The welfare of a person or family depends not only on the income and on the ability to spend the money wisely, but also on NATIONAL BANK the ability to increase the means of earning money. People who set aside a portion of their income may deposit it in a savings bank for safekeeping. The bank lends the money to others at a sufficiently high rate of interest to be able to pay interest at a lower rate to the depositor. A bank may lend money receiving 6% a year, and pay 3% or 31% a year for the use of the money. Thus, the money saved earns more money for the depositor. EXERCISES 1. Find the interest for 1 year on the following: $500 at 6%; $800 at 4%; $650 at 31%; $p at 5%; $700 at r%; $p at r%. 2. State the formula for finding the interest for 1 year on p dollars at r%. 3. The interest on $500 for one year is $10. Find the rate. 4. A sum invested at 5% yields interest equal to $90 a year. How large is the sum? 154. How to use the interest formula in problems. The interest is equal to the rate per cent times the principal times the number of years. Denote the interest by i, the rate by r, the principal by p, the time by t, and state the interest formula. EXERCISES 1. Find the interest on $300 at 6% for 3 years. 2. Find the interest on $3850 at 5% for 2 years; on $4200 at 7% for 3 years; at 8% for 6 months; on $68 on $250 at 7% for 1 year, 3 months; on $325 at 51% for 1 year, 6 months. 3. Find the interest on $2400 at 4% for 4 months 12 days. 4. Find the interest on $250 at 4% for 3 years 2 months 10 days; on $3500 at 31% for 2 years 3 months 15 days. 5. What sum of money put at interest at 5% yields an income of $800 in 3 years? 6. What sum of money at 4% will yield an income of $500 in 3 years? 7. A sum of money is invested at 5%. In 2 years, after the interest is added to the investment, the total amounts to $3500. Find the principal. Solve this equation to determine x. 8. What principal must be invested at 6% in order that one may have $3200 at the end of the third year? 9. A mortgage is the offering of property as security for the payment of a loan, which becomes void upon payment. Failure to pay back the loan on the date due gives the holder of the mortgage the right to have the property sold at public auction. From this sale he is paid his full amount, the remainder going to the owner of the property. Any legal property may be mortgaged, e.g., furniture, automobiles, even crops which are planted but not yet grown. On a house worth $8500 the bank holds a mortgage for $2300 at 51%, interest to be paid every 6 months (semiannually). How much must be paid each time? 10. What is the semi-annual payment a man must make to a bank holding a mortgage of $1800 on his house, if 6% interest is charged? 11. Mr. James bought an automobile of Mr. Crane, paying $165 in cash and giving a note for $420 payable one year after date at 6% interest. Find the total cost of the automobile. 155. Graphical representation of the interest formula. The graph of the interest formula is similar to that of the percentage formula. Hence, in making the graph, the suggestions of §148 should be followed. EXERCISES 1. Using the same axes, and the arrangement shown in §148, make graphs of the equations i=(.03)p; i=(.05)p; i=(.06)p. 2. From the graph find the yearly interest at 5% of $75; of $125; of $275. A STUDY OF EQUATIONS 156. Uses of the equation. The equation has been used for various purposes. First it was used to express the equality of two numbers. Thus if x and y are the measures of two angles (Fig. 266), the equation x=y expresses briefly the statement: "The two angles are equal." The equation was also used to express a relation, or interdependence, between numbers. For example, if a, b, and c are the measures of the angles of a triangle (Fig. 267), the relation given by the C equation a+b+c=180 makes it possible to determine one angle if the others are known. Similarly, the equation c=πd expresses the relation between diameter and circumference, and d=rt states a relation between time, distance, and rate. Furthermore, the equation is a FIG. 267 b used as a brief statement of a problem. The statement, "If a number is increased by 4 times itself, the result is 60," is written briefly x+4x=60. The solution of this equation determines the unknown number. Because equations are important we must learn to solve them and to understand the laws used in the process of solving. For example, we know that the laws known as the multiplication and division axioms enable х us to solve such equations as 5x=20 and 25. In the 2 first, we divide each member by 5 to find the required value of x; in the second, we multiply each member by 2. 157. Use of the subtraction axiom in solving equations. An equation may be regarded as expressing bal ance between two numbers. It may be compared with a pair of scales (Fig. 268). If a bag of unknown weight w together with a 5-lb. weight just balances a 10lb. and a 5-lb. weight, we may express this fact by means of the equation w+5=15. If 5 lb. are taken from each pan, the bag in one pan just balances 10 lb. in the other. In the form of an equation this may be written w=10. Just as a 5-lb. weight may be taken from each pan without destroying the balance, the number 5 may be subtracted from both members of the equation |