Multiplying Decimals

Short Course 

This is a short course (slide show below) on how to multiply any size numbers.  

Setting up the Calculation

Setting up the multiplication calculation that involves decimal numbers is almost exactly the same as it is with integer numbers. We will set the multiplicand and multiplier in exactly the same way as we did in the previous sections. But in order to do this, there are two new things we need to do first:

• 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 007.46000 has several zeros that can be removed. On the left side of the decimal, the first 2 zeros can be removed. The digit 7 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 6 can be removed. The digit 6 is the last significant digit on right so any zeros to the right of it can be removed. So the number 007.46000 is the same as 7.46.

  • Another example is the number 050.030.  The most significant digit on the left is the digit 5, in the tens 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 3 in the hundredths column. Any zeros to the right of this digit can be removed.  However, the zero in the tenths column is significant and can’t be removed.  So the number 050.030 is the same as 50.03.

  •  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 0.07495. 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 7495.

• The second task is to remove (temporarily) the decimal point. When we enter both the multiplicand and multiplier, we will set integer numbers just as we did in the previous chapters. In the previous examples, 7.46 will be entered as 746, 50.03 will be entered as 5003 and .07495 will be entered as 7495.

Why have we included these 2 new steps? By removing the extra zeros from both the multiplicand and multiplier and the decimal point we can condense the numbers into integers. First, this allows us to determine if we can actually set both the multiplicand and the multiplier. As you know, the Cranmer abacus has 13 columns. If the multiplicand has 4 digits, then the multiplier can only have 4 digits. If the multiplicand has more digits, the multiplier must have fewer digits. By condensing the multiplicand and multiplier, we may be able to perform calculations with numbers with more decimal digits.  

Now, set up the multiplicand. The multiplicand is the first of the two numbers in the multiplication calculation and is set on the left most columns of the Cranmer abacus. Set the multiplicand, from left to right, starting in column 13.

Then, set up the multiplier. Count the number of digits in the multiplier and multiplicand. Then add the two numbers together and add 1 for the process of multiplication.  This number is the column you should start to set the multiplier from left to right.

Multiplying Decimal Numbers

As in previous sections, the result of multiplication calculation is the product. It is calculated by multiplying the digits of the multiplicand times the digits of the multiplier. The result is added number to the right of the multiplier. When all the digits of the multiplier and multiplicand have been multiplied, the result is called the final product without the decimal point.

The process for calculating the final product is the same process used in the previous chapters with one step added at the end of the process. Each digit of the multiplicand is multiplied by each digit of the multiplier. Then we calculate the position of the decimal point and insert it into the product for the final product. This is the process in more detail:

• • • First, we multiply the rightmost digit of the multiplier and the leftmost digit of the multiplicand. Set the product in the two columns immediately to the right of the rightmost digit of the multiplier, first position.

    • Then, we multiply the same digit of the multiplier to next digit of the multiplicand. Add this product to the partial product after shifting 1 column to the right. The partial product is completed when all digits of the multiplicand have been multiplied by the digit in the multiplier. Lastly, clear this digit in the multiplier. A new digit will be used to multiply each digit in the multiplicand. This means that the new first position of the partial product has shifted one column to the left. 

    • Continue multiplying each digit of the multiplier and adding the product to the partial product after shifting the position one column to the left. When you have completed multiplying each digit of both numbers and adding the product to the partial product, all the digits of the multiplier will be cleared.  The result will be the final product without the decimal point.

    • Lastly, add the number of digits to the right of the decimal point that were used in the calculation for both the multiplicand and the multiplier. This is the location of the decimal point.   

Examples

In these examples, since both the multiplier and the multiplicand have more than 1 digit, the number of digits in the multiplicand will limit the size of the multiplier. If the multiplicand has 4 digits, then the multiplier can only have 4 digits. If the multiplicand is smaller, then the multiplier can be a larger number. 

• Example: 7.2 * 9.7

• Example: 88.1 * 4.48

• Example: 988.0 * 31.3

• Example: 56.6 * 5.45

• Example: 230.9 * 13.8

• Example: 0.1738 * 1.74

• Example: 43.8 * 8.790

• Example: 2.795 * 4.387

• Example: 600.0 * 0.0171

Build Your Skill

Now you are ready to try some calculations on your own. Click on problems and their solutions to test your skill.