THE MEANING OF DIVISIONLesson 10 Section 2 Four properties of division
(45 × 100) ÷ 100 = 45. And (4500 ÷ 100) × 100 = 4500. The parentheses indicate to do the operation they enclose first. We will explain below why that is true .
Also,
For an explanation why that is true, see below. Example 1. The divisor a decimal. How many times is .2 contained in 6?
The divisor must be a whole number. Therefore, to make .2 into a whole number, multiply it by 10. Multiply 6 by 10 also:
This example illustrates that, even though we say we are dividing decimals, we can really divide only whole numbers and then correctly place the decimal point. We will see that again in Lesson 12.
Why did we multiply by 100? Because that makes the divisor .03 into a whole number. In other words: If the divisor is a decimal, multiply it by 10, or 100,
on dividing 12 by 2. Or,
on dividing 8 by 2. Or, finally,
These three possibilities imply: The order in which we multiply or divide does not matter. Also, if we decompose the dividend into factors, then we can sometimes find a multiple of the divisor.
For the explanation why Property 3 is true, see below. The following is perhaps the most important property of division, because it leads to a simple method for dividing mentally. It is called decomposing the dividend.
For example, say that you did not know 42 ÷ 3. Then you could decompose 42 -- chop it up
Why? Because you know that 30 is made up of ten 3's, and that 12 is made up of four 3's. Therefore, you would know that 42 is made up of fourteen 3's.
If a number is a divisor of two numbers, then it will also be a divisor In other words, we go from what we know to what we do not know. That is the essence of all mental calculation. |
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Example 2. Divide $92 equally among 4 people. Answer. We have to divide 92 by 4. To do that, we will go from what we know to what we do not know. We will break 92 up into two numbers that are obviously divisible by 4. Now, what multiple of 4 is close to 92? 80, for example. To make 92 we need to add 12. Therefore, we will decompose 92 into 80 + 12:
92 is made up of twenty-three 4's (which is equal to 4 twenty-three's). Each person will get $23. Alternatively, we could have broken up 92 as 100 − 8. If we divide each of those by 4, we get 25 − 2 = 23. Example 3. 265 is made up of how many 5's? Answer. Again, we go from what we know to what we do not know. Now, which number divisible by 5 is closest to the first two digits of 265? 25. And since 25 is divisible by 5, so is 250. Therefore, decompose 265 as 250 + 15. Then,
265 is made up of Fifty 5's + Three 5's: Fifty-three 5's. With a little practice, this will be a mental calculation. Example 4. 6 CD's that cost the same, together cost $114. How much did each one cost?
Each one cost $19. In this case, it was convenient to decompose 114 as the difference, 120 − 6. Example 5. A business spent $2,580 on items that cost $6 each. How many such items did they buy? Answer. What number times 6 is 2,580? We must divide 2580 by 6. Since 24 is divisible by 6, then so is 2400.
We can check this -- and any division -- by multiplying. 430 × 6 = 400 × 6 + 30 × 6 = 2400 + 180 = 2580. (Lesson 8) Example 6. You have $840 from which you have to make monthly payments of $75. How many $75 payments can you make? Solution. There are at least ten 75's in 840: 10 × 75 = 750. And 750 plus another 75 is 825; plus 15 more will make 840.
You could make 11 payments of $75. And $15 would remain. Example 7. Inexact division.
"5 goes into 22 four (4) times with 2 left over." Write the remainder 2 as the numerator of the fraction.
In the next Lesson, we will see that it is on this principle of decomposing the dividend that the historical written method is based. Explanation of the properties of division Arithmetic is the first science. We look at facts themselves. We will explain the properties of division by looking at the arithmetical meanings of multiplication and division, and the relationship between them. This is not algebra.
To illustrate that, let us start with 5, and then multiply it by 3:
3 × 5 And now, let us divide by 3. Let us repeatedly subtract 3's. But since 15 is now composed of 5's, how can we do that? According to the order property of multiplication, 3 × 5 = 5 × 3.
And so the product is made up of 3's -- it is made up of five 3's. 15 ÷ 3 = 5. Hence we are back where we started, at 5. The student should realize that the multiplication table -- 3 × 5 = 15 -- is not the point here. The point is to understand -- to see -- that while 3 × 5 is a sum of 5's, we can still subtract 3's. This property, after all, holds for any numbers. (206 × 19) ÷ 19 = 206. Understanding that has nothing to do with knowing the "answer" to 206 × 19 Next, let us start with 15 and divide it by 5:
We will get 3, because 3 × 5 = 15. But according to the order property, 5 × 3 = 15.
Therefore, if we now multiply 3 by 5, we will get back to 15
The quotient is the number of times the divisor is contained in the dividend. As many times, then, as we increase or decrease the dividend -- and keep the divisor the same -- the quotient will increase or decrease that same number of times. If we double the dividend, the divisor will go in twice as many times, that is, the quotient will double. If we triple the dividend, the quotient will triple. On the other hand, if we divide the dividend by 2, that is, is we take half of it -- -- then quotient will also be half. And so on. Next, as many times as we increase or decrease the divisor -- and keep the dividend the same -- the quotient will decrease or increase that same number of times. If we double the divisor, the quotient will be half. If we triple the divisor, the quotient will be one third as much. On the other hand, if we divide the divisor by 2 -- -- it will go in twice as many times. If we divide the divisor by 3, the quotient will triple. And so on. Therefore, if we increase or decrease the dividend and the divisor the same number of times, the quotient will not change. In other words: The quotient will not change if we multiply the dividend and divisor by the same number, or if we divide them both by the same number.
(3 × 20) ÷ 5 = 3 × (20 ÷ 5) That is, to divide 3 × 20 by 5, we may first divide 5 into 20, and then multiply by 3. Here is (3 × 20) ÷ 5 -- Three 20's divided by 5:
But that is the same as (20 ÷ 5) --
-- added three times! Therefore, (3 × 20) ÷ 5 = 3 × (20 ÷ 5). Please "turn" the page and do some Problems. or Return to Section 1. or Continue on to the next Lesson. Introduction | Home | Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2001-2008 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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-- it into the sum, 30 + 12.
