Lesson 9 Section 2 Decomposing the multiplicand:




In each case, we "distributed" 2 to both 100 and 5. We cannot emphasize enough the importance of the distributive property. It is the basis not only of mental calculation, as we are about to see, but it is the basis of the traditional written method as well. Moreover, it is a theorem that we can prove. 

 
We expanded 24 into 20 + 4, and then "distributed" 3 to each one. Look:
The repeated addition of 24 is equal to the repeated addition of 20, plus the repeated addition of 4. (For a general proof, see Appendix 3.) Example 2. Multiply 5 × 37 mentally. Technique. Distribute 5 to 30 + 7. Say: "150 plus 35 is 185." Multiply the numbers as you read them, from left to right. The last number you say is the answer. Example 3. Multiply mentally 8 × 46. "320 + 48 = 368." Example 4. 800 × 460 Ignore the final 0's and multiply 8 × 46. But we just saw that 8 × 46 = 368. Therefore: 800 × 460 = 368,000. This is "368" with three 0's. Example 5. Multiply 6 × 7.30. (Treat problems with decimal points as dollars and cents.) Technique. Expand 7.30 mentally into 7 + .30 Then
Example 6. What is the price of five items that cost $3.25 each? Answer. Since 4 × $.25 = $1.00, then 5 × $.25 = $1.25. Say, "5 × 3.25 = 15 + 1.25 = 16.25" Example 7. Multiply 2 × 438 mentally.
The point is to say each partial sum. Look at 2 × 438 and say, "860 + 16 is 876." (Again, in 438, the 4 signifies 400, and the 3 signifies 30. Lesson 2.) Example 8. Multiply 4 × 709.
Note: Any number times 0, or 0 times any number, is 0. Therefore, to calculate 4 × 709, simply ignore the 0 and say: "2800 + 36 is 2836." Example 9. Multiply 8,000 × 4,310. Technique. Ignore the final 0's:
Now replace the four 0's: 8,000 × 4,310 = 34,480,000 Example 10. How much is 20% of $68? Solution. 10% of $68 is $6.80. (Lesson 4.) Therefore, 20% is 2 × $6.80 = $12 + $1.60 = $13.60. Example 11. How many hours are there in one week? How many minutes are there? Solution. There are 24 hours in one day, and there are 7 days in a week. Therefore, 7 × 24 = 140 + 28 = 168 hours. Now, in each hour there are 60 minutes. To multiply 60 × 168, ignore the 0 and multiply
Now replace the 0 we ignored: 10080. In one week, then, there are 10,080 minutes. At this point, please "turn" the page and do some Problems. or Continue on to Section 3: Multiplying by rounding off Introduction  Home  Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? Email: themathpage@nyc.rr.com 