Analysis.-The factor 45=5X9; hence the dividend is composed of the factors 84X5X9. We may therefore cancel 9. which is common both to the divisor and dividend, and 84×5, the other factors of the dividend, will be the answer required. Note. We cancel the factors 6 and 3 in the dividend and 18 in the divi sor; for 6X3=18. Canceling the same or equal factors in the divisor and dividend, is dividing them both by the same number, and therefore does not affect the quotient. (Arts. 146, 148.) Hence, 151. When the divisor and dividend have common factors. Cancel the factors common to both; then divide the product of those remaining in the dividend by the product of those remaining in the divisor. 8. Divide 15×7×12 by 5×3×7×2. 9. Divide 27X3X4×7 by 9X12×6. -10. Divide 75X15X24 by 25×3×6×4×5. Note. The further development and application of the principles of Cancelation, may be seen in reduction of compound fractions to simple ones; in multiplication and division of fractions; in simple and compound proportion, &c. 151.a. The four preceding rules, viz: Addition, Subtraction, Multiplication, and Division, are usually called the FUNDAMENTAL RULES of Arithmetic, because they are the foundation or basis of all arithmetical calculations. OBS. Every change that can be made upon the value of a number, must necessarily either increase or diminish it. Hence, the fundamental operations in arithmetic are, strictly speaking, but two, addition and subtraction; that is, increase and decrease. Multiplication, we have seen, is an abbreviated form of addition; division of subtraction. (Arts. 80, 114.) QUEST.-151. a. Name the fundamental rules of Arithmetic. Why are these rules called fundamental? APPLICATIONS OF THE FUNDAMENTAL RULES. 152. When the sum of two numbers and one of the numbers are given, to find the other number. From the given sum, subtract the given number, and the remainder will be the other number. Ex. 1. The sum of two numbers is 87, one of which is 25: what is the other number? Solution.-87-25-62, the other number. (Art. 72.) 2. A and B together own 350 acres of land, 95 of which belong to A: how many does B own? 3. Two merchants bought 1785 bushels of barley together, one of them took 860 bushels: how many bushels did the other have? 153. When the difference and the greater of two numbers are given, to find the less. Subtract the difference from the greater, and the remainder will be the less number. 4. The greater of two numbers is 72, and the difference between them is 28: what is the less number? Solution.-72-28=44, the less number. (Art. 72.) 5. A man bought a horse and chaise; for the chaise he gave 265 dollars, which was 75 dollars more than he paid for the horse: how much did he give for the horse? 6. A traveler met two droves of sheep; the first contained 1250, which was 125 more than the second had: how many sheep were there in the second drove ? 154. When the difference and the less of two numbers are given, to find the greater. QUEST.-152. When the sum of two numbers and one of them are given, how is the other found? 153. When the difference and the greater of two numbers are given, how is the less found? 154. When the difference and the less of two numbers are given, how is the greater found? Add the difference and the less number together, and the sum will be the greater number. (Art. 73. Obs.) 7. The difference between two numbers is 12, and the less number is 45: what is the greater number? Solution.-45+12=57, the greater number. 8. A is worth 1890 dollars, and B is worth 350 dollars more than A: how much is B worth? 9. A man's expenses are 2561 dollars a year, and his income exceeds his expenses 875 dollars: how much is his income? 155. When the sum and difference of two numbers are given, to find the two numbers. From the sum subtract the difference, divide the remainder by 2, and the quotient will be the smaller number. To the smaller number thus found, add the given difference, and the sum will be the larger number. 10. The sum of two numbers is 48, and their difference is 18: what are the numbers? Solution.-48-18=30, and 30÷2=15, the smaller number. And 15+18=33, the greater number. PROOF.-33+15=48, the given sum. (Ax. 11.) 11. The sum of the ages of two men is 173 years, and the difference between them is 15 years: what are their ages? 12. A man bought a span of horses and a carriage for 856 dollars; the carriage was worth 165 dollars more than the horses: what was the price of each? 156. When the product of two numbers and one of the numbers are given, to find the other number. Divide the given product by the given number, and the quotient will be the number required. (Art. 91.) QUEST.-155. When the sum and difference of two numbers are given, how are the numbers found? 156. When the product of two numbers and one of them are given, how is the other found? 13. The product of two numbers is 144, and one of the numbers is 8 what is the other number? Solution.-144÷8=18, the required number. (Art. 120.) 14. The product of A and B's ages is 50 years what is the age of A? is 3250 years, and B's age 15. The product of the length of a field multiplied by its breadth is 15925 rods, and its breadth is 91 rods: what is its length? 157. When the divisor and quotient are given, to find the dividend. Multiply the given divisor and quotient together, and the product will be the dividend. (Art. 121.) 16. If a certain divisor is 12, and the quotient is 30, what is the dividend? Solution.-30X12=360, the dividend required. PROOF.-360-12-30, the given quotient. (Art. 120.) 17. If the quotient is 275 and the divisor 683, what must be the dividend? 18. If the divisor is 1031 and the quotient 1002, what must be the dividend? 158. When the dividend and quotient are given, to find the divisor. Divide the given dividend by the given quotient, and the quotient thus obtained will be the number required. (Art. 122.) 19. A certain dividend is 864, and the quotient is 12: what is the divisor? Solution.-864-12-72, the divisor required. (Art. 120.) 20. A gentleman handed a purse containing 1152 shillings, to QUEST.-157. When the divisor and quotient are given, how is the dividend found? 158. When the dividend and quotient are given, how is the divisor found? Add the difference and the less number together, and the sum will be the greater number. (Art. 73. Obs.) 7. The difference between two numbers is 12, and the less number is 45 what is the greater number? Solution.-45+12=57, the greater number. 8. A is worth 1890 dollars, and B is worth 350 dollars more than A: how much is B worth? 9. A man's expenses are 2561 dollars a year, and his income exceeds his expenses 875 dollars: how much is his income? 155. When the sum and difference of two numbers are given, to find the two numbers. From the sum subtract the difference, divide the remainder by 2, and the quotient will be the smaller number. To the smaller number thus found, add the given difference, and the sum will be the larger number. 10. The sum of two numbers is 48, and their difference is 18: what are the numbers? Solution.-48-18-30, and 30÷215, the smaller number. And 15+18=33, the greater number. PROOF.-33+15=48, the given sum. (Ax. 11.) 11. The sum of the ages of two men is 173 years, and the difference between them is 15 years: what are their ages ? 12. A man bought a span of horses and a carriage for 856 dollars; the carriage was worth 165 dollars more than the horses: what was the price of each ? 156. When the product of two numbers and one of the numbers are given, to find the other number. Divide the given product by the given number, and the quotient will be the number required. (Art. 91.) QUEST.-155. When the sum and difference of two numbers are given, how are the numbers found? 156. When the product of two numbers and one of them are given, how Is the other found? |