the number 0.008 is equivalent to the ratio of 8 to what number
| Multiplying a decimal by a whole number |
| Multiplying decimals by decimals |
| Dominion for multiplying decimals | Quick Quiz |
Multiplication of decimals relies on knowing how to multiply whole numbers, understanding identify value, and appreciating the various multiplication situations involving decimals. The countless base ten concatenation shows the consistency of the base ten identify value system from whole numbers to decimals.
Equally with whole number multiplication, the club of the numbers being multiplied does not affect the product. This is chosen the commutative belongings of multiplication.
Multiplying a decimal by a whole number
Multiplying decimals relies on adapting multiplication of whole numbers. When one of the numbers beingness multiplied is reduced to a 10th of its original size, the product is also reduced to a tenth of its original size. The post-obit chart demonstrates this human relationship.
| Example | Working out | Regrouping | Answer |
| 3 10 eight = 24 | Three groups of eight ones is 24 ones. | 24 ones is the same as 2 tens and iv ones. | 24 |
| 3 10 0.eight = 2.iv | 3 groups of viii tenths is 24 tenths. | 24 tenths is the same as 2 ones and four tenths. | 2.4 |
| 3 x 0.08 = 0.24 | Three groups of eight hundredths is 24 hundredths. | 24 hundredths is the same as 2 tenths and iv hundredths. | 0.24 |
| 3 10 0.008 = 0.024 | Three groups of viii thousandths is 24 thousandths. | 24 thousandths is the same as 2 hundredths and four thousandths. | 0.024 |
Multiplying decimals by decimals
When multiplying a number by a decimal less than ane, the production will exist smaller than the number being multiplied. This is because we are finding a partial amount of a quantity. For example, 0.1 ten 0.8 = 0.08, because the question is asking us to find ane tenth of eight tenths. A tenth of a tenth (or a tenth multiplied by a tenth) is a hundredth, thus i tenth of eight tenths is eight hundredths.
Viewing decimal multiplication problems in common fraction and extended decimal form may help us better understand the answer. We tin link multiplication of decimals to common fractions considering we know that 0.1 (ane tenth) is the same every bit 1/10, or 1 ÷ 10. Similarly, nosotros tin can expand the fashion nosotros record decimals to proceeds an agreement of the size of answers because we know that 0.1 is the same as 1 x 0.i, or 1 ÷ 10. We tin rewrite 0.1 x 0.8 equally:
- ane/10 x eight/10 = eight/100 = 0.08 (using rules for fraction multiplication)
- 0.1 x (0.one x 8) = 0.01 x 8 = 0.08 (using rules for decimal multiplication)
- one tenth of eight tenths is 8 hundredths (using common sense)
The human relationship betwixt the identify value of the numbers being multiplied and the product is summarised by the following chart. Equally we move to the right or downwards the chart, the numbers increase by a factor of ten, so the answers increase past a factor of ten.
Rule for multiplying decimals
A mutual way to multiply decimals is to treat them as whole numbers, then position the decimal point in the product. The number of digits subsequently the decimal points in the factors determines where the decimal point is placed in the respond. For example, 0.3 x 0.8 = 0.24. There is one digit later the decimal point in both 0.iii and 0.8, ii digits altogether, so the answer will have two digits subsequently the decimal point. We know that 3 x 8 = 24, so we can place the decimal bespeak in front of 2 digits, giving 0.24. This method relies on a sound agreement of place value to bank check that answers are reasonable.
| Example | Correct placement of the decimal point | Place value thinking |
| 0.4 x 0.v | This is just similar 4 x 5 4 10 5 = 20 0.4 has ane decimal place, 0.five has one decimal identify, so my answer will have two decimal places. My answer is 0.20 | This is 4 tenths of v tenths. Tenths by tenths gives hundredths, so my answer will accept two decimal places. 4 tenths 10 five tenths = 20 hundredths 0.4 x 0.five = 0.twenty |
Case 1: Full explanation of one.8 ten 2.3 = 4.14
A wire expands to 1.viii times its length when heated. The wire is 2.3 cm long. How long volition information technology exist when heated?
| Case | Working out | Thinking |
| 1.eight x two.3 | i.viii x 2.three = four.xiv The wire will be four.xiv cm long | This is nearly 2 times 2.three cm, so the answer will be about 4 cm. The easiest way to do this instance is to utilize whole number multiplication so adjust our answer accordingly. To be able to multiply 18 by 23 nosotros multiply 1.8 past x and ii.3 past ten then have actually multiplied by 100. (1.8 x ten x 2.3 x 10 = 18 ten 23 x 100) Once we have washed the whole number multiplication, we must and then divide our respond by 100 to recoup for this. 18 10 23 = 414 Out final answer is one hundredth of 18 x 23. So 1.8 x 2.3 = 4.14 This answer makes sense according to my initial interpretation . |
Example 2: Placing the decimal signal, 23 10 0.67 = 15.41
Petrol costs $0.67 per litre. Khumalo pumps 23 litres of petrol into his car. How much does information technology cost him? To discover this out I demand to multiply 23 litres by $0.67. 0.67 is over half of one dollar, so my answer will exist over $11 (half of $22).
Example three: Full explanation of 0.234 x 0.07 = 0.01638
| Example | Working out | Thinking |
| 0.234 x 0.07 | | Nosotros can utilise whole number multiplication to solve this problem if we are aware of how decimals relate to whole numbers. 0.234 can be expressed as 234 thousandths and 0.07 can exist expressed as vii hundredths. This can be confirmed using a number expander (discussed in Meaning and Models). Nosotros so practice the multiplication as whole number multiplication. Considering we are multiplying thousandths and hundredths, our respond will be in hundred-thousandths. Nosotros now demand to express 1638 hundred-thousandths as a decimal. If nosotros are not certain how to do this we tin use the number expander. |
The post-obit movie shows a 'procedural' method being used to solve this problem.
Rows of zeros contribute naught to the final answer and therefore are a waste material of time and ink. The decimal point tin can exist placed without multiplying the rows of zeros. 234 x vii is 1638, but 0.237 ten 0.07 will have five decimal places in the answer, then it volition be 0.01638.
Caution: if you use the rule of counting decimal places in the question to find the number of decimal places in the respond, y'all MUST non discard zeros besides early on. For example, to multiply 0.25 by 0.4, you need 2+1 decimal places and the multiplication without a decimal indicate gives 0100. The answer is therefore 0.100. This is equal to 0.1, but the right-well-nigh decimal places cannot be discarded until afterward the multiplication rule has been used.
| Why is it not necessary to line up the decimal point to multiply? |
| In addition and subtraction, it is of import to line upwardly the identify value columns because nosotros can but add or subtract digits in the same place value columns. Nosotros can, yet, multiply digits in ane place value column by those in another, so we do not need to line up the columns. |
Multiplication Quiz
ane. Given that 4857 x 6 = 29142, find:
| (a) | 4857 ten 0.6 |
| (b) | 48.57 x 0.half-dozen |
| (c) | 485.7 x 0.0006 |
| two. | 1.5 x 7.62 |
| 3. | 0.4 10 0.06 |
| 4. | 0.002 ten 0.05 |
Actress questions:
i. Find the production of 7.15 and i.9
2. Evaluate 7.04 ten 1.1
three. Steve'southward stride is 84.vi cm. How many centimetres volition he travel if he takes 200 strides?
4. The length of my front end garden bed is 4.three times longer than the length of my back garden bed. If the length of my back garden bed is 156 cm, how long is my front garden bed?
5. Is the following statement true or fake: 9.6734 ten 0.nine > nine.6734?
To view the quiz answers, click hither.
If you lot would like to practice some more than questions, click here to go to the mixed operations quiz at the finish of the sectionalization department.
Source: https://extranet.education.unimelb.edu.au/SME/TNMY/Arithmetic/decimals/operations/multidec.html
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