Dividing Decimals
Single Abacus Approach
5.0401 divided by 0.8940 = 5.6376 (to 4 decimal places)
Short Course
This is a short course (slide show) in long division with decimal divisors. All of the examples in this show use tw0 or more digits in the divisors.
Setting up the Calculation
Setting up the division calculation that involves decimal numbers is almost exactly the same as it is with integer numbers. We will set the divisor and dividend in exactly the same way as we did in the previous chapters. But in order to do this, there are two things we need to do first (this is the same technique we used when we multiplied two decimal numbers):
First, we must remove any extra zero digits.
Extra zeros can be found on either side of the decimal. They don’t change the value of number if they are removed. For example:
The number 008.54000 has several zeros that can be removed. On the left side of the decimal, the first 2 zeros can be removed. The digit 8 is a significant digit but the leading zeros don’t change the number at all. On the right side of the decimal, the zeros to right of the digit 4 can be removed. The digit 4 is the last significant digit on right so any zeros to the right of it can be removed. So, the number 008.54000 is the same as 8.54.
Another example is the number 050.090. The most significant digit on the left is the digit 5, in the ten’s column. Any zeros to left of this digit can be removed. However, the zeros to the right of digit are also significant and can’t be removed. To the right of the decimal point, the least significant digit is 9 in the hundredth’s column. Any zeros to the right of this digit can be removed. However, the zero in the tenth’s column is significant and can’t be removed. So the number 050.090 is the same as 50.09.
For our calculations, we can also remove zeros from the right of the decimal point when there are no digits on the left side of the decimal. For example, one of our calculations uses the number 0.04695. In this case, we can remove the zero to the left of the decimal point and the first zero. For our calculation, we can use the digits 4695.
The second task is to remove (temporarily) the decimal point. When we enter both the divisor and dividend, we will set integer numbers just as we did in the previous chapters. In the previous examples, 8.54 will be entered as 854, 50.09 will be entered as 5009 and .04695 will be entered as 4695.
Why have we included these two steps? By removing the extra zeros from both the divisor and dividend and the decimal point we can condense the numbers into integers. First, this allows us to determine if we can actually set both the divisor and dividend. As you know, the Cranmer abacus has 13 columns. If the dividend has 8 digits, then the divisor can only have 3 digits (assuming that quotient will be an integer). If we allow for 3 decimal places, the dividend can only have 6 digits. We also need to avoid the situation where we need to place the first digit in the quotient in a column that is already occupied by a digit in the divisor. By condensing the divisor and dividend, we may be able to perform calculations with numbers with more decimal digits.
The process for setting up the division calculation with the condensed numbers remains the same. First, set up the dividend. Enter this number from on the right side of the Cranmer abacus. If the dividend is a 3-digit number, then from left to right, set each digit starting in column 3 (for an integer remainder). If you want the remainder to a decimal, move the starting position to the left to accommodate the decimal digits. We will perform our division calculations to 3 decimal positions so the dividend will be entered 3 additional columns to the left.
Then, set up the divisor. Start from the left side of your Cranmer abacus, in column 13, and set each digit of the divisor. If the divisor has 2 digits, it will be set in columns 13 and 12.
The result of the division calculation is the quotient. We will enter this number in the in the columns to the left of the dividend using the “equal, shift 2 left” and “not equal, shift 1 left” principles.
Dividing Decimal Numbers
The process for calculating the final quotient is the same process used in the previous chapters with two steps added at the end of the process. In this process we use a “higher divisor” or a “lower quotient” and upwardly adjust the quotient when needed.
The location of the partial quotient is determined by the first digit of the divisor.
If the number of digits we use in the divisor equals the number of digits we use in the dividend, we apply the “equal, shift 2 left” principle. This means we move 2 columns to the left of the dividend and then set the result of the division calculation in this column.
If the number of digits we use in the divisor doesn’t equal the number of digits we use in the dividend, we apply the “not equal, shift 1 left” principle. In this case we move 1 column to the left and set the results of the division calculation in this column.
Then we multiply the number that you just entered by each digit of the divisor. Remember, in this book, the product of two single digits is always a 2-digit number (3 * 3 = 09). Then subtract the result from the correct columns of the dividend. You will also remember that each product is moved one column to the right as we multiply each digit of the divisor. The result gives you a partial dividend.
If the partial dividend is greater than the divisor, repeat the division process by dividing the first digit of the divisor into either the first digit or first 2 digits of the new partial dividend, setting the result in the correct column, multiplying the result by the divisor and subtracting the product from the partial dividend. The division process ends when the partial dividend is 0 or less than the value of the divisor.
Now, we need to determine the zero point. Count the number of digits in the divisor and add 1 for the process of division. For example, 7.268 divided by 0.92 (7268 divided by 92) is 79. If we performed this calculation without allowing for 3 decimal places, the quotient would-be set-in columns 5 and 4. The zero point would be between columns 3 and 4 (2 for the number of digits in the divisor and 1 for the process of division).
Lastly, we need to locate the decimal place. Subtract the number of decimal places in the divisor from the number of decimal places in the dividend. If the number is positive, move the decimal point to the left. If the number is negative, move the decimal point to the right. In the example above, 7268 divided by 92 is 79. Where will the decimal point be located? Calculating the difference between 3 decimal places in the dividend and the 2 decimal places in the divisor is +1. The decimal place is 1 column to the left of the zero point, between columns 5 and 4. So, the final quotient is 7.9.
Examples
Now, let’s look at some more examples of this process. In these examples, the dividend will have many digits, the divisor will have 3 or more digits and the remainder will be a decimal (to 3 places). Click on the link to show the steps to calculate the answer.
Example: 133.4 divided by 4.1
Example: 7.443 divided by 0.45
Example: 51.625 divided by 4.07
Example: 4,348.77 divided by 0.026
Example: 5.0401 divided by 0.8940