VI LITERAL NUMBERS 98. Arithmetical Equation.-Mathematicians use a sort of shorthand in making statements. These statements consist of mathematical symbols and letters. The letters taking the places of words are usually the first of the word. Thus, in the place of Minuend minus subtrahend equals remainder is written M Similarly, S = R (1) total price equals number of articles times price of one. Each expression, as (1) and (2), is called an equation. It states that two numbers, or sets of numbers, are equal; that is, they have the same value. The parts on each side of the equals sign are called the members of the equation. Every letter in the equation represents some number. An equation shows at a glance the relation between the numbers it contains. Equations will be used to simplify the solution of problems. Hence, learn thoroughly how to make and to use these shorthand statements. To express that Henry is 15 yr. older than George, write The small H and G tell whose age is meant. Small letters or numerals placed at the lower right-hand corner are called subscripts. EXERCISES Write each of the following statements in full. Then rewrite them in this shorthand mathematics. State the right and the left hand members of each equation. 1. Minuend equals remainder plus subtrahend. 2. Dividend equals divisor times quotient plus remainder. (Use D and d for dividend and divisor.) 3. Dividend minus divisor times quotient equals remainder. 4. The larger of two numbers equals the smaller plus (Use N and n.) six. 5. The larger of two numbers is four times the smaller. 6. The larger of two numbers less the smaller equals five times the smaller number. Express each of the following as an equation: 7. The larger of two numbers exceeds twice the smaller by 13. 8. Profits equal total sales less total expenses. 9. Total sales less profits equal total expenses. 10. Total sales less profits less expenses equal nothing. Total profits equal total sales times rate of gain. 11. 12. Loss equals total expenses less total sales. 13. Give the equations for losses corresponding to the statements 9, 10, and 11 for profits. 14. Mary is twice as old as Jane. 15. Henry is 5 yr. older than William. 16. 17. 18. 19. Elizabeth is 6 yr. more than twice as old as May. John weighs 6 lb. less than twice as much as Harry. William weighs as much as Henry and John together. The numerator of a fraction is 3 more than 5 times the denominator. 20. 99. Evaluations.-Replacing letters by their numerical values in an equation is called evaluation or substitution. Thus, if in D=dXq+r (1) the quotient is 4, divisor 7, and remainder 3, the dividend is found by replacing the letters in the equation by these numerical values when 1. Find the dividend when the quotient is 8, divisor 6, and remainder 4; quotient 17, divisor 9, and remainder 7. 2. In division we also have r = D-dXq. Find r when D = 532, d = 25, and q = 21. If N and n are two numbers, what is the equivalent English expression of the equations: 7. Find N for each equation when n = 6; n = 15. 8. What would the letters M, S, and R stand for in subtraction? Fill in the right-hand members of the following equations using these meanings of the letters: 12. Play a number game, using equations and substitu = 100. Solution of Equations. If an equation contains only one literal number, its numerical value can be found. Finding this value is called solving the equation. The numerical value of the literal number found is called the root of the equation. The equation stating that five times a number equals 30 is 5 x n = 30. (1) Hereafter 5 X n, 7 × n, etc., will be written 5n, 7n, etc. 5n, 7n, etc., mean that there are 5 n's, 7 n's, etc. We get equation (3) from equation (2) because if 5 n's are worth 30 then one n is worth, 30 ÷ 5 EXERCISES = 6. 1. If stands for cents, what is meant by 5¢? by 9¢? If a stands for apples, what is meant by 12a? by 8a? If d stands for dogs, what is meant by 7d? by 35d? by 12d? 2. If 8 books cost 72 cents, what will one book cost? Also solve the equation: 8b = 72. 3. If 7 tables cost $350, what does one table cost? Also solve the equation: If you know the value of any certain number of articles, how can you find the value of one article? If you know how much any number times a certain letter equals, how can you find the value of that letter? Solve the following equations: 101. Solution of Equations.-The equation says that 4 2n+37 the scales will still balance. If the same number of blocks be taken from each pan it will also balance. All the blocks are taken from one pan and the same number from the other pan. This leaves the pans as in the picture to the right. The num ber of blocks and weights in each pan is shown by the equation above the scales. If the known weights in the pan with the blocks be removed shown. The corresponding equation is like the equations solved on the last page. How many known weights does each block weigh? |