Multiplying Larger Numbers with Single Digits
Double Abacus Approach
Short Course
This is a short course (slide show) on how to multiply larger numbers with single digits. Just click on the arrow on the right to start the show.
Multiplying Larger Numbers with Single Digits
Let’s look at the process of calculating the product using larger numbers. The setup of the calculation is the same as when you multiplied by single digits. Remember, in this system, the first number is called the multiplicand and the second number is the multiplier. The multiplicand is entered on the left side of the top abacus. The multiplier is entered in the right side of the top abacus. The result (or product) is entered and added in the bottom abacus. If you switch the two numbers, you will still get the same result. The steps you use to calculate the product will be different, but the result (product) will be the same.
It is important to understand two principles:
Multiplying 2 digits always results in a 2-digit result. If we multiply 3 time 3 the product (result) is 09. If we multiply 4 times 4, the product is 16.
Multiplying two numbers, using the Double-A method, is always performed from left to right. While this method is different from what you learned in school, it produces the answer more quickly and preserves the place value of the result.
As you learned earlier, each digit in a number has a place value. The value of each digit increases as we move from right to left. For example, in the number, 43,567, we have 4 ten thousands, 3 thousands, 5 hundreds, 6 tens and 7 ones. The most significant digit (the digit with the highest value) is the 4 in the ten thousands column. When we multiply two numbers, we multiply the most significant digits first. Then moving from left to right, we multiply the next digit (the digit) to the right by the multiplier and add the result to the product in the bottom abacus. Each time we multiply a new digit, we shift one column to the right until we have multiplied all the digits in the multiplicand.
Let’s see how this works in an example, 43,567 times 3. Each digit of the multiplicand (with its place value) is multiplied by the multiplier:
(40000 * 3) = 120000 + Partial sum = 120,000
(3000 * 3) = 09000 + Partial sum = 129,000
(500 * 3) = 1500 + Partial sum = 130,500
(60 * 3) = 180 + Partial sum = 130,680
(7 * 3) = 21 Final sum = 130,701
When we use our top and bottom abacus in the Double-A method, the starting column for the partial product in the bottom abacus is determined by adding the number of digits we are multiplying and the number of digits in the multiplier. In this example, the multiplication of each digit is:
(40000 * 3) = 120000 + Partial sum = 120,000 Starting column is 5 + 1 = 6
(3000 * 3) = 09000 + Partial sum = 129,000 Starting column is 4 + 1 = 5
(500 * 3) = 1500 + Partial sum = 130,500 Starting column is 3 + 1 = 4
(60 * 3) = 180 + Partial sum = 130,680 Starting column is 2 + 1 = 3
(7 * 3) = 21 Final sum = 130,701 Starting column is 1 + 1 = 2
As you can see, the starting columns for the product of each digit is one less the than the previous starting column. In other words, we shift one column to the right (in the bottom abacus) each time we multiply a digit.
Examples
Now, let’s look at some more examples of this process. In these examples, the multiplicand will have many digits, but the multiplier will always be a single digit. Click on the link to show the steps to calculate the answer.
Example: 85 times 5
Example: 119 times 7
Example: 2,573 times 8
Example: 93,447 times 4
Example: 153,245 times 5