Explainer: Multiplying Decimals

In this explainer, we will learn how to multiply multidigit decimals.

Definition: Decimal

A decimal is a number that contains a decimal point. The digits to the left of the decimal point represent the whole number part of the number, and the digits to the right of the decimal point represent the decimal part of the number.

We can multiply decimals using the same methods for multiplying whole numbers. Common multiplication methods include the column method, the grid method, and the lattice method. In this explainer, we will review these methods for multiplication and use them to multiply decimals. However, any method for multiplication of whole numbers can also be used for decimals. Let us now look at how to multiply decimals.

How to Multiply Decimals

  • Remove the decimal point from the numbers to be multiplied and consider them as whole numbers.
  • Multiply these whole numbers using the desired multiplication method.
  • Put the decimal point into the answer; the answer will have as many decimal places as the sum of the decimal places in the original decimal numbers.

Example 1: Multiplying Two Decimals Using the Grid Method

Calculate 0.39×5.6.

Answer

To begin, we remove the decimal points from 0.39 and 5.6 and consider them as the whole numbers 39 and 56. We can now calculate 39×56 using any multiplication method. Here, we will use the grid method.

×309
501,500450
618054

In the grid above, we expanded 39 into 30 and 9 in the row heading and expanded 56 into 50 and 6, writing these in the column heading. Each entry in the table is the product of the number in the corresponding row and column headings.

The solution to 39×56 is the sum of the table entries, 1,500, 450, 180, and 54. Therefore, 39×56=1,500+450+180+54=2,184.

Next, to find the solution to 0.39×5.6 we identify that 0.39 has two decimal places and 5.6 has one decimal place. Our answer will have as many decimal places as the sum of the decimal places in the two numbers. Therefore, our answer will have 3 decimal places.

Hence, 0.39×5.6=2.184.

Example 2: Multiplying Two Decimals Using the Lattice Method

Calculate 0.71×0.53.

Answer

To begin, we remove the decimal points from 0.71 and 0.53 and consider them as the whole numbers 71 and 53. We can now calculate 71×53 using any multiplication method. Here, we will use the lattice method.

To use the lattice method, we draw a grid and write the digits in the numbers to be multiplied along the top and down the right side of the lattice. We draw diagonal lines through each section of the grid. In each of the grid entries, we calculate the product of the digit at the top of the row and the digit on the right edge. When our product is a single digit, such as in the case of 5×1=5, we insert 05. We can then sum the digits in the lattice along the diagonal strips.

Here, we have 71×53=3,763.

To find the solution to 0.71×0.53, we add the decimal places in 0.71 (2 decimal places) and 0.53 (2 decimal places) to give 4 decimal places. Therefore, our answer will have 4 decimal places. Putting a 0 as a place holder before the decimal point gives us 0.71×0.53=0.3763.

Example 3: Multiplying Decimals Using the Column Method

Calculate 7.106×0.29.

Answer

To begin, we remove the decimal points from 7.106 and 0.29 and consider them as the whole numbers 7,106 and 29. We can now calculate 7,106×29 using any multiplication method. Here, we will use the column method. We begin our multiplication by multiplying the 9 in the ones column by each of the digits in 7,106, which gives us 7106×2963954

We then continue the calculation by multiplying 7,106 by the 2 tens, remembering to add a 0 in the ones column. We then add our two lines, which gives us 7106×2963954+142120206074

Therefore, 7,106×29=206,074.

To calculate 7.106×0.29, we add the decimal places in 7.106 (3 decimal places) and in 0.29 (2 decimal places) to give us 5 decimal places. So, our answer will have 5 decimal places.

Therefore, 7.106×0.29=2.06074.

We may often see a problem where we are given a whole-number multiplication, which we can use to solve a decimal multiplication. First, we identify that the digits in the numbers in our given calculation match those in the decimal multiplication. Then, we can use the given whole-number answer to work out the decimal answer by calculating how many decimal places it should have. Let us now look at some examples of this type of problem.

Example 4: Calculating a Decimal Multiplication given a Whole-Number Multiplication

Given that 225×248=55,800, what is 2.25×24.8?

Answer

Recall that the answer to two decimals multiplied will have as many decimal places as the sum of the decimal places in the original numbers. Therefore, since 2.25 has 2 decimal places and 24.8 has 1 decimal place, we add 2 and 1 to give 3 decimal places in our answer.

Since we are given 225×248=55,800, we can write 55,800 as a number with 3 decimal places as 2.25×24.8=55.800.

We can also write this without the final zeros, giving us 2.25×24.8=55.8.

Example 5: Calculating a Decimal Multiplication given a Whole-Number Multiplication

Given that 636×572=363,792, find the missing number to complete the calculation ×57.2=3.63792.

Answer

Recall that the answer to two decimals multiplied will have as many decimal places as the sum of the decimal places in the original numbers.

Therefore, since the answer 3.63792 has 5 decimal places and 57.2 has 1 decimal place, the missing number in our calculation must have 4 decimal places, since 51=4.

Since we are given 636×572=363,792, we need to write the number 636 as a number that has 4 decimal places.

It can be helpful to write 636 into a place value grid so that the final digit is in the 4th decimal place.

We need to write a zero as a place holder before the decimal place and another one after the decimal place before the digits 636.

We can now read the decimal number 0.0636 from the grid, and so our answer is 0.0636×57.2=3.63792.

Key Points

  • We can multiply decimals using the same techniques that we use to multiply whole numbers, including the grid method and the column method.
  • When we multiply decimals, we consider the numbers as whole numbers by removing the decimal points and multiplying the whole numbers. We then put the decimal point into the answer so that it has the same number of decimal places as the sum of the decimal places in the original numbers.

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