Algorithm No. 1
To solve simple problems without parentheses, and calculate roots and fractions, remember the following rules:
- We perform all actions from left to right.
- First multiply and divide, then add and subtract.
How to use these rules?
Example No. 1. Calculate: 15 – 3 + 7.
First, perform all actions from left to right:
- 15 – 3 = 12
- 12 + 7 = 19
Answer: 15 – 3 + 7 = 19.
Example No. 2. Calculate: 10 ÷ 2 x 8.
Here we also perform operations from left to right:
- 10 ÷ 2 = 5
- 5 x 8 = 40
Answer: 10 ÷ 2 x 8 = 40.
Example No. 3. Calculate: 5 x 4 – 8 ÷ 2.
Also perform calculations from left to right, but remember that multiplication and division occur first. So:
- 5 x 4 = 20. This is the multiplication that occurs first when viewed from left to right.
- 8 ÷ 2 = 4. This is division, and although it follows subtraction, it has priority.
- 20 – 4 = 16. Now the usual order: after multiplication and division, we move on to subtraction.
Answer: 5 x 4 – 8 ÷ 2 = 16.
The bright sideIf you have multiple actions to perform, you can number them so you remember the correct order.
Important: There is no need for parentheses if addition and subtraction are done from left to right. For example, instead of (4 – 2) + 3, you could simply write 4 – 2 + 3. There's also no need for parentheses when prioritizing actions anyway. For example, instead of 5 + (4 x 3), you could just write 5 + 4 x 3 because multiplication takes precedence anyway.
The bright sideAlgorithm No. 2
A mathematical expression can contain parentheses that change the usual order in which operations are performed. To make correct calculations, remember the following rules:
- All operations in parentheses are performed first.
- Then perform the remaining actions from left to right.
- Multiplication and division are always done first, followed by subtraction and addition.
- The same rules apply to brackets.
- If fractures or fractures are present, they should be evaluated (if possible) before all other operations.
Example No. 1. Calculate: 5 x (8 – 4) ÷ 2.
Do the operations in parentheses first, then follow the usual order:
- 8 – 4 = 4. Knowing the result in parentheses, we can write 5 x (8 – 4) ÷ 2 = 5 x 4 ÷ 2. Then follow the order:
- 5 x 4 = 20
- 20 ÷ 2 = 10
So 5 x (8 – 4) ÷ 2 = 10.
Answer: 5 x (8 – 4) ÷ 2 = 10.
Example No. 2. Calculate and compare the results: 7 – 3 + 2 and 7 – (3 + 2).
Calculate the result of the first expression: 7 – 3 + 2 = 6. Now calculate the result of the second expression: 7 – (3 + 2) = 7 – 5 = 2. The parentheses in the second example have changed their order of operations, so the results are different.
The bright sideExample No. 3
Example No. 3. Calculate 8 – 2 x (15 – 4 x 3) + (7 + 3 x 2).
At first glance, this expression seems complicated. To simplify the process, break it down into separate activities:
- Do the operations in parentheses first. To get the right result on the first slide, it's important to remember which activities are a priority. So first calculate 4 x 3 and then subtract the result from 15. The answer is 3. Do the same with the second bracket – calculate 3 x 2 and add 7. The result is 13.
- Now write 8 – 2 15.
Important: You may sometimes encounter expressions in which parentheses are inside other parentheses. The principle is the same: first calculate everything between the inner parentheses, then the outer ones, and then do the rest of the calculations. Different parentheses are used – in most cases this is (), but you can also use { } and [ ].
The bright sideThe multiplication symbol before the parentheses has been removed, which may cause errors
- The multiplication symbol before the parentheses has been removed, which may cause errors.
For example, we have the following expression: 8 + 4(3 – 1). When solving this task, it is easy to get confused in the order. The correct order is as follows: First calculate the result in parentheses, then multiply it by 4 and add the result to 8. Thus we get: 8 + 4(3 – 1) = 8 + 4 x (3 – 1) = 8 + 4 x 2 = 8 + 8 = 16.
In this case, things get more complicated: 8 ÷ 4(3 – 1). But the order is the same. First the parentheses, then division and multiplication from left to right: 8 ÷ 4 x (3 – 1) = 8 ÷ 4 x 2 = 2 x 2 = 4.
The bright sideIncorrect removal of parentheses preceded by a minus sign
- Incorrect removal of parentheses preceded by a minus sign
It happens that in some expressions you need to remove the parentheses to simplify them. In this case, if there is a minus sign before the arc, all pluses and minuses should be replaced with their opposites. For example: 6 + 5 – (4 + 3 – 2) becomes 6 + 5 – 4 – 3 + 2. People often make mistakes when dealing with complex variables and equations such as 3 + 2(x + 1) — 2(x – 1). Since we don't know the variable, we can't calculate the result inside the parentheses, so we have to remove the parentheses and convert the expression to 3 + 2x + 2 — 2x + 2 = 7. If you make a mistake, it will look like this: 3 + 2x + 2 — 2x — 2 = 3.
- Calculations performed on a calculator
Not all calculators can perform operations in the correct order, although some models distinguish simple operations from complex operations. How to check your calculator? Try calculating 1 + 5 x 7. If your result is 36, the calculator will be able to solve complex expressions.
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