Mental calculation
Decomposing the dividend

Lesson 10  Section 2

Back to Section 1

We say that one number is a divisor of another if it can go into it exactly.  Thus 4 is a divisor of 12.  5 is not a divisor of 12.

That is another sense of the word divisor.

The following is one of the most important properties of division, because it leads not only to the traditional written method, but to a method for dividing mentally as well.

If a number is a divisor of two numbers, then it is also a divisor
of their sum and difference.

For example, say that you do not know 42 ÷ 3.  Then you could decompose 42 -- break it up -- it into the sum, 30 + 12.

Why?  Because you know 30 ÷ 3 and you know 12 ÷ 3.  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.

Again:

If a number is a divisor of two numbers, then it will also be a divisor
of their sum.

As always in mental calculation, we go from what we know to what we do not know.

Example 1.   Divide $92 equally among 4 people.

Answer.  We have to divide 92 by 4.  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 2.   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 3.   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 4.   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 5.   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.

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.

At this point, please "turn" the page and do some Problems.

or

Continue on to the next Section: Three properties of division.

Section 1 of this Lesson.

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