What is 6 times 5 plus 2
Dot before line and calculation in brackets
In this section we deal with two very important rules of mathematics: the dot-before-line calculation and the use of brackets. What this is all about and how it works should be clear to you after reading the article. There are also exercises / written exams including solutions.
When working on tasks, it is important in which order you do the calculations. To determine this, mathematicians have defined two rules: the point before line calculation and the use of brackets.
These two rules determine the order in which calculation steps are carried out. In the following we want to take a closer look at these two rules. Of course, there are also explanations, examples and exercises with solutions.Show:
Point before line calculation
Let's start with the dot before dot calculation. This rule says: First calculate multiplication and division and only then add or subtract. An example will now show how to use this rule (and how to get it wrong).
- a) 5 + 2 4 = 5 + 8 = 13 (correct solution)
- b) 5 + 2 4 = 7 4 = 28 (wrong solution)
Explanation: In both cases the result of the task 5 + 2 · 4 is sought. In case a) the problem was solved correctly. For this purpose, the multiplication was calculated first. 2 · 4 = 8. Then 5 + 8 = 13 was calculated. As you can see: First the multiplication was carried out. Then the two numbers were added. In case b) was not correct Calculated: Here it was first added and then multiplied. The result will be calculated incorrectly.
Another small note: The left of the "=" must always be the same as the right of the "=". If you do the math as shown in the examples, it will happen automatically to you. I only mention this because a later chapter - the so-called equations with unknowns - will discuss this in more detail. At this point, however, you should just try to follow the examples here and apply the concept to the exercises yourself.
Go over the example from just now. A number of other examples then follow. Take a close look at each one and try to follow the calculation. And also here applies: First multiplication or division, then addition or subtraction.
- Exercise: 8 3 + 2 =?
- Solution: 8 3 + 2 = 24 + 2 = 26
- Exercise: 2 + 3 4 =?
- Solution: 2 + 3 4 = 2 + 12 = 14
- Exercise: 6: 3 + 2 =?
- Solution: 6: 3 + 2 = 2 + 2 = 4
- Exercise: 5 - 8: 4 =?
- Solution: 5 - 8: 4 = 5 - 2 = 3
- Exercise: 5 3 + 8: 4 =?
- Solution: 5 3 + 8: 4 = 15 + 2 = 17
Explanations: In the first four examples, the point-before-line rule was consistently observed. First it was multiplied or divided, then added or subtracted. In the last example, both a multiplication and a division had to be carried out. It doesn't matter what you calculate first. Only the addition has to be calculated because of the point before the line at the end. That means in good German: First calculate 5 · 3 and 8: 4 and then add the two results.
Tip: Only by practicing can you become confident with dot before dot calculation. So by all means do the exercises for this chapter (to be found at the end of this article).Show:
Calculate with brackets
We have just dealt with point before line calculation. However, there is a small catch: How do I do it when an addition is to be carried out on a task and then a multiplication? The answer is: put a bracket. The mathematical rule is then quite simply "A bracket is calculated first". In order not to get confused with all the brackets, dot before line, etc., here is a short guide:
- If present in an exercise, the brackets are calculated first
- Then multiplication and division are calculated
- Lastly, addition and subtraction
Try to memorize the guide from above and then understand the following examples:
- Exercise: (2 + 3) 5 =?
- Solution: (2 + 3) 5 = 5 5 = 25
- Exercise: 8 (3 + 4) =?
- Solution: 8 (3 + 4) = 8 7 = 56
- Exercise: 2 (1 + 2) + 5 =?
- Solution: 2 (1 + 2) + 5 = 2 3 + 5 = 6 + 5 = 11
- Exercise: 3 (8 + 2): 10 + 1 =?
- Solution: 3 (8 + 2): 10 + 1 = 3 10: 10 + 1 = 30: 10 + 1 = 3 + 1 = 4
Explanations: The last two tasks in particular look daunting. But let's start with the first two tasks: In both cases, the brackets were calculated first. The result is then multiplied by the number behind or in front of it. In the third exercise, the parentheses are calculated first, then the multiplication and then the addition.
We come to the last task: Here, too, the brackets are calculated first. The result in brackets can then either be multiplied or divided first, which is mathematically irrelevant. In my calculation, I first multiplied, then divided. And finally added up. Going through the task one more time, piece by piece, should become clearer to you.
Priority for brackets / multiple brackets
Last but not least, there is one small nasty thing that needs to be addressed: What happens if there are multiple brackets and nested brackets? First two small examples including solutions. Then the explanation follows.
- Exercise: (5 + 3) + (1 + 2) =?
- Solution: (5 + 3) + (1 + 2) = 8 + 3 = 11
- Exercise: ((3 + 4) + 2) + 1 =?
- Solution: [(3 + 4) + 2] + 1 = (7 + 2) + 1 = 10
Explanation: In the first example there are two brackets. Which of them calculates first does not matter. So calculate both individually and then add the two results. Let's come to the second example: Here two brackets are nested inside each other. The calculation rule is very simple: Always calculate the inner brackets first.
For a better overview with the brackets, they usually look a little different: The inner bracket is usually marked with "(" and ")". The outer bracket with "[" and "]". And of course, as always, the following applies: In order to really be able to use the arithmetic rules, you should work on the exercises.
Exercises / written exams with solutions
Both the dot before stroke and the bracket rule require some practice in order to be able to use them correctly. For this reason, we have put together a series of exercises and old exam questions for both topics.
First calculate this for yourself and then take a look at our sample solution.
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