Thanks to the knowledge of the distributive property of multiplication and addition, it is possible to verbally solve complex, at first glance, examples. This rule is studied in algebra lessons in grade 7. Tasks using this rule are found at the exam and exam in mathematics.
Distribution property of multiplication
In order to multiply the sum of some numbers, you can multiply each term individually and add the results.
Simply put, a × (b + c) = av + ac or (b + c) × a = av + ac.
Also, to simplify the solution, this rule also works in the reverse order: a × b + a × c = a × (b + s), that is, the common factor is taken out of the brackets.
Using the distribution property of addition, the following examples can be solved.
- Example 1: 3 × (10 + 11). Multiply the number 3 by each term: 3 × 10 + 3 × 11. Add: 30 + 33 = 63 and write down the result. Answer: 63.
- Example 2: 28 × 7. Imagine the number 28 as the sum of two numbers 20 and 8 and multiply by 7, like this: (20 + 8) × 7. Perform the calculations: 20 × 7 + 8 × 7 = 140 + 56 = 196. Answer: 196.
- Example 3. Solve the following problem: 9 × (20 - 1). Multiply by the number 9 and the decremented 20, and the subtracted 1: 9 × 20 - 9 × 1. Calculate the results: 180 - 9 = 171. Answer: 171.
The same rule applies not only to the sum, but also to the difference of two or more expressions.
Distribution property of multiplication with respect to difference
In order to multiply the difference by a number, you should multiply the decrement by it, and then subtract it and perform the calculation of the results.
a × (b - s) = a × b - a × s or (b - s) × a = a × b - a × s.
Example 1: 14 × (10 - 2). Using the distribution law, multiply 14 by both numbers: 14 × 10 -14 × 2. Find the difference in the values obtained: 140 - 28 = 112 and write down the result. The answer is 112.
Example 2: 8 × (1 + 20). This task is solved similarly: 8 × 1 + 8 × 20 = 8 + 160 = 168. Answer: 168.
Example 3: 27 × 3. Find the value of the expression using the studied property. Imagine 27 as the difference of two numbers 30 and 3, like this: 27 × 3 = (30 - 3) × 3 = 30 × 3 - 3 × 3 = 90 - 9 = 81. Answer: 81.
Applying a property for more than two terms
The distribution property of multiplication is used not only for two terms, but for absolutely any quantity, in which case the formula has this form:
a × (b + s + d) = a × b + a × s + a × d.
a × (b - s - d) = a × b - a × s - a × d.
Example 1: 354 × 3. Imagine 354 as the sum of three numbers: 300, 50 and 3: (300 + 50 + 3) × 3 = 300 × 3 + 50 × 3 + 3 × 3 = 900 + 150 + 9 = 1059. Answer: 1059.
Simplify a few expressions using the property mentioned earlier.
Example 2: 5 × (3x + 14y). Open the brackets using the distribution law of multiplication: 5 × 3x + 5 × 14y = 15x + 70y. You cannot add 15 x and 70 y, since the terms are not similar and have a different letter part. Answer: 15x + 70u.
Example 3: 12 × (4s - 5d). Given the rule, multiply by 12 and 4s and 5d: 12 × 4s - 12 × 5d = 48s - 60d. Answer: 48s - 60d.
Using when solving examples, the distribution property of addition and multiplication:
- complex examples are easily solved, their solution can be reduced to an oral account;
- time is noticeably saved when solving complex, at first glance, tasks;
- thanks to the knowledge gained, one can easily simplify expressions.