| 4.319 + 221.8 | 10 − 0.063 | ||
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Line up the decimal points and add zeroes. |
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Line up the decimal points and add zeroes. |
To add or subtract decimals, make sure to line up the decimal points. Then add zeroes to make the right sides of the decimals the same length:
| 4.319 + 221.8 | 10 − 0.063 | ||
|
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Line up the decimal points and add zeroes. |
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Line up the decimal points and add zeroes. |
Addition & Subtraction: Line up the decimal points
To multiply decimals, ignore the decimal point until the very end. First, multiply the numbers as you would if they were whole numbers. Then count the total number of digits to the right of the decimal point in the factors. The product should have the same number of digits to the right of the decimal point:
| 0.02 × 1.4 | 14 × 2 28 |
Multiply normally |
There are three digits to the right of the decimal point in the factors (the digits 0 and 2 in the first factor and the digit 4 in the second factor). Therefore, move the decimal point three places to the left in the product: 28 → 0.028.
Multiplication: In the factors, count all the digits to the right of the decimal point—then put that many digits to the right of the decimal point in the product.
If the product ends with 0, count it in this process: 0.8 × 0.5 = 0.40, because 8 × 5 = 40. Thus, 0.8 × 0.5 = 0.4.
If you are multiplying a very large number and a very small number, the following trick works to simplify the calculation: move the decimals in the opposite direction the same number of places.
0.0003 × 40,000 = ?
Move the decimal point right four places on the 0.0003: 3
Move the decimal point left four places on the 40,000: 4
0.0003 × 40,000 = 3 × 4 = 12
The reason this technique works is that you are multiplying and then dividing by the same power of 10. In other words, you are trading decimal places in one number for decimal places in another number. This is just like trading decimal places for powers of 10, as we saw earlier.
If there is a decimal point in the dividend (the inner number) only, you can simply bring the decimal point straight up to the answer and divide normally:
Ex. 12.42 ÷ 3 = 4.14
However, if there is a decimal point in the divisor (the outer number), you should shift the decimal point to the right in both the divisor and the dividend to make the divisor a whole number. Then, bring the decimal point up and divide. Be sure to shift the decimal in both numbers before dividing.
Ex: 12.42 ÷ 0.3: 124.2 ÷ 3 = 41.4
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Move the decimal one space to the right to make 0.3 a whole number. Then, move the decimal one space in 12.42 to make it 124.2. |
Division: Always shift the decimals on top and bottom so you are dividing by whole numbers.
You can always simplify division problems that involve decimals by shifting the decimal point in the same direction in both the divisor and the dividend, even when the division problem is expressed as a fraction:
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Move the decimal four spaces to the right to make both the numerator and the denominator whole numbers. |
Note that this is essentially the same process as simplifying a fraction. You are simply multiplying the numerator and denominator of the fraction by a power of 10—in this case,
.
Keep track of how you move the decimal point. To simplify multiplication, you can move decimals in opposite directions. However, to simplify division, you move decimals in the same direction.
Equivalently, by adding zeroes, you can express the numerator and the denominator as the same units, then simplify:
= 45 ten-thousandths ÷ 900 ten-thousandths =
62.8 + 4.5768 = ?
7.125 − 4.309 = ?
0.00018 × 600,000 = ?
85.702 ÷ 0.73 = ?