DOUBLE RULE OF THREE. DEFINITION.-The Double Rule of Three teaches to solve such questions as require two statings in the Single Rule of Three; five numbers are given to find a sixth. RULE.-State the question by placing that number, which is the principal cause of gain or loss, in the first place; that number which represents time, distance, &c. (that the first is gaining, or losing) in the second place; that number which represents the gain or loss in the third place; then place the other two numbers under those of the same name; and if the term sought, or the blank place, fall under the third term; then multiply the three last terms together for a dividend, and the other two, for a divisor; but if the blank place fall under the first, or second term, then multiply the first, second and last terms together for a dividend, and the other two for a divisor. NOTE. The sixth term or answer always will come of the same name, and of the same denomination, of the term directly over the blank place. To solve questions that belong in the DOUBLE RULE of THREE, by two statements in the SINGLE RULE of THREE. RULE. Make the number which is the principal cause of gain, or loss, the first term; the gain, or loss the second; the number, which is the demand of the question the third; the answer to this first statement will show what the third term gained, or lost, in a time that the first term was gaining or losing, (which time is always mentioned in the question) there fore say in the second stating, as the space of time, or distance, &c. mentioned in the question, is to the answer of the first statement, so is the required time or distance, &c. to the answer, NOTE. I have done the questions in the Double Rule of Three, first by stating the five numbers at once; and then by two statings in the Rule of Three Direct; the learner may follow my examples or not, as may best suit his inclination. Examples. 1. If 200 dollars in 12 months will gain 12 dollars; I demand how many 50 dollars will gain in six months ? dolls. mo, dolls. 200 12 :: 12 50 : 6 Three last terms 50 x 6×123600. dividend. 3600 2400 $1.50 Ans. The same question done by two statements. divisor. 2. If 4-men spend 60 dollars, going 300 miles; I demand what is sufficient for the expense of 20 men, and one boy 700 miles, allowing the boy, one half a man's expense? Ans. $717-50 dolls. As 300 : 307.5 :: 700 : 717:5 Ans. 7 men would 3. If 3 men can build 360 rods of wall in 24 days; I demand how many rods build in 27 days? Ans. 945 rds. 4. If 3 men in 24 days can build 360 rods of wall; I demand the number of men necessary to build 945 rods in 27 days? Ans. 7 men. The same question done by two statements. Ans. 7 men. 5. If 50 men can build a bridge in 144 days; I demand the number of men necessary to build a like bridge in 720 days? Ans. 10 men. The same question done by two statements. Days. bridge. days. bridges. As 144 : 1. :: 720 5 6. A man lent a friend 600 dollars, for 6 months for which he received 9 dollars interest: I demand the sum that will gain the same interest in 2 months? Ans. $1800. The same question done by two statements. mo. As 6 dolls. mo. dalls. 3 dolls. dolls. dolls. dolls. As 3 :. 600 :: 9: 1800 Ans. 7.A. received of B. 9 dollars, for the loan of 600 dollars six months; now B. wishes to hire of A. 1800 dollars until the loan should amount to the same sum ; how long may he keep it? Ans. 2 months. The same question done by two statements. dolls. interest dolls. interest. 9: 1800 : 27 VULGAR FRACTIONS. DEFINITION.-Vulgar Fractions are parts of whole numbers, and are expressed thus,,, &c. The top figure is a remainder left after division, and is called the numerator; the bottom figure is the divisor used in division, and is called the denominator. Fractions are read thus,, is read one tenth; and, is read six sevenths, &c. Fractions are proper, improper, compound, or mixed; a Proper Fraction has its numerator the smallest; an Improper Fraction has its numerator, equal, or the largest; a Compound Fraction is the fraction of a fraction, and is coupled by the word of; a mixed number is a whole number and fraction. A proper fraction is written thus,,,,, &c. Α An improper fraction is written thus, 1,, or f, 7,&c. A compound fraction is written thus, of of of. A mixed number is written thus, 124, 6, 5 &c. CASE I. To find the greatest common measure; RULE. Divide the denominator by the numerator, and the last divisor by the last remainder; the last divisor used is the common measure; if 1 is the last divisor, the fraction is in its lowest terms. The last divisor being two, it is the greatest com mon measure of the fraction. 2. What is the greatest common measure of 864? Ans. 8. 3. What is the greatest common measure of? Ans. 4. CASE II. To reduce fractions to their lowest terms. di RULE. Find a common measure by case first; vide the numerator and denominator of the given fraction, by the common measure, the quotient is the fraction in its lowest terms. To reduce a mixed number to an improper fraction. RULE.-Multiply the whole number by the denominator of the fraction, to the product add the numerator of the fraction; this sum placed over the given denominator will form the fraction required. Examples. 1. Reduce 10 to an improper fraction. 10X4 40+3=43 numerator, 43 Ans.. 2. Reduce 112 to an improper fraction. 112×3+1=33 Ans. 3. Reduce 28% to on improper fraction. -Ans. 292 ΤΟ |