Multiplying two three digit decimals is very similar to the previous procedures. The following explanation will not go step by step but will only show the work that would be done.
* Place one decimal above the other so that they are lined up on the right side. Draw a line under the bottom number. Temporarily disregard the decimal points and multiply the numbers like you would multiply a three digit number by a three digit number.
0.529
0.467
3703
3174
2116
0.247043
* At the start we disregarded the decimal points. In the answer we counted up the decimal places and moved the decimal place to its proper location. We have three decimal places in both numbers so we move the decimal six places to the left to give the final answer of 0.247043.
Division of Decimals by Whole Numbers
The procedure for the division of decimals is very similar to the division of whole numbers.
How to divide a four digit decimal number by a two digit number (e.g. 0.4131 ÷ 17).
* Place the divisor (17) before the division bracket and place the dividend (0.4131) under it.
17)0.4131
* Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
0.0243
17)0.4131
Adding Fractions with Different Denominators
How to Add Fractions with different denominators:
* Find the Least Common Denominator (LCD) of the fractions
* Rename the fractions to have the LCD
* Add the numerators of the fractions
* Simplify the Fraction
Example: Find the Sum of 2/9 and 3/12
* Determine the Greatest Common Factor of 9 and 12 which is 3
* Either multiply the denominators and divide by the GCF (9*12=108, 108/3=36)
* OR - Divide one of the denominators by the GCF and multiply the answer by the other denominator (9/3=3, 3*12=36)
* Rename the fractions to use the Least Common Denominator(2/9=8/36, 3/12=9/36)
* The result is 8/36 + 9/36
* Add the numerators and put the sum over the LCD = 17/36
* Simplify the fraction if possible. In this case it is not possible
Multiplying Fractions
To Multiply Fractions:
* Multiply the numerators of the fractions
* Multiply the denominators of the fractions
* Place the product of the numerators over the product of the denominators
* Simplify the Fraction
Example: Multiply 2/9 and 3/12
* Multiply the numerators (2*3=6)
* Multiply the denominators (9*12=108)
* Place the product of the numerators over the product of the denominators (6/108)
* Simplify the Fraction (6/108 = 1/18)
*
* The Easy Way. It is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.
* For example: 2/9 * 3/12 = (2*3)/(9*12) = (1*3)/(9*6) = (1*1)/(3*6) = 1/18
Multiplying Fractions by Whole Numbers
* Multiplying a fraction by an integer follows the same rules as multiplying two fractions.
* An integer can be considered to be a fraction with a denominator of 1.
* Therefore when a fraction is multiplied by an integer the numerator of the fraction is multiplied by the integer.
* The denominator is multiplied by 1 which does not change the denominator.
Dividing Fractions by Whole Numbers
To Divide Fractions by Whole Numbers:
* Treat the integer as a fraction (i.e. place it over the denominator 1)
* Invert (i.e. turn over) the denominator fraction and multiply the fractions
* Multiply the numerators of the fractions
* Multiply the denominators of the fractions
* Place the product of the numerators over the product of the denominators
* Simplify the Fraction
Example: Divide 2/9 by 2
* The integer divisor (2) can be considered to be a fraction (2/1)
* Invert the denominator fraction and multiply (2/9 ÷ 2/1 = 2/9 * 1/2)
* Multiply the numerators (2*1=2)
* Multiply the denominators (9*2=18)
* Place the product of the numerators over the product of the denominators (2/18)
* Simplify the Fraction if possible (2/18 = 1/9)
*
* The Easy Way. After inverting, it is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.
* For example: 2/9 ÷ 2 = 2/9 ÷ 2/1 = 2/9*1/2 = (2*1)/(9*2) = (1*1)/(9*1) = 1/9
Dividing Fractions by Fractions
To Divide Fractions:
* Invert (i.e. turn over) the denominator fraction and multiply the fractions
* Multiply the numerators of the fractions
* Multiply the denominators of the fractions
* Place the product of the numerators over the product of the denominators
* Simplify the Fraction
Example: Divide 2/9 and 3/12
* Invert the denominator fraction and multiply (2/9 ÷ 3/12 = 2/9 * 12/3)
* Multiply the numerators (2*12=24)
* Multiply the denominators (9*3=27)
* Place the product of the numerators over the product of the denominators (24/27)
* Simplify the Fraction (24/27 = 8/9)
*
* The Easy Way. After inverting, it is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.
* For example: 2/9 ÷ 3/12 = 2/9*12/3 = (2*12)/(9*3) = (2*4)/(3*3) = 8/9
Adding decimals
How to add Decimals that have different numbers of decimal places
* Write one number below the other so that the bottom decimal point is directly below and lined up with the top decimal point.
* Add each column starting at the right side.
Example: Add 3.2756 + 11.48
3.2756
11.48
14.7556
How to add three or more decimal numbers that have different numbers of decimal places.
* Write the numbers in a column so the decimal points are directly lined up.
* Add each column starting at the right side.
Example: Add 23.143 + 3.2756 + 11.48
23.143
3.2756
11.48
37.8986
Division of Decimals by Decimals
The procedure for the division of decimals is very similar to the division of whole numbers. Make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places.
How to divide a four digit decimal number by a two digit decimal number (e.g 0.4131 ÷ 0.17).
* Place the divisor before the division bracket and place the dividend (0.4131) under it.
0.17)0.4131
* Multiply both the divisor and dividend by 100 so that the divisor is not a decimal but a whole number. In other words move the decimal point two places to the right in both the divisor and dividend
17)41.31
* Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
2.43
17)41.31
Division of Decimals by Decimals
The procedure for the division of decimals is very similar to the division of whole numbers. Make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places.
How to divide a four digit decimal number by a two digit decimal number (e.g. 0.424 ÷ 0.8).
* Place the divisor before the division bracket and place the dividend (0.424) under it.
0.8)0.424
* Multiply both the divisor and dividend by 10 so that the divisor is not a decimal but a whole number. In other words move the decimal point one place to the right in both the divisor and dividend
8)4.24
* Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
0.53
8)4.24
4 0
24
0
Ordering Decimal Numbers
Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them.
Example: If we start with the numbers 4.3 and 8.78, the number 5.2764 would come between them, the number 9.1 would come after them and the number 2 would come before both of them.
Example: If we start with the numbers 4.3 and 4.78, the number 4.2764 would come before both of them, the number 4.9 would come after them and the number 4.5232 would come between them.
The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Rounding Decimals
27.17469 rounded to the nearest whole number is 27
36.74691 rounded to the nearest whole number is 37
12.34690 rounded to the nearest tenth is 12.3
89.46917 rounded to the nearest tenth is 89.5
50.02139 rounded to the nearest hundredth is 50.02
72.63539 rounded to the nearest hundredth is 72.64
46.83531 rounded to the nearest thousandth is 46.835
9.63967 rounded to the nearest thousandth is 9.640
Rules for rounding decimals.
1. Retain the correct number of decimal places (e.g. 3 for thousandths, 0 for whole numbers)
2. If the next decimal place value is 5 or more, increase the value in the last retained decimal place by 1.
Estimate the difference between two decimals
How to Estimate a difference by rounding.
* Round each decimal term that will be subtracted.
* Subtract the rounded terms
Example: Estimate the difference of 0.988 - 0.53
* Round 0.988 - 0.53 to 1 - 0.5
* Subtract the rounded numbers to obtain the estimated difference of 0.5
* The actual difference of 0.988 - 0.53 is 0.458
Some uses of rounding are:
* Checking to see if you have enough money to buy what you want.
* Getting a rough idea of the correct answer to a problem
Estimate the sum of two decimals
How to Estimate a sum by rounding.
* Round each decimal term that will be added.
* Add the rounded terms
Example: Estimate the sum of 0.988 + 0.53
* Round 0.988 + 0.53 to 1 + 0.5
* Add the rounded numbers to obtain the estimated sum of 1.5
* The actual sum of 0.988 + 0.53 is 1.518
Some uses of rounding are:
* Checking to see if you have enough money to buy what you want.
* Getting a rough idea of the correct answer to a problem
Reciprocals
The product of a number and its reciprocal equals 1.
The reciprocal of 4 is 1/4.
The reciprocal of 2/3 is 3/2.
The reciprocal of 1 is 1.
The number 0 does not have a reciprocal because the product of any number and 0 equals 0.
Multiplying Mixed Numbers
Mixed numbers consist of an integer followed by a fraction.
* Multiplying two mixed numbers: Convert each mixed number to an improper fraction.
* Multiply the two numerators together.
* Multiply the two denominators together.
* Convert the result back to a mixed number if it is an improper fraction.
* Simplify the mixed number.
Example: 6 2/8 * 3 5/9 =
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