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In this lesson, we will learn how to multiply multidigit decimals.
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We can multiply decimals using the same methods for multiplying whole numbers.
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In this video, we will look at three different methods, the column method, the grid method, and the lattice method.
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However, it’s important to note that any method for multiplication of whole numbers can also be used for decimals.
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Let’s begin with the definition of a decimal.
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A decimal is a number that contains a decimal point.
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The digits to the left of the decimal point represent the whole part of the number.
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And the digits to the right represent the decimal part of the number.
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We can consider this in more detail using place value columns.
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In this example, to the left of the decimal point, we have the thousands, hundreds, tens, and ones columns.
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To the right of the decimal point, we have the tenths, hundredths, and thousandths.
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If we consider the number 4736.528, then the digits to the left of the decimal point, 4736, represent the whole part.
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The digits five, two, and eight represent the decimal part.
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We have five tenths, two hundredths, and eight thousandths.
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Before looking at our three different methods, we will look at some general rules in how to multiply decimals.
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Our first step is to remove the decimal point from the numbers we want to multiply and consider them as whole numbers.
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Next, we need to multiply these whole numbers using our preferred multiplication method.
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As previously mentioned, we will look at examples using the grid method, the lattice method, and the column method.
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Finally, we put the decimal point back into the answer.
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The answer will have as many decimal places as the sum of the decimal places in the original numbers.
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If there are a total of three numbers after the decimal points in the question, there will be three numbers after the decimal point in the answer.
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We will now look at some examples of multiplying two decimals.
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Calculate 0.39 multiplied by 5.6.
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We will answer this question using the grid method.
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Our first step is to remove the decimal points.
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The calculation, therefore, becomes 39 multiplied by 56.
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Our second step is to multiply the whole numbers.
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We will do this as mentioned using the grid method.
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We split both of our numbers into their tens and ones or units component.
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39 becomes 30 and nine, and 56 becomes 50 and six.
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We then need to carry out four multiplication calculations.
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Firstly, we multiply 50 by 30.
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Five multiplied by three is equal to 15.
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Adding the two zeros means that 50 multiplied by 30 is 1500.
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Next, we’ll multiply 50 by nine.
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Five multiplied by nine is equal to 45.
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Therefore, 50 multiplied by nine is 450.
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Next, we will multiply six by 30.
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This is equal to 180, as six multiplied by three is 18.
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Finally, six multiplied by nine is equal to 54.
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This means that the solution to 39 multiplied by 56 is the sum of 1500, 415, 180, and 54.
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1500 plus 180 is 1680.
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450 plus 54 is equal to 504.
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Adding these two values gives us 2184.
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Therefore, 39 multiplied by 56 is 2184.
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Our final step is to put the decimal points back in.
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Remember, the answer will have as many decimal places as the sum of the decimal places in the question.
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0.39 had two digits after the decimal point.
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5.6 had one number after the decimal point.
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Two plus one is equal to three.
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Therefore, our answer needs to have three numbers after the decimal point.
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The digits one, eight, and four will all be after the decimal point.
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We can therefore say that 0.39 multiplied by 5.6 is equal to 2.184.
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We will now look at the second example using a different method.
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Calculate 0.71 multiplied by 0.53.
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We will solve this calculation using the lattice method.
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Our first step when multiplying two decimals is to remove the decimal points.
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In this question, our calculation becomes 71 multiplied by 53.
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Next, we need to multiply the two whole numbers.
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In this case, as previously mentioned, we will use the lattice method.
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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.
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In this case, we have seven and one on the top and five and three on the right.
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We draw diagonal lines through each section of the grid.
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In each of the grid entries, we then calculate the product of the digit at the top of the row and the digit on the right edge.
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We can begin by multiplying seven and five.
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Seven multiplied by five is equal to 35.
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We put a three in the top half and a five in the bottom half.
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Seven multiplied by three is equal to 21.
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Therefore, we have two in the top half and one in the bottom half.
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Next, we need to multiply one and five.
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One multiplied by five is equal to five.
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And as this is less than 10, we put a zero in the top half and five in the bottom half.
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One multiplied by three is equal to three.
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So, once again, we have a zero in the top half and a three in the bottom half.
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To work out the answer using the lattice method, we then sum the digits in the lattice along the diagonal strips.
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The first strip just has the number three.
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The second strip has a five, a zero, and a one.
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These numbers sum to six.
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The third strip has a zero, a five, and a two.
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Zero plus five plus two is equal to seven.
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The final strip, once again, only has a three.
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Reading in the direction of the arrow, our answer is 3763.
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We can therefore say that 71 multiplied by 53 is equal to 3763.
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Our final step, in order to calculate 0.71 multiplied by 0.53, is to put the decimal point back in.
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Remember that our answer will have as many decimal places as the sum of the decimal places in the question.
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The number 0.71 had two digits after the decimal point.
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The number 0.53 also had two numbers after the decimal point.
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As two plus two is equal to four, our answer will have four digits after the decimal point.
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All four of the digits three, seven, six, and three will be after the decimal point.
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We can therefore conclude that 0.71 multiplied by 0.53 is equal to 0.3763.
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From this question comes an important point when multiplying decimals.
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When we multiply two decimals that are less than one, the answer will also be less than one.
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We will now look at a third example using a different method.
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Calculate 7.106 multiplied by 0.29.
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We will answer this question using the column method.
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When multiplying two decimals together, our first step is to remove the decimal points.
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In this question, our calculation will become 7106 multiplied by 29.
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Our next step is to multiply the whole numbers.
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In this case, as mentioned, we will use the column method.
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We begin by multiplying the top number by the ones digits in the bottom number.
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We multiply 7106 by nine.
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Nine multiplied by six is equal to 54.
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We put the four in the ones column and carry the five.
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Nine multiplied by zero is equal to zero.
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Adding the five that we carried gives us five.
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Nine multiplied by one is equal to nine.
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Finally, nine multiplied by seven is equal to 63.
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7106 multiplied by nine is equal to 63954.
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Our next step is to multiply 7106 by 20.
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As we’re multiplying by 20, we can put a zero in the ones column and then multiply by two.
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Two multiplied by six is equal to 12.
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We put a two in the tens column and carry the one.
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Two multiplied by zero is equal to zero.
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Adding the one we carried gives us one.
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Two multiplied by one is equal to two.
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And finally, two multiplied by seven is equal to 14.
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7106 multiplied by 20 is equal to 142120.
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Our final step to multiply 7106 by 29 is to add these two rows.
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Four plus zero is equal to four.
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Five plus two is equal to seven.
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Nine plus one is equal to 10, so we need to carry the one.
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Three plus two plus the one we carried is six.
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Six plus four is equal to 10.
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And finally, one plus one is equal to two.
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7106 multiplied by 29 is equal to 206074.
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Our final step is to put the decimal points back in.
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At this point, it’s important to remember that the answer will have as many decimal places as the sum of the decimal places in the question.
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7.106 has three digits after the decimal point.
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0.29 has two digits after the decimal point.
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This means that our answer will have five digits after the decimal point.
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7.106 multiplied by 0.29 is equal to 2.06074.
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Our answer has five decimal places.
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We will now look at a couple of different situations where we’re given a whole number multiplication.
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Given that 225 multiplied by 248 is equal to 55800, what is 2.25 multiplied by 24.8?
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Before trying to answer this question, we need to recall one of our key facts when multiplying decimals.
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That is, that the answer to two decimals multiplied will have as many decimal places as the sum of the decimal places in the original numbers.
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The number 2.25 has two digits after the decimal point.
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The number 24.8 has one digit after the decimal point.
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This means that our answer will have three digits after the decimal point.
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The digits in our original calculation, 225 and 248, are the same as the digits in the new calculation.
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This means that the digits in the answer will be five, five, eight, zero, zero.
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As already mentioned, we need three digits after the decimal point.
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This means that 2.25 multiplied by 24.8 is equal to 55.800.
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We can rewrite this ignoring the final zeros.
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Therefore, our answer is 55.8.
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An alternative way of looking at this question would be to consider how we get from our original numbers to the numbers in the second calculation.
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225 divided by 100 is equal to 2.25.
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248 divided by 10 is equal to 24.8.
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This means that we have divided the calculation by 100 and by 10.
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Dividing by 100 and then dividing by 10 is the same as dividing by 1000.
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Dividing 55800 by 1000, once again, gives us 55.800.
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When dividing by 1000, all the digits move three places to the right.
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This confirms that the answer to 2.25 multiplied by 24.8 is 55.8.
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This type of problem where we’re given a whole number multiplication which can be used to solve a decimal multiplication is often given in examinations.
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We do this by working out how many decimal places the answer should have.
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We’ll now look at the summary of the key points of this lesson on multiplying decimals.
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Firstly, we can multiply decimals using the same techniques that we use to multiply whole numbers.
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In this lesson, we have looked at the column method, the lattice method, and the grid method.
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When we multiply decimals, we firstly consider the numbers as whole numbers by removing the decimal points.
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Once we’ve multiplied the two numbers, we put the decimal point back into the answer so that it has the same number of decimal places as the sum of the decimal places in the original numbers.