worth $40, and pays the rest in cash; how much money does he pay? 11. If the subtrahend be 750 and the remainder 964, what is the minuend? 12. If the minuend be 60,402 and the remainder 29,475, what is the subtrahend? 13. The difference of two numbers is 607 and the greater number is 1,005; what is the less number? 14. 7,96354,923 +27,984 - 64,937 = ? 15. 22,786 (10,342 +5,684)=? 16. A man worth $18,000 left $4,287 to his elder son, $3,754 to his younger son, $3,219 to his daughter, and the remainder to his wife; what was the wife's portion? REVIEW QUESTIONS. What is a unit? What is a number? What is an abstract number? What is a concrete number? What is a simple number? What is a compound number? Define notation. Define numera tion. Give all the methods of notation. Write seven thousand two hundred and fifty-one by means of the Arabic notation. Write five hundred and seventy-eight by means of the Roman notation. Numerate 7,852,643,827,462, and read the number. What is addition? What numbers can be added together? Give the rule for addition. How do you prove addition? Make the sign of addition. Of equality. Make the sign of dollars. When dollars and cents are written, how many orders of units are occupied by cents? What sign is put between dollars and cents? When dollars, cents, and mills are written, how many orders of units do mills occupy? What is subtraction? Define minuend. Define subtrahend. Define remainder. Make the sign of subtraction, name it, and tell how it is used. Give the rule for subtraction. Work the following example: 178,462—(6,895 +18,754). When the difference and the greater of two numbers are given, how do you find the less? Suppose the sum of three numbers and two of them are given, how will you find the third? Construct a problem illustrating each of the above questions. MULTIPLICATION. 51. 1. In this picture there are two groups of boys and 3 boys in each group; how many boys are there in the picture? SOLUTION.-Since there are 3 boys in one group, in 2 groups there are 2 times three boys, 3+3=6, 3 taken twice = 6. 2. In one group, each boy has 3 apples; how many apples have the 3 boys? SOLUTION.-Since each boy has 3 apples, 3 boys have 3 times 3 or 9 apples, 3+3+3=9, or 3 taken 3 times 9. 3. In the other group, each boy has 4 pears; how many pears have the 3 boys. SOLUTION.-Since each boy has 4 pears, 3 boys will have 3 times four or 12 pears, 4+4+4 = 12, or 4 taken 3 times = 12. 4. How many trees in 4 rows, if there are 5 trees in each row? 4 times 5 trees are how many trees? SOLUTION. 5+5+5+5=20, or 5 taken 4 times = 20. 5. How many hands have 6 boys? SOLUTION. 2+2+2+2+2+2: = 12, or 2 taken 6 times 6. How many feet have 6 horses? SOLUTION. 4+4+4+4+4+4 = 24, or 4 taken 6 times = 24. 7. John's father gave him 6 five-cent pieces; how many cents did the father give John ? SOLUTION. 5+5+5+5+5+5 = 30, or 5 taken 6 times = 30. 8. Bought 3 lbs. of sugar at 10 cts. a pound; how much did the sugar cost? 3 times 10 cts. = ? SOLUTION. 10+10+10= 30, 10 taken 3 times = 30. In the first example we find the sum of 2 threes, or 3 taken twice. In the second we find the sum of 3 threes, or 3 units taken 3 times. In the fourth we find the sum of 3 fours, or 4 taken 3 times, and so on. In the first example, how many times is 3 taken? What is the result? In the fifth example, how many times is the number 2 taken? In the sixth example, how many times is 4 taken? What number is taken 6 times in the seventh example? How many times is 10 taken in the eighth ? 52. The operation of taking a number a certain number of times is called Multiplication. The number to be taken is called the Multiplicand, and the number which shows how many times the multiplicand is taken is called the Multiplier. DEFINITIONS. 53. Multiplication is the operation of taking one number as many times as there are units in the other. 54. The Multiplicand is the number to be taken or multiplied. 55. The Multiplier is the number which shows how many times the multiplicand is to be taken, or what part of it is to be taken. 56. The Product is the result of the multiplication. What is multiplication? What is the multiplicand? What is the multiplier? In each of the eight examples on pages 60 and 61, tell which number is the multiplicand? Which is the multiplier? In Example 1st, 6 is the product; in Example 2d, 9 is the product; in Example 3d, 12 is the product. Tell what is the product in each of the other examples. 57. The multiplicand and multiplier are called Factors of the product. 58. The following is the Sign of Multiplication, x. When placed between two numbers it is read multiplied by; thus, 3 × 2 is read 3 multiplied by 2. The value of the product does not depend on the order in which the factors are taken. Thus, 4 times 5 is the same as 5 times 4, as shown in the diagram; for, if we take the stars by rows, we have 4 stars taken 5 times; * if we take them by columns, we have 5 stars taken 4 times; in either case there are 20 stars. The multiplier, however, must always be considered an abstract number. The multiplicand and product are like numbers, and may be abstract or concrete. |