Overview:
The Distributive Property of Multiplication over Addition is a good way to simplify calculations. It combines two different operations in a shortcut to make solving variables easier.
The Distributive Property of Multiplication over Addition
The general rule for this property is for any numbers a, b, and x , ax + bx = (a +b)x. Imagine one rectangle with area ax and add it to another rectangle with area bx. The length of one side would equal x and the length of the other side would equal a + b. To make the problem less abstract, numbers could be substituted. The constant a could equal 4, the constant b could equal 3, and the variable x could equal 2. Does 4(2) + 3(2) equal (4 +3)2? It does, because 8 + 6 =14.
The Distributive Property of Multiplication over Subtraction
The general rule for this property is for any numbers a, b, and x, ax – bx = (a-b)x. This can also be illustrated with numbers substituting for the letters, as in the example above. Does 4(2) – 3(2) equal (4-3)2? It does, because 8-6 =2.
Solving Repeating Decimals Using the Distributive Property
Repeating decimals can be solved using the distributive property of multiplication over subtraction. For example, let x = .3333… and multiply both sides by 10 to isolate the repeating part, so that 10 x equals 3.333. Now 10x-x equals 3.333… minus 333…. Using the Distributive Property, (10-1)x = 3.333 – .333, so 9x =3, or x = 3/9, or 1/3 in simplest terms.
Use the Distributive Property in Mental Math
The distributive property is a useful tool to solve problems mentally. For example, suppose one ream of paper costs 5.50 and 12 reams are needed. Now 12 times 5 is $60.00, and half of 12 (the 50 cents) is 6, so add the figures together to get the cost of $66.00. In a number sentence, that problem will read 12(5.50) = 12(5) + 12(.50)= 66.
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