6. What number added to one third of itself will give a sum equal to 12? 7. What number added to one fourth of itself will give a sum equal to 20? 8. What number added to a fifth of itself will make 24 ? 9. What number diminished by one half of itself will leave 4? Why? 10. What number diminished by one third of itself will leave 6? 11. James gave one seventh of his marbles to William, and then had 24 left. How many had he at first ? 12. What number added to two thirds of itself will give a sum equal to 20 ? 13. What number diminished by three fourths of itself will leave 9 ? 14. What number added to five sevenths of itself will make 24 ? 15. What number diminished by seven eighths of itself will leave 4? 16. What number added to eight ninths of itself will make 34 ? 17. What number diminished by eleven twelfths of itself will leave 5 ? 18. Margaret gave nine tenths of her apples to her sister, and then had 6 left. How many had she at first ? 19. What number added to 3 times one ninth of itself will give 72? 20. Henry had a certain number of cents. He lost one third of them, and had 15 left. How many had he at first ? CHAPTER I. DEFINITIONS. 1. Quantity is anything which can be increased, diminished, and measured ; as number, distance, weight, time, etc. To measure a thing is to find how many times it contains some other thing of the same kind, taken as a standard. The assumed standard is called the unit of measure. 2. Mathematics is the science which treats of the measurement, properties, and relations of quantities. In pure mathematics there are but eight kinds of quantity, and consequently but eight kinds of units; viz., units of number, units of currency, units of length, units of surface, units of volume, units of weight, units of time, and units of angular measure. 3. Algebra is a branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed are indicated by signs. 4. The quantities employed in algebra are of two kinds, – known and unknown. Known quantities are those whose values are given. They are generally represented by the leading letters of the alphabet; as, a, b, c, etc. - Unknown quantities are those whose values are required. They are generally represented by the final letters of the alphabet; as, x, y, z, etc. When an unknown quantity becomes known, it is often denoted" by the same letter, with one or more accents; as, 3. D. N. E. A. 33 X', x", x'". These symbols are read, “x prime," " X second," " x third." 5. The sign of addition (+) is called plus. When placed between two quantities, it indicates that the second is to be added to the first. Thus, a + b is read "a plus b," and indicates that b is to be added to a. If no sign is written, the sign + is understood. The sign + is sometimes called the positive sign; and the quantities before which it is written are called positive quantities, or additive quantities. 6. The sign of subtraction (-) is called minus. When placed between two quantities, it indicates that the second is to be subtracted from the first. Thus, the expression c-d, read “c minus d,” indicates that d is to be subtracted from c. If a stands for 6, and d for 4, then a - d is equal to 6 – 4, which is equal to 2. The sign – is sometimes called the negative sign; and the quantities before which it is written are called negative quantities, or subtractive quantities. - 7. The sign of multiplication (x) is read “multiplied by," or “multiplied into." When placed between two quantities, it indicates that the first is to be multiplied by the second. Thus, a x 6 indicates that a is to be multiplied by b. If a stands for 7, and b for 5, then a x b is equal to 7 x 5, which is equal to 35. The multiplication of quantities is also indicated by simply writing the letters one after the other, and sometimes by placing a point between them. Thus, a x b signifies the same thing as ab or as a.b; a Xb-X c signifies the same thing as abc or as a .6.c. 8. A factor is any one of the multipliers of a product. Factors are of two kinds, — numeral and literal. Thus, in the expression 5 abc there are four factors, - the numeral factor 5, and the three literal factors, a, b, and c. 9. The sign of division (+) is read “divided by." When written between two quantities, it indicates that the first is to be divided by the second. There are three signs used to denote division. Thus, a+b, g, alb, denote that a is to be divided by b. 10. The sign of equality (=) is read “equal to." When written between two quantities, it indicates that they are equal to each other. Thus, the expression a+b=c indicates that the sum of a and b is equal to c. If a stands for 3, and b for 5, c will be equal to 8. 11. The sign of inequality (> or <) is read "greater than " or “less than.” When placed between two quantities, it indicates that they are unequal, the greater one being placed at the opening of the sign. Thus, the expression a > b indicates that a is greater than b, and the expression c< d indicates that c is less than d. 12. The sign :: means “therefore," or "consequently." 13. A coefficient is a number written before a quantity, to show how many times the quantity is taken additively. Thus, in the expression a tatatata=5a, 5 is the coefficient of a. A coefficient may be denoted either by a number or by a letter. Thus, 5x indicates that x is taken 5 times, and ax indicates that w is taken a times. If no coefficient is written, the coefficient 1 is understood. Thus, a is the same as la. 14. An exponent is a number written at the right and a little above a quantity, to indicate how many times the quantity is taken as a factor. Thus, |