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Lesson 8 THE MEANING OF MULTIPLICATIONMental CalculationThe student by now should have mastered the multiplication table. In this Lesson, we will answer the following:




3 × 5 means Three 5's: 5 + 5 + 5. 5 is the unit. More precisely, multiplication is the repeated addition of one number, called the multiplicand, as many times as there are 1's in another number, called the multiplier. Since 3 is three 1's, 3 = 1 + 1 + 1, then 3 × 5 = 5 + 5 + 5. 3 is the multiplier, 5 is the multiplicand  the number that is being repeatedly added. (In the United States we write the multiplier on the left.) As for 5 × 3, it means five 3's: 5 × 3 = 3 + 3 + 3 + 3 + 3 (We will come to the most general definition of multiplication in Lesson 26.) The result of multiplication is called the product. 4 × 8 = 32 4 is the multiplier. 8 is the multiplicand. 32 is the product. The order property a) What multiplication does this illustrate? Answer. 3 × 6 cm = 18 cm. 6 cm are the multiplicand. They are being repeatedly added. b) What multiplication does this illustrate? Answer. 6 × 3 cm = 18 cm. 3 cm are the multiplicand. They are being repeatedly added. This illustrates the order property of multiplication. If we add 3 cm six times, we will get the same number as when we add 6 cm three times. 3 × 6 cm = 6 × 3 cm This is a property that we can prove. And it is not dependent on knowing that 3 × 6 = 18 Here again is 3 × 6 cm. Do you see 6 × 3 cm ? Look at the 1st cm in each 6. There are 3 cm. Now look at the 2nd. There's another 3. And so on, up to the 6th in each group. There are 6 × 3 cm. And it was not necessary to name the product (In algebra, this property of multiplication has the strange 19th century name of the commutative law. For the indication of a formal proof, see the end of the following section.) We now turn our attention to certain skills. The order of any factors When numbers are multiplied, they are called the factors of the product. 2 × 3 × 5 The factors are 2, 3, and 5. The order of any number of factors does not matter. 2 × 3 × 5 = 5 × 2 × 3 = 5 × 3 × 2 And so on. In each case, we will get 30. We may group the factors as we please. Example 1. Multiply as easily as you can: 7 × 2 × 9 × 5 Solution. Take advantage of factors that produce a multiple of 10: 7 × 2 × 9 × 5 = (2 × 5) × (7 × 9) = 10 × 63 = 630. Here is another essential skill: 



Example 2. 200 × 30 = 6000. Ignore the final 0's and simply multiply 2 × 3 = 6 Now we ignored a total of three 0's. Therefore we must put back those three 0's: 200 × 30 = 6000 When we ignore final 0's, we have in effect divided For a number does not change if we divide it and then multiply the quotient by the same number. See Lesson 10. Example 3. 9 × 20 = 180 Example 4. Multiply 3 × $.40 mentally. Answer. Ignore the decimal point for the moment, and we have 3 × 40 = 120 Now put back those two decimal places: 3 × $.40 = $1.20 (We will explain in the next Lesson about ignoring decimal points and then replacing them.) Example 5. 7 × $.80 = $5.60 Expanding the multiplier Since multiplication is repeated addition, we can often go from what we know to what we do not know. Say, for example, that you do not know 12 × 8. But Twelve 8's are composed of 10 × 8 = 80, 2 × 8 = 16. Therefore, 12 × 8 = 80 + 16 = 96. 8 is the unit. We are counting 8's. Example 6. Calculate 16 × 5 mentally by expanding 16 into its tens and ones.
Example 7. Calculate 34 × 6 mentally by expanding 34. Solution. 34 × 6 = 30 × 6 + 4 × 6 = 180 + 24 = 204 "Thirtyfour 6's are equal to Thirty 6's plus four 6's." Example 8. Multiplying by 11. 11 × 45. Multiplication by 11 is particularly easy, because 11 = 10 + 1. Therefore, 11 × 45 is equal to 10 × 45 + 1 × 45: 450 + 45 = 495. Example 9. 11 × 6.2 = 62 + 6.2 = 68.2 To do this or any of these problems in writing   is nothing more than a trick to get the right answer. Example 10. How much is 13 × 12? Answer. If you know that 12 × 12 = 144, then 13 × 12 is simply one more 12: 13 × 12 = 144 + 12 = 156 Or, we could count the 12's like this:
This is the property of adding multiples of a number, which we can express it as follows: A sum of units does not depend on the unit. In other words, 2 units + 3 units = 5 units, no matter what the unit. 2 chairs + 3 chairs = 5 chairs. And 2 nines + 3 nines = 5 nines. (Since this is true in arithmetic, the righthand distributive law of algebra may be applied to arithmetic.) How to square a number mentally A square figure is a foursided figure in which all the sides are equal, and all the angles are right angles. To square a number means to multiply it by itself. "9 squared" = 9 × 9 = 81. To find the area of a square figure  that is, the space enclosed by the four straight lines  square the length of a side. (See Lesson 9.) If each side were measured by number 6, for example, then the area would be measured by 6 × 6 = 36. Now, say that we want to square a twodigit number such as 24, mentally. The following geometrical arithmetic illustrates how. Look at this square figure in which each side is 24. Then the area of that square will be 24 × 24. But we can break up 24 into 20 + 4. Therefore the entire square will be composed of the following: The square of 20, which is 400. Two rectangles, each with area 20 × 4 = 80. And the square of 4, which is 16. In other words, the square of 20 + 4 is equal to The square of 20, plus Two times 20 × 4, plus The square of 4. 400 + 160 + 16 = 576. This is not a difficult mental calculation. Example 11. Square 52. Solution. 52 = 50 + 2. The square of 50 + 2 is equal to The square of 50, plus Two times 50 × 2, plus The square of 2. 2500 + 200 + 4 = 2704. Problem 1. Square 35. To see the answer, pass your mouse over the colored area. The square of 30 + 5 is equal to: The square of 30 + Two times 30 × 5 + The square of 5. 900 + 300 + 25 = 1225. Example 12. Square 48. Solution. Rather than square 40 + 8, it is simpler to square 50 − 2. In this case, the square of 50 − 2 equals The square of 50, minus Two times 50 × 2, plus The square of 2. 2500 − 200 + 4 = 2304. The entire square is the square of 50. It is composed of the square of 48, plus two rectangles, 50 × 2  which we must subtract. But we should not subtract the square of 2 twice. Therefore we add it at the end. Problem 2. Square 39. The square of 40 − 1 is equal to: The square of 40 − Two times 40 × 1 + The square of 1. 1600 − 80 + 1 = 1521. At this point, please "turn" the page and do some Problems. or Continue on to the Section 2: Introduction  Home  Table of Contents Please make a donation to keep TheMathPage online. Copyright © 20012007 Lawrence Spector Questions or comments? Email: themathpage@nyc.rr.com 