FINDING NUMBERS FROM THEIR SUM AND DIFFERENCE 16. One half the sum of the sum and difference of two numbers equals the larger number. One half the difference of the sum and difference of two numbers equals the smaller number. 122 10 (difference of sum and difference). 102 5 (the smaller number). = 17. Solve: ORAL AND WRITTEN EXERCISES 1. The sum of two numbers is 80; their difference is 20. What are the numbers? How 2. The number of votes cast for two candidates was 3,200. The successful candidate had 200 more than the other. many votes did each receive? 3. Luther and Richard together have 120 marbles. Luther has 10 marbles more than Richard. How many has each? 4. John and Henry in a relay race ran 7 miles. John ran 2 miles farther than Henry. How far did each run? 5. Mr. Allison gave his sons, Alfred and Charles, $28, giving each $1 for every year of his age. He gave Charles $6 more than Alfred. How old was each boy? 6. Mary and Lucy together picked 66 qt. of strawberries. Mary picked 6 qt. more than Lucy. How many did each pick? 7. John and James have 80 cents. John has 20 cents more than James. How many cents has each? ADDITION AND SUBTRACTION 9 'ADDITION AND SUBTRACTION WRITTEN EXERCISES 18. The fifteen universities in the United States having the largest number of students in the year 1909 were: 1. How many students were enrolled in Harvard, Yale, and Columbia? How many in Columbia, Cornell, and Syracuse? 2. How many students were enrolled in the eastern universities? How many in the western universities? (Stars show western universities.) 3. How many were enrolled in the fifteen universities? 4. Which of the western universities had the largest enrollment? 5. How much larger was it than that of each of the other western universities? 6. How many more were enrolled in Columbia than in each of the others? 7. Which had the more students, the eastern or western universities? How many more? He 8. January 1, Mr. Brown had in the bank $1263.75. made the following deposits during the month; $180, $67.73, $180, $24.30, $635.02, $873.60, $765, $937.50. (1) How much did he deposit ? (2) How much did he have in the bank during the month? 9. February 1, he called for a statement of his account with the bank, and together with the paid checks he received a slip of paper on which were written the amounts of the checks drawn. (1) Add these amounts to find how much money he had drawn from the bank. (2) How may the amount which Mr. Brown has in the bank February 1 be found? (3) Find the amount of his balance in the bank on that date. $101.50 9.00 88.79 17.00 10.00 25.00 25.00 84.35 3.14 103.15 60.00 10.00 124.20 21.23 2.80 92.75 7.25 212.75 18.93 211.45 1.40 21.00 2.52 (1) What was the total production of iron for each year? (2) How much larger was the production of 1906-7 than that of each of the other years? (3) How much greater was the production of 1908–9 than that of 1907-8? MULTIPLICATION 19. How many days are there in 4 years? 365+365 + 365 + 365 = 365 × 4 = 1,460. Multiplication is a short method of finding the sum of equal numbers. Define multiplier, multiplicand, product. The multiplier and the multiplicand are factors of the product. The multiplicand (one of the equal addends) may be a concrete number, as 7 days, 34 years. The multiplier is always an abstract number and tells simply how many times the multiplicand is taken, or how many addends are to be added. For example: To find out how many days there are in four weeks, we take 7 days 4 times, not 4 weeks times. If the number to be multiplied is smaller than the multiplier, it may be simpler to perform the multiplication as if both were abstract numbers and to multiply the larger by the smaller, adding the name afterwards. Thus: How many days are 365 times 17 days? There are 6,205 days in 365 times 17 days. 1. Explain what you do with each cipher in the multiplicand and in the multiplier. 2. Why is 6, the right-hand figure of the second partial product, placed under the 5 of the first? TESTING MULTIPLICATION 21. To test a product, divide it by either the multiplier or the multiplicand. If it is correct, the quotient should equal the other term. NOTE. For testing multiplication by casting out the nines, see page SHORT METHODS OF MULTIPLICATION 22. TO MULTIPLY BY 5, 50, 500, OR 5,000. 1. How is an integer multiplied by 10? By 100? By 1,000? 2. What is the ratio of 10 to 5? Of 100 to 50? Of 1,000 to 500? 5. Multiply 5,763 by 50. By 500. 6. Make a rule for a short method of multiplying by 5. By 50. By 500. |