Speak
"7 goes into 25 three (3) times (21) with 4 left over."
Write the remainder 4 beside the 2. Continue:
"7 goes into 42 six (6) times exactly."
Compare the simplicity of that with long division:
In long division, we bring down the 2 and write it next to the remainder 4. In short division, we simply write the remainder next to 2. Long division, moreover, de-emphasizes the spoken nature of arithmetic. It gives the false impression that, like algebra, arithmetic is a written skill.
Long division now properly belongs to the history of mathematics.
For no other reason than tradition, long division is still taught. Since both long and short division are nothing but methods, neither one requires a true understanding of division. That comes only with decomposing the dividend -- which in any case is the principle on which all methods are based.
Begin, "5 goes into 17 three (3) times (15) with 2 left over."
Write 3 over the 7 (not over the 1), and write the remainder 2 next to the 9.
Continue: "5 goes into 29 five (5) times (25) with 4 left over.
Write 5 over the 9, and write the remainder 4 next to the 8.
Finally, "5 goes into 48 nine (9) times (45) with 3 left over."
Write 9 over the 8. The final remainder is 3.
This problem will illustrate the following point:
Over every digit in the dividend
we must write a digit in the quotient.
We proceed one digit at a time.
|
We will write a digit over the 1, then over the 6, then over the 0, and so on, until finally we write a digit over the 3.
Begin,
"4 goes into 21 five (5) times (20) with remainder 1 ."
Next, "4 goes into 16 four (4) times exactly."
Next, "4 goes into 0 zero (0)."
Whenever the partial dividend is less than the divisor
-- 0 is less than 4 -- write 0 in the quotient.
Next, we must write a digit over the 2: "4 goes into 2
zero (0)."
Now the 2 remains. It is the remainder.
Whenever the quotient is 0, that digit beneath it
in the dividend is the remainder.
"4 goes into 24 six (6) times exactly."
Finally, "4 goes into 3 zero (0)."
3 is the final remainder.
Again, whenever the quotient is 0, the digit beneath it in the dividend is the remainder.
"3 goes into 15 five (5) times. 3 goes into 2 zero (0)."
2 is the remainder.
That is, 152 = 50 × 3 + 2.
We use short division whenever it is easy to multiply the divisor.
Example 4. Harold has a debt of $3,164. He is able to pay $25 a week. How many weeks will it take him to pay the debt?
Solution. How many 25's will equal 3,164. To find out, we must divide:
"25 goes into 31 one (1) time (25) with 6 left over."
"25 goes into 66 two (2) times (50) with 16 left over."
"25 goes into 164 six (6) times (150) with 14 left over."
At the end of 126 weeks, then, the debt will be almost paid. $14 will remain. Therefore, it will take Harold 127 weeks.
Problem. The multiples of 8 that are less than 100, are what percent of all the numbers less than 100?
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
The multiples of 8 are the numbers 8, 16, 24, 32, and so on. How many multiples of 8 are contained in 100 -- that is, what number is 100 ÷ 8?
100 ÷ 8 = 12 R 4. That means, there are 12 eights less than 100. (12 × 8 = 96.) So the question is: 12 is what percent of 100?
The student should know immediately that the answer is 12% -- because that should be the first lesson in percent!
Lesson 4.
*
We will now consider the case in which the dividend is a decimal.
The divisor, we have seen (Lesson 11), must always be a whole number. (In Lesson 13 we will consider the case in which the divisor is a decimal.)
|
