24. Solving formulas. The pupil has seen that the formula is a very helpful means for finding the value of the letter in its left member when the values of the letters in its right member are given. Thus, if the two bases and the altitude of a trapezoid are known to be 12 ft., 8 ft., and 3 ft., the area of the trapezoid may be found by substituting these values in the formula

T=h(b+b').

But a problem may be given in which the area, the altitude, and the upper base are known and the length of the lower base desired. We then need a formula in which the upper base, b, stands alone in the left member. We may get such a formula by using the formula T=1⁄2h(b+b') as an equation in which b is the unknown number and the other letters treated as known numbers. It will be seen from the examples given below that any letter in a formula may be regarded as the unknown, and its value may be found in terms of the other letters by solving the equation for the unknown letter.

EXAMPLE 1. From the formula s=c+g get a formula for the cost, c, when the selling price, s, and the gain, g, are known.

SOLUTION.

Subtract g from each member, then

s=c+g.

s-g=c,

or c=s-g.

EXAMPLE 2. Solve the formula T=h(b+b') for b.

Subtract hb' from each member. 2T-hb' = hb.

Exercise 25

1. From the formula s=c+g get a formula for finding the gain when the selling price and the cost are known.

2. Write the formula for the area of a rectangle when the base and the altitude are known. From it get a formula for finding the base when the area and the altitude are known. 3. Solve the formula Tab for a; for b. SUGGESTION. First multiply each member by 2.

4. Use one of the formulas found in the preceding exercise to find the altitude of a triangle whose area is 36 sq. in. and whose base is 4 in.

5. Find the base of a triangle whose area is 141 sq. rd. and altitude 77 yd.

6. Write the formula for finding the interest on a given sum of money for a given time at a given rate. Solve this formula for the principal, p, supposing the interest, i, the

time, t, and the rate, r, to be known.

7. State the rule given by the formula

p=

i

=

Find the

rt

principal which will produce $400 interest in 2 yr. 6 mo. at 5%.

8. Solve the formula i=prt for r. Find the rate of interest at which $456 will produce $95.76 interest in 3 yr.

9. Solve the formula i=prt for t. Find the time required for $1275 to produce $56 interest at 6%.

10. State as a rule the formula for finding the time in which a given principal will produce a given interest at a given rate.

11. A man bought a bond for $110 and after two years sold it for $115. During that time his income from the bond was $4 a year. He also bought a farm for $1500 which he kept for 14 years and sold for $1550. His net income from the farm during the time that he held it was $114. On which

investment did he receive the higher rate of interest on the money invested?

12. Write the formula for finding the dividend, D, when given the divisor, d, the quotient, q, and the remainder, r. Solve this formula for r for for d.

13. Make a formula for finding the rectangle whose length is I and width w. for l; for w.

14. Solve the formula a=2x+x for x.

15. Solve the formula s=b+rb for b.

perimeter, p, of a

Solve the formula

HINT. First add the terms containing b. 1b+rb=(1+r)b. 16. Solve the formula a=p+prt for p.

17. Use the new formula you found in the preceding problem to find the principal which will amount to $660 if put at interest for 2 yr. at 5%.

18. Solve for r the formula a=p+prt.

If $2000 is put at interest for 4 yr. 6 mo. at a certain rate it amounts to $2360. Find the rate.

19. Solve the formula FC+32 for C.

20. Use the formula given in the preceding exercise and find F when C=0; when C-16.

21. Use the result you obtained in exercise 19 and find C when F-98; when F=32.

22. The formula for the average, a, of the two numbers r and s is a

=

rts. Solve this equation for s.

2

23. The average of two numbers is 85. One of the numbers is 76. What is the other?

24. The average of two numbers is . One of the numbers is. What is the other?

25. State in words the rule for finding the average of two numbers.

31. In a room there are m men, w women, and 17 children. How many persons in all? If 8 persons leave, how many remain? If a more persons leave, how many remain?

32. A boy has c cents.

get to have one dollar?

How many more cents must he

33. A boy has x cents. How many more must he get to have n dollars?

34. How many hours are required to walk 12x2 miles at the rate of 3x miles per hour?

35. How many hours are required to walk x miles at the rate of 5m miles per hour?

36. What must be added to 4y-9 to give 4y? To a-b to give a?

37. What does it mean to say that a number satisfies an equation?

38. Define root of an equation.

39. What does it mean to solve an equation?

40. What does it mean to check a solution?

41. State four principles used in solving equations. Give equations in the solution of which these principles are used. 42. Define division.

Solve the following equations :

53. This formula is used in finding the load that may safely be placed upon wooden pillars:

In this formula L represents tons and l and d represent inches. Find L if l=96 and d=8; also if l=120 and d=8.

54. Show that the area of Figure 8 is given by the formula A = dt+(s+y)z. Find A if d= 12 in., t=2 in., z=4 in., y = 11⁄2 in., and s= in.

55. I=0.0491(a1-b4). Find I when a=6 and b=5; also when a=1.5, and b=1.2, correct to 0.0001.

56. If y=5x2-4x+6, find the value of y for each of the following values of x: 0, 1, 5, 10, 20.

y

FIG. 8