All the Math You'll Ever Need. Steve Slavin
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Название: All the Math You'll Ever Need
Автор: Steve Slavin
Издательство: John Wiley & Sons Limited
Жанр: Математика
isbn: 9781119719199
isbn:
You'll do the same sort of regrouping with the 12 tens. Ten of those tens make 1 hundred, leaving 2 tens in the tens place. You can do this without using the expanded form. Add 6 + 8 to get 14. Put down the 4 and carry the one ten.
Add 1 + 5 + 6 to get 12. Put down the 2 (tens) and carry the 1 (hundred).
Add 1 + 7 + 9 to get 17. The 7 goes in the hundreds place and the 1 (thousand) slides into the thousands place.
Problem 1:
Add 312 and 423.
Solution:
All that's necessary is adding the digits in each column: 2 + 3 = 5, 1 + 2 = 3, and 3 + 4 = 7.
Problem 2:
What is the result when 459 is added to 1,276?
Solution:
This one requires a little bit of regrouping. Add 6 + 9 to get 15, put down the 5 and carry 1 to the next column. Then 7 + 5 is 12, plus the 1 you carried is 13. Put down the 3 and carry the 1. You can think of the rest as 2 + 4 + 1 = 7 and the 1 thousand comes down unchanged, or you can think of it as 12 + 4 is 16, plus 1 you carried is 17.
Problem 3:
What is the combined total of 9,671 and 2,859?
Solution:
Here again you're regrouping. In the ones column, 1 + 9 is 10, so put down the 0 and carry the 1. Then 7 + 5 is 12 plus 1 you carried makes 13. Put down the 3 and carry the 1. Add 6 + 8 + 1 to get 15. Put down the 5 and carry the 1. Finally, 9 + 2 + 1 is 12.
2 MULTIPLICATION
Multiplication is repeated addition. For instance, you probably know 4 × 3 is 12 because you searched your memory for that multiplication fact. There's nothing wrong with that.
Another way to calculate 4 × 3 is to think of it as adding four threes, or adding three fours.
What about 5 × 7? Maybe you know it's 35, but you could always do this:
You do multiplication instead of addition because it's shorter—sometimes much shorter. Suppose you needed to multiply 78 × 95. If you set this up as an addition problem, you'd have to write 78 copies of 95 before you could even start adding.
Let's set this up as a regular multiplication problem and take a look at the expanded form.
The key to this multiplication is you have to multiply 8 × 5 and 8 × 90 and then multiply 70 × 5 and 70 × 90, and add up all the results. Don't get discouraged, because there is a condensed form.
The first set of numbers we'd multiply would be 8 × 5. You probably know, or can figure out, that's 40. (We'll focus on all the multiplication facts you should memorize in Chapter 3, “Focus on Multiplication.”) Then we'd multiply 8 × 90, which just means multiplying 8 × 9 and putting a zero at the end. Whenever you multiply a number that ends in zero, you can deal with the non-zero parts and add the zero at the end. (See Chapter 5, “Mental Math” for more on that shortcut.) 8 × 9 =72 so 8 × 90 = 720. Next would come 70 × 5. 7 × 5 = 35 so 70 × 5 =350. The last multiplication would be 70 × 90. Multiply 7 × 9 = 63, and then add a zero for the 70 and another zero for the 90. 70 × 90 = 6,300. Add up 6,300 + 350 + 720 + 40 to get 7,410.
Here's how to write it more compactly. Multiply 8 × 5 = 40, put down the 0 and carry the 4. 8 × 9 = 72 and the 4 we carried makes 76. Write the 76 in front of that 0 you put down and you see 760. This 760 is the 40 and the 720 combined. Now, you need to multiply 95 by 70, which means multiply by 7 and add a zero. So put the zero down first, under the 0 of the 760. Then 7 × 5 = 35. Put down the 5 to the left of the 0 and carry the 3. 7 × 9 = 63 plus the 3 you carried is 66. Write the 66 in front of the 50 and you've got 6,650, which is the 350 and 6300 combined. Add the two lines, and you're done.