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Compilations of Problems in Algebra

1. Ten less than four times a certain number is 14. Determine the number.

2. The hypotenuse of a right triangle is 34 cm. Find the length of the two legs if one leg is 14 cm longer than the other.

3. Find the equation whose roots are the reciprocals of the roots of the equation 2x^2 - 3x – 5 = 0.

4. The sum of two numbers is 21 and one number is twice the other. Find the numbers.

5. For a particular experiment, you need 5 liters of a 10% solution. You find 7% and 12% solution on the shelves. How much of the 7% solution should you mix with the appropriate amount of the 12% solution to get 5 liters of a 10% solution.

6. A circle with a radius of 6 has half of its area removed by a cutting border of uniform width. Find the width of the border.

7. Two triangles have equal bases. The altitude of one triangle is 3 units more than its base while the altitude of the other is 3 units less than its base. Find the altitudes if the areas of the triangle differ by 21 square units.

8. The roots of the quadratic equation are 1/3 and 1/4. What is the equation?

9. Find the 30th term of the progression 4, 7, 10, …

10. The denominator of a certain fraction is three more than twice the numerator. If 7 is added to both terms of the fraction, the resulting fraction is 3/5. Find the original fraction.

11. A piece of wire is shaped to enclose a square whose area is 169 sq. cm. It is then reshaped to enclose a rectangle whose length is 15 cm. Find the area of the rectangle.

12. Find the sum of 6, -2, 2/3, …

13. In the expansion of (x+4y)^12, find the numerical coefficient of the 5th term.

14. Find the sum of the first 10 terms of the progression 2, 4, 8, 16, …

15. Find the ratio of an infinite geometric progression if the sum is 2 and the first term is 1/2.

16. Find the 30th term of the sequence 4, 7, 10, …

17. A man rows downstream at the rate of 5 mph and upstream at the rate of 2 mph. How far downstream should he go if he is to return in 7/4 hours after leaving?

18. Find the mean proportional of 4 and 36.

19. Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as Ana is now. How old is Ana now?

20. Determine x so that x, 2x+7, 10x–7 will be a geometric sequence.

21. Mike, Louis and Joy can mow the lawn in 4, 6 and 7 hours, respectively. What fraction of the yard can they mow in 1 hour if they work together?

22. Given: (x) = (x+3) (x–4) + 4. When f(x) is divided by (x–k), the remainder is k. Find k.

23. The sum of the digits of a two-digit number is 11. If the digits are reversed, the resulting number is seven more than twice the original number. What is the original number?

24. The time required for the examinees to solve the same problem differ by two minutes. Together, they can solve 32 problems in one hour. How long will it take for the slower problem solver to solve a problem?

25. Find the sum of the roots of 5x^2 - 10x + 2 = 0.

26. In a box there are 25 coins consisting of quarters, nickels and dimes with a total amount of $2.75. If the nickels were dimes, the dimes were quarters and the quarters were nickels, the total amount would be $3.75. How many quarters are there?

27. A man travels in a motorized banca at the rate of 15 kph from his barrio to the poblacion and come back to his barrio at the rate of 12 kph. If his total time of travel back and forth is 3 hours, find the distance from the barrio to the poblacion.

28. One leg of a right triangle is 20 cm and the hypotenuse is 10 cm longer than the other leg. Find the length of the hypotenuse.

29. Three times the sine of a certain angle is twice of the square of the cosine of the same angle. Find the angle.

30. A man is 41 years old and his son is 9. In how many years will the father be three times as old as his son?

31. A tank is filled with an intake pipe that fills it in 2 hours and an outlet pipe that empty it in 6 hours. If both pipes are left open, how long will it take to fill the empty tank?

32. A piece of wire of length 50 m is cut into two parts. Each part is then bent to form a square. It is found that the total area of the square is 100 m^2. Find the difference in length of the sides of the two squares.

33. A purse contains $11.65 in quarters and dimes. If the total number of coins is 70, find how many dimes are there?

34. How many liters of water must be added to 35 liters of 89% HCl solution to reduce its strength to 75%?

35. Find the value of m that will make 4x^2 - 4mx + 4m + 5 a perfect square trinomial.

36. Ana is 5 yrs. older than Beth. In 5 yrs., the product of their ages is 1.5 times the product of their present ages. How old is Beth now?

37. Find the coefficient of the term involving b^4 in the expansion of (a^2 - b^2)^10.

38. The seating section in a Coliseum has 30 seats in the first row, 32 seats in the second row, 34 seats in the third row, and so on, until the tenth row is reached, after which there are ten rows each containing 50 seats. Find the total number of seats in the section.

39. Pedro started running at a speed of 10 kph. Five minutes later, Mario started running in the same direction and catches up with Pedro in 20 minutes. What is the speed of Mario?

40. The sum of two numbers is 35 and their product is 15. Find the sum of their reciprocals.

41. The ten’s digit of a certain two digit number exceeds the unit’s digit by four and is one less than twice the unit’s digit. Find the number.

42. One pipe can fill a tank in 6 hours and another pipe can fill the same tank in 3 hours. A drain pipe can empty the tank in 24 hours. With all three pipes open, how long will it take to fill in the tank?

43. A piece of paper is 0.05 inch thick. Each time the paper is folded into half, the thickness is doubled. If the paper was folded twelve times, how thick in feet the folded paper be?

44. A man invested part of Php 20,000 at 18% and the rest at 16%. The annual income from 16% investment was Php 620 less than three times the annual income from 18% investment. How much did he invest at 18%?

45. If 3^x=9^y  and 27^y=81^z, find x/z.

46. It takes an airplane one hour and forty five minutes to travel 500 miles against the wind and covers the same distance in one hour and fifteen minutes with the wind. What is the speed of the airplane?

47. An airplane travels from points A and B with a distance of 1500 kms and a wind along its flight line. If it takes the airplane 2 hours from A to B with the tailwind and 2.5 hours from B to A with the headwind, what is the velocity?

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