Exercise 66

1. If one-half of a number is added to five-sixths of the number the sum is 56. Find the number.

2. Two-fifths of a certain number is 3 more than onethird of the number. Find the number.

3. In a mixture of corn and oats there are 11⁄2 times as much corn as oats. There are 65 bu. of the mixture. How many bushels of each kind of grain?

4. In a mixture of cement, sand and crushed rock there are 2 times as much sand as cement and 4 times as much crushed rock as cement. Find how much there is of each kind of material in 300 cu. ft. of the mixture.

5. A 1:3:5 mixture of cement, sand and gravel is used in making the foundation of a storeroom. The foundation is 60 ft. by 38 ft., outside measurement, 7 ft. deep, and 15 in. thick. Find how much of each kind of material will be required to make this foundation.

6. What number must be added to the numerator of so that the resulting fraction equals ?

7. What number must be added to the denominator of so that the resulting fraction equals ?

8. What number added to the numerator and also to the denominator of gives a result equal to ?

9. One kind of war bread contained as much potato flour as wheat flour. If a baker used at one time 140 lb. of the mixed flour, how much of each kind was used?

10. If to a number there be added of it, of it, and of it the result is 200. Find the number.

11. A line 36" long is divided into two segments having the ratio of 4:5. How long is each segment?

12. A line is divided into two segments having the ratio 37. The longer segment is 28". How long is the line?

13. The three sides of one triangle are 8', 12′, and 15′. The longest side of a similar triangle is 20'. Find the other two sides of the second triangle.

14. In a ration of bran, oats, and oil meal the ratio is 32. How much of each in 30 bushels of the mixture?

15. Thirty per cent cream means cream that contains 30% butter fat. If a quart of 30% cream is mixed with a quart of 20% cream what is the per cent of cream in the mixture? If 3 qt. of 30% cream are mixed with 4 qt. of 20% cream what is the per cent of cream in the mixture?

16. How much 40% cream must be mixed with 4 qt. of 20% cream to get a mixture of 24% cream?

HINT. Let x = the number of quarts of 40% cream.

How many quarts of the mixture? How much butter fat in the 20% cream? How much butter fat in the x quarts of 40% cream? How much butter fat in the mixture? What two things are equal?

17. How much 25% cream must be mixed with 12 qt. of 40% cream to make a mixture of 35% cream?

18. How much 4.5% milk (milk containing 4.5% butter fat) must be mixed with 10 gallons of 2.4% milk to get 3.1% milk?

19. How much water must be added to a pint of a 10% solution of a certain medicine to reduce it to a 1% solution?

20. How much water must be added to a quart of 90% alcohol to reduce it to 60% alcohol? Ninety per cent alcohol means alcohol that contains 10% water.

60. Equations involving other letters besides the unknown. The pupil has already learned how to solve formulas for different letters. The following will give further practice with certain important formulas.

11. Solve for r, С=2πr. Find the radius of a circle whose circumference is 132 in. In this and the following problems use =3, unless told otherwise.

12. Solve for a, V = r2a. Find the altitude of a cylinder whose radius is 2 in. and whose volume is 100 cu. in.

13. The standard bushel is the volume of a cylinder 18 in. in diameter. Find its height. Use =3.1416.

14. Solve for r, A=2πra. Find the radius of a cylinder whose lateral surface is 3520 sq. in. and whose altitude is 35 in.

15. Solve C=rd for d. Find the diameter of a circle whose circumference is 8 ft.

16. A piece of wire 4 ft. 6 in. long is bent into the form of a circle. Find the diameter of the circle.

17. Solve C=(F-32) for F. Find the centigrade temperature, C, when the Fahrenheit temperature, F, is -22°. Find the Fahrenheit temperature when the centigrade temperature is 29°.

Find the sum of the

18. Solve S=(n-2) 180 for n.

angles of a polygon of 12 sides. Find the number of sides of a polygon if the sum of the angles is 2520°.

19. Solve for n, A

(n-2)180

=

n

Find the number of

degrees, A, in one of the angles of a regular polygon of 10 sides. Find the number of sides of a regular polygon if the number of degrees in one of the angles is 90; also if the number of degrees in one of the angles is 135.

20. In the set of numbers 3, 8, 13, 18, 23, . . ., each number is 5 greater than the preceding number. In the set of numbers a, a+d, a+2d, a+3d, a+4d,..., each number is d greater than the preceding number. It can be proved that if there are n such numbers in a set of which a is the first and I is the last, then l=a+(n−1)d. Find l if a=6, n=10, and

d=4. Find a if l=70, n=13, and d=7.

21. Solve the formula in the preceding problem for d and find d if l=2396, a= −4, and n = 25.

22. Solve the same formula for n and find n if a=0, l=88, and d=8.

23. A bookkeeper accepts a position at a salary of $900 a year. If his salary is increased $75 each year what will be his salary the tenth year?

24. If a body falls 16 ft. the first second, 48 ft. the second second, 80 ft. the third second, and so on, how far will it fall the tenth second?

25. A ball rolls down an inclined plane at the rate of 3 ft. the first second, 7 ft. the second second, 11 ft. the third second, and so on. In what second will the ball be rolling. at the rate of 43 ft. a second?

61. Equations containing the square of the unknown. The formula for finding the area of a square of side s is A = s2. This equation may be solved for s by taking the square root of both sides of the equation, which gives √A=s, or s=√Ā. This gives a formula for finding the side of a square when the area is given. For example, if A=625, s=√625=25.

There are other useful formulas which we shall wish to solve and which involve the square of the unknown.

1. Solve r2=k for r. .0256.

Exercise 68

Find r when k=169; when k=

2. Solve y2=2px for y. Find y if p=6 and x = 12; also if p=.9 and x = = 20.

3. If x2+y2=r2, show that x=√r2-y2. Solve the equation for y. Solve it for r. Find x if r = 14 and y = 12.

4. Show that x, y, and r in the preceding exercise may represent the three sides of a right triangle.

5. What is the diagonal of a rectangle whose sides are 25 yd. and 60 yd.?

6. A rope that is known to be 80 ft. long is attached to the top of a pole. When the rope is stretched it touches the ground 30 ft. from the foot of the pole. Find the height of the pole.

7. Solve A=πr2 for r. Find the radius of a circle whose area is 616 sq. ft. Use π=34.

8. What must be the radius of a circular flower bed to contain the same area as a rectangular one which is 6 ft. by 16 ft.? Use π=34.

9. The area of the State of New York is 49,204 sq. mi. Find the radius of a circle of the same area, correct to .1 mi. Use T=3.1416.