worded Problems(Motion Problems)

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✔An airplane, flying with the wind, took 2 hours for a 1000-mile flight, and 2.5 hours for the return flight. Find the wind velocity and the speed of the airplane in still air.
✔A motor boat can travel 15 miles per hour downstream and 9 miles per hour upstream on a certain river. Find the rate of its current and the rate at which the boat can travel in still water.
✔An airplane flew 660 miles with the wind, and then took 40 minutes longer for the return flight against the wind. If the plane flies 200 mph in still air, find the speed of the wind.
✔A motorboat takes 4 hours to travel 30 miles downstream and then 18 miles upstream on a river whose current flows at the rate of 3 miles per hour. How fast can the boat travel in still water?
✔A boat, which can travel 8 miles per hour in still water, travels upstream for a certain time in a river whose current flows 2 miles per hour. Then, the boat returns to its starting point. If the trip, up and back, consumed 8 hours, how long did the boat travel up-stream?
✔A boat travels 60 miles upstream in 15 hours. The same boat makes a return trip downstream in 6 hours. Find the speed of the boat int the river
✔A jet fighter travels 2000 miles in 3.5 hours with the tail wind. The return trip, into the wind takes 4 hours. Find the wind speed and the jet speed.

Worded Problems(Work Problems)

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✔When A and B both work, they can paint a certain house in 8 days. Also, they could paint this house if A worked 12 days and B worked 6 days. How long would it take each to paint the house alone?
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✔A tank has one supply pipe in which could fill the tank in 5 hours and another which could fill it in 9 hours. How many hours will it take to fill the tank if both pipes are used simultaneously?
✔How long will it take workers A and B, together, to finish a job which can be done by A alone in 6 days and by B alone in 4 days?
✔In what time would A, B and C together do a piece of work if A alone could do it in 6 hrs. more , B alone in 1 hr more, and C alone in twice the time?

Ratio and Proportion

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✔The sides of a triangle are 10, 9 and 15 inches long. In a similar triangle, the longest side is 21 inches long. Find the other sides.
✔Divide 80 into two parts such that the ratio of one part, decreased by 4, to the other part, decreased by 8, is 1:3.
✔A line 20 inches long is divided into two parts whose lengths have the ratio 3:7. Find the lengths of the parts.
✔A basketball player is 6 feet 8 inches tall. How far should he stand from the light which is 15 feet above the ground, in order to cast a shadow 25 feet long?

Worded Problems(Number Problems)

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✔The sum of the reciprocals of two numbers is 7. The larger reciprocal exceeds the smaller by \(\frac{7}{ 3}\). Find the numbers.
✔Find two consecutive odd integers whose product is 1023.
✔Separate 27 into two parts whose product is 92.
✔Find the two consecutive positive integers whose product is 506.

Worded Problems(Digit Problems)

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✔In an integer between 10 and 100, the units' digit is 3 greater than the tens' digit. Find the integer if it is 4 times as large as the sum of its digits.

Quadratic Equations

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✔The floor of a basement game room is covered by 480 square asphalt tiles of a certain size. If 3 inches is added to the length of a side of each tile, the floor could be covered by 270 tiles. Find the length of a side of the tile on the floor.
✔If \(A\) is a cross-section area of a chimney, its so-called effective area \(E\) is the smallest root of the equation \(E^2 -2AE+A^2-0.36A =0\). Solve \(E\) interms of \(A\). From the result, find \(E\) if \(A = 30\) square feet.
✔If an object is shot vertically from the surface of the earth with an initial velocity of \(v\) feet per second, and if air resistance and other disturbing factors are neglected, it is found that \(s = vt - 0.5gt^2\), where \(s\) feet is the height of the object above the surface at the end of \(t\) seconds and \(g = 32\), approximately. (a) Solve for \(t\) in terms of \(s\) (b) If \(v=300 \text{fps}\), use (a) to find where \(s=450\) and \(s=0\).
✔If an object is shot vertically from the surface of the earth with an initial velocity of v feet per second and if the air resistance and other disturbing factors are neglected, it is found that \(s = vt -0.5gt^2\), where s feet is the height of the object above the surface at the end of t seconds and g = 32 feet per second square. (a) Solve t in terms of s. (b) If v = 300 feet per second use (a) to find when s =450 and s= 0.
✔The base of a triangle is 4 feet shorter than its altitude, and the area of the triangle is 126 square feet. Find the base and the altitude.
✔An open rectangular gutter is to be made by turning up the sides of a piece of metal 18 inches wide. If the cross-section area of the resulting gutter is 36 square inches, find the gutter's depth.
✔After plowing a border inside a rectangular field 60 rods wide and 80 rods long, a farmer finds that one half of the field remains to be plowed. How wide is the border?

Geometric Progression (Solutions)

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✔What is the sum of the infinite geometric series of 1, -1/5, 1/25, ...?
✔A rubber ball is dropped from a height of 50 inches. On each rebound, the ball rises to 3/4 of the height from which it last fell. Find the distance travelled by the ball in coming to rest.
✔If the 6th term of a geometric progression (g.p.) is 4 and the 10th term is 4/81, find r.

Equations in one variable

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✔A student in an algebra class has test grades of 81 and 92. What must he score on the fourth test in order to have an average of 90?

Cubic Equations

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✔Graph the curve \(C = 0.0031t^3-0.21t^2+5.54t+19.1\), where \(0\pm t \pm 23\).

Modeling with Algebra

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✔A Ferris wheel has a radius of 75 ft. You board a car at the bottom of the Ferris wheel, which is 10 feet above the ground, and rotate 255 degrees counter clockwise before the ride temporarily stops. How high above the ground are you when the ride stops? If the radius of the Ferris wheel is doubled, is your height above the ground doubled? Explain your reasoning.

Worded Problems(Mixture Problems)

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✔How much copper and how much iron should be added to 100 pounds of an alloy containing 25% copper and 40 % iron in order to obtain an alloy containing 30% copper and 50% iron.
✔An alloy contains 20% silver and 30% lead. How much silver and how much lead should be added to 100 pounds of alloy in order to obtain an alloy containing 25% silver and \(33\frac{1}{3}\%\) lead?
✔We wish to obtain a 40% solution of nitric acid by mixing a 20% solution in a 70% solution of acid. What percentage of the final solution should be taken from the 20% and 70% solutions?
✔How much of a 40% solution of alcohol and how much of an 80% solution should be mixed to give 40 gallons of 50% solution?
✔A jewelry craftsman needs 100 grams of gold alloy, of 75 % pure gold for his products. Only two alloys of fold are available from the supplier. The first is 80 % pure gold and the other is 60% pure gold. How many grams of 80% gold alloy must he use to suit his requirement?
✔A 10 kg salt solution originally \(4 \%\) by weight NaCl in water is evaporated until the concentration is 5% by weight NaCl. What percentage of water in the original solution is evaporated?

Arithmetic Progression

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✔A man invests 500 dollars at the end of each year for 16 years at 5% simple interest. What is the accumulated value of his investments just after the 16th investment. if no interest has been withdrawn.
✔Arithmetic Progression

System of Linear Equation

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✔Of the 316 people watching a movie, there are 78 more children than women and 56 more women than men. The number of men in the movie house is ______.
✔It takes a motorboat \(1\frac{1}{3}\) hours to go 20 miles downstream and \(2\frac{2}{9}\) hours to return. Find the rate of the current and rate of the boat in still water.
✔When we placed a boy 4 feet from the fulcrum, and a girl 12 feet from the fulcrum on one side of a teeterboard, they balanced a man weighing 150 pounds who is seated 6 feet from the fulcrum on the other side. Also, the board is balanced if the boy moves to 8 feet and the girl to 6 feet from the fulcrum on their side. Find their weight.
✔At present John's age is 30% of his father's age. Twenty years from now, Jon's age will by 58% of his father's age. How old are they?
✔If the smaller dimension of a rectangle is increased by 3 feet and the larger dimension by 5 feet, one dimension becomes \(\frac{3}{5}\) of the other, and the area is increased by 135 square feet. Find the original dimensions.
✔A contactor has a daily payroll of 277 dollars when employing some men at 8 dollars per day and the others at 9 per day. On increasing those at 8 dollars by \(50\%\), and doubling those at 9 dollars, he creates a daily payroll of 474. Find how many he originally employed at each wage?
✔Two angles are supplementary (their sum is 180^o). If one is doubled, the result is 69^o less than the other angle. Find the other angles.