Students are introduced to fractions in the 5th grade. Previously, people who knew how to perform operations with fractions were considered very smart. The first fraction was 1/2, that is, half, then 1/3 appeared, etc. For several centuries the examples were considered too complex. Now detailed rules have been developed for converting fractions, addition, multiplication and other operations. It is enough to understand the material a little, and the solution will be easy.
An ordinary fraction, called a simple fraction, is written as the division of two numbers: m and n.
M is the dividend, that is, the numerator of the fraction, and the divisor n is called the denominator.
Identify proper fractions (m< n) а также неправильные (m >n).
A proper fraction is less than one (for example, 5/6 - this means that 5 parts are taken from one; 2/8 - 2 parts are taken from one). An improper fraction is equal to or greater than 1 (8/7 - the unit is 7/7 and one more part is taken as a plus).
So, one is when the numerator and denominator coincide (3/3, 12/12, 100/100 and others).
Operations with ordinary fractions, grade 6
You can do the following with simple fractions:
- Expand a fraction. If you multiply the upper and lower parts of the fraction by any identical number (just not by zero), then the value of the fraction will not change (3/5 = 6/10 (simply multiplied by 2).
- Reducing fractions is similar to expanding, but here they divide by a number.
- Compare. If two fractions have the same numerators, then the fraction with the smaller denominator will be larger. If same denominators, then the fraction with the largest numerator will be larger.
- Perform addition and subtraction. With the same denominators, this is easy to do (we sum up the upper parts, but the lower part does not change). If they are different, you will have to find a common denominator and additional factors.
- Multiply and divide fractions.
Let's look at examples of operations with fractions below.
Reduced fractions grade 6
To reduce is to divide the top and bottom of a fraction by some equal number.
The figure shows simple examples of reduction. In the first option, you can immediately guess that the numerator and denominator are divisible by 2.
On a note! If the number is even, then it is divisible by 2 anyway. Even numbers- this is 2, 4, 6...32 8 (ends with an even number), etc.
In the second case, when dividing 6 by 18, it is immediately clear that the numbers are divisible by 2. Dividing, we get 3/9. This fraction is further divided by 3. Then the answer is 1/3. If you multiply both divisors: 2 by 3, you get 6. It turns out that the fraction was divided by six. This gradual division is called successive reduction of fractions by common divisors.
Some people will immediately divide by 6, others will need to divide by parts. The main thing is that at the end there is a fraction left that cannot be reduced in any way.
Note that if a number consists of digits, the addition of which results in a number divisible by 3, then the original one can also be reduced by 3. Example: number 341. Add the numbers: 3 + 4 + 1 = 8 (8 is not divisible by 3, This means that the number 341 cannot be reduced by 3 without a remainder). Another example: 264. Add: 2 + 6 + 4 = 12 (divisible by 3). We get: 264: 3 = 88. This will make it easier to reduce large numbers.
In addition to the method of sequentially reducing fractions by common divisors, there are other methods.
GCD is the most big divisor for number. Having found the gcd for the denominator and numerator, you can immediately reduce the fraction to the desired number. The search is carried out by gradually dividing each number. Next, they look at which divisors coincide; if there are several of them (as in the picture below), then you need to multiply.
Mixed Fractions Grade 6
All improper fractions can be converted into mixed fractions by separating the whole part from them. The whole number is written on the left.
Often you have to make a mixed number from an improper fraction. The conversion process is shown in the example below: 22/4 = 22 divided by 4, we get 5 integers (5 * 4 = 20). 22 - 20 = 2. We get 5 integers and 2/4 (the denominator does not change). Since the fraction can be reduced, we divide the upper and lower parts by 2.
It is easy to turn a mixed number into an improper fraction (this is necessary when dividing and multiplying fractions). To do this: multiply the integer by the lower part of the fraction and add the numerator to it. Ready. The denominator does not change.
Calculations with fractions 6th grade
Mixed numbers can be added. If the denominators are the same, then this is easy to do: add the integer parts and numerators, the denominator remains in place.
When adding numbers with different denominators, the process is more complicated. First, we reduce the numbers to one smallest denominator (LSD).
In the example below, for the numbers 9 and 6, the denominator will be 18. After this, additional factors are needed. To find them, you should divide 18 by 9, this is how you find the additional number - 2. We multiply it by the numerator 4 to get the fraction 8/18). They do the same with the second fraction. We already add the converted fractions (integers and numerators separately, we do not change the denominator). In the example, the answer had to be converted to a proper fraction (initially the numerator turned out to be greater than the denominator).
Please note that when fractions differ, the algorithm of actions is the same.
When multiplying fractions, it is important to place both under the same line. If the number is mixed, then we turn it into simple fraction. Next, multiply the upper and lower parts and write down the answer. If it is clear that fractions can be reduced, then we reduce them immediately.
In the above example, you didn’t have to cut anything, you just wrote down the answer and highlighted the whole part.
In this example, we had to reduce the numbers under one line. Although you can shorten the ready-made answer.
When dividing, the algorithm is almost the same. First, we turn the mixed fraction into an improper fraction, then we write the numbers under one line, replacing division with multiplication. Don’t forget to swap the top and bottom parts of the second fraction (this is the rule for dividing fractions).
If necessary, we reduce the numbers (in the example below we reduced them by five and two). We convert the improper fraction by highlighting the whole part.
Basic fraction problems 6th grade
The video shows a few more tasks. Used for clarity graphic images solutions that will help you visualize fractions.
Examples of multiplying fractions grade 6 with explanations
Multiplying fractions are written under one line. They are then reduced by dividing by the same numbers (for example, 15 in the denominator and 5 in the numerator can be divided by five).
Comparing fractions grade 6
To compare fractions, you need to remember two simple rules.
Rule 1. If the denominators are different
Rule 2. When the denominators are the same
For example, compare the fractions 7/12 and 2/3.
- We look at the denominators, they do not match. So you need to find a common one.
- For fractions, the common denominator is 12.
- We first divide 12 by the lower part of the first fraction: 12: 12 = 1 (this is an additional factor for the 1st fraction).
- Now we divide 12 by 3, we get 4 - extra. factor of the 2nd fraction.
- We multiply the resulting numbers by the numerators to convert fractions: 1 x 7 = 7 (first fraction: 7/12); 4 x 2 = 8 (second fraction: 8/12).
- Now we can compare: 7/12 and 8/12. It turned out: 7/12< 8/12.
To better represent fractions, you can use pictures for clarity where an object is divided into parts (for example, a cake). If you want to compare 4/7 and 2/3, then in the first case the cake is divided into 7 parts and 4 of them are selected. In the second, they divide into 3 parts and take 2. With the naked eye it will be clear that 2/3 will be greater than 4/7.
Examples with fractions grade 6 for training
You can complete the following tasks as practice.
- Compare fractions
- perform multiplication
Tip: if it is difficult to find the lowest common denominator for fractions (especially if their values are small), then you can multiply the denominator of the first and second fractions. Example: 2/8 and 5/9. Finding their denominator is simple: multiply 8 by 9, you get 72.
Solving equations with fractions 6th grade
Solving equations requires remembering operations with fractions: multiplication, division, subtraction and addition. If one of the factors is unknown, then the product (total) is divided by the known factor, that is, the fractions are multiplied (the second is turned over).
If the dividend is unknown, then the denominator is multiplied by the divisor, and to find the divisor you need to divide the dividend by the quotient.
Let's present simple examples of solving equations:
Here you only need to produce the difference of fractions, without leading to a common denominator.
The answer came out as an improper fraction. It can be converted to 1 whole and 3/5.
In the second method, the numerator and denominator were multiplied by 4 to cancel out the bottom portion rather than flipping the denominator.
Now that we have learned how to add and multiply individual fractions, we can look at more complex designs. For example, what if the same problem involves adding, subtracting, and multiplying fractions?
First of all, you need to convert all fractions to improper ones. Then we sequentially perform the required actions - in the same order as for ordinary numbers. Namely:
- Exponentiation is done first - get rid of all expressions containing exponents;
- Then - division and multiplication;
- The last step is addition and subtraction.
Of course, if there are parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.
Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:
Now let's find the value of the second expression. Here fractions with whole part no, but there are parentheses, so we do the addition first, and only then the division. Note that 14 = 7 · 2. Then:
Finally, let's consider the third example. There are brackets and a degree here - it is better to count them separately. Considering that 9 = 3 3, we have:
Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately, the denominator.
You can decide differently. If we recall the definition of a degree, the problem will be reduced to the usual multiplication of fractions:
Multistory fractions
Until now, we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.
But what if you put a more complex object in the numerator or denominator? For example, another numerical fraction? Such constructions arise quite often, especially when working with long expressions. Here are a couple of examples:
There is only one rule for working with multi-level fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash means the standard division operation. Therefore, any fraction can be rewritten as follows:
Using this fact and following the procedure, we can easily reduce any multi-story fraction to an ordinary one. Take a look at the examples:
Task. Convert multistory fractions to ordinary ones:
In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is 12 = 12/1; 3 = 3/1. We get:
IN last example the fractions were canceled before the final multiplication.
Specifics of working with multi-level fractions
There is one subtlety in multi-level fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:
- The numerator contains the single number 7, and the denominator contains the fraction 12/5;
- The numerator contains the fraction 7/12, and the denominator contains the separate number 5.
So, for one recording we got two completely different interpretations. If you count, the answers will also be different:
To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the line of the nested fraction. Preferably several times.
If you follow this rule, then the above fractions should be written as follows:
Yes, it's probably unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:
Task. Find the meanings of the expressions:
So, let's work with the first example. Let's convert all fractions to improper ones, and then perform addition and division operations:
Let's do the same with the second example. Let's convert all fractions to improper ones and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:
Due to the fact that the numerator and denominator of the basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.
I will also note that in both examples the fraction bar actually replaces the parentheses: first of all, we found the sum, and only then the quotient.
Some will say that the transition to improper fractions in the second example was clearly redundant. Perhaps this is true. But by doing this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.
The next action that can be performed with ordinary fractions is subtraction. As part of this material, we will look at how to correctly calculate the difference between fractions with like and unlike denominators, how to subtract a fraction from natural number and vice versa. All examples will be illustrated with problems. Let us clarify in advance that we will only examine cases where the difference of fractions results in a positive number.
How to find the difference between fractions with like denominators
Let's start right away with a clear example: let's say we have an apple that has been divided into eight parts. Let's leave five parts on the plate and take two of them. This action can be written like this:
As a result, we have 3 eighths left, since 5 − 2 = 3. It turns out that 5 8 - 2 8 = 3 8.
Thereby simple example We saw exactly how the subtraction rule works for fractions whose denominators are the same. Let's formulate it.
Definition 1
To find the difference between fractions with the same denominators, you need to subtract the numerator of the other from the numerator of one, and leave the denominator the same. This rule can be written as a b - c b = a - c b.
We will use this formula in the future.
Let's take specific examples.
Example 1
Subtract the common fraction 17 15 from the fraction 24 15.
Solution
We see that these fractions have the same denominators. So all we need to do is subtract 17 from 24. We get 7 and add the denominator to it, we get 7 15.
Our calculations can be written as follows: 24 15 - 17 15 = 24 - 17 15 = 7 15
If necessary, you can reduce complex fraction or select an entire part from an incorrect one to make it easier to count.
Example 2
Find the difference 37 12 - 15 12.
Solution
Let's use the formula described above and calculate: 37 12 - 15 12 = 37 - 15 12 = 22 12
It is easy to notice that the numerator and denominator can be divided by 2 (we already talked about this earlier when we looked at the signs of divisibility). Shortening the answer, we get 11 6. This is an improper fraction, from which we will select the whole part: 11 6 = 1 5 6.
How to find the difference of fractions with different denominators
This mathematical operation can be reduced to what we have already described above. To do this, we simply reduce the necessary fractions to the same denominator. Let's formulate a definition:
Definition 2
To find the difference of fractions for which different denominators, it is necessary to bring them to the same denominator and find the difference between the numerators.
Let's look at an example of how this is done.
Example 3
Subtract the fraction 1 15 from 2 9.
Solution
The denominators are different, and you need to reduce them to the smallest overall value. In this case, the LCM is 45. The first fraction requires an additional factor of 5, and the second - 3.
Let's calculate: 2 9 = 2 5 9 5 = 10 45 1 15 = 1 3 15 3 = 3 45
We have two fractions with the same denominator, and now we can easily find their difference using the algorithm described earlier: 10 45 - 3 45 = 10 - 3 45 = 7 45
A short summary of the solution looks like this: 2 9 - 1 15 = 10 45 - 3 45 = 10 - 3 45 = 7 45.
Do not neglect reducing the result or separating an entire part from it, if necessary. IN in this example we don't need to do that.
Example 4
Find the difference 19 9 - 7 36.
Solution
Let's reduce the fractions indicated in the condition to the lowest common denominator 36 and get 76 9 and 7 36, respectively.
We calculate the answer: 76 36 - 7 36 = 76 - 7 36 = 69 36
The result can be reduced by 3 and get 23 12. The numerator is greater than the denominator, which means we can select the whole part. The final answer is 1 11 12.
A short summary of the entire solution is 19 9 - 7 36 = 1 11 12.
How to subtract a natural number from a common fraction
This action can also be easily reduced to a simple subtraction ordinary fractions. This can be done by representing a natural number as a fraction. Let's show it with an example.
Example 5
Find the difference 83 21 – 3 .
Solution
3 is the same as 3 1. Then you can calculate it like this: 83 21 - 3 = 20 21.
If the condition requires subtracting an integer from an improper fraction, it is more convenient to first separate the integer from it by writing it as a mixed number. Then the previous example can be solved differently.
From the fraction 83 21, when separating the whole part, you get 83 21 = 3 20 21.
Now let's just subtract 3 from it: 3 20 21 - 3 = 20 21.
How to subtract a fraction from a natural number
This action is done in a similar way to the previous one: we rewrite the natural number as a fraction, bring both to a single denominator and find the difference. Let's illustrate this with an example.
Example 6
Find the difference: 7 - 5 3 .
Solution
Let's make 7 a fraction 7 1. We do the subtraction and transform the final result, separating the whole part from it: 7 - 5 3 = 5 1 3.
There is another way to make calculations. It has some advantages that can be used in cases where the numerators and denominators of the fractions in the problem are large numbers.
Definition 3
If the fraction that needs to be subtracted is proper, then the natural number from which we are subtracting must be represented as the sum of two numbers, one of which is equal to 1. After this you need to subtract the required fraction from one and get the answer.
Example 7
Calculate the difference 1 065 - 13 62.
Solution
The fraction to be subtracted is a proper fraction because its numerator is less than its denominator. Therefore, we need to subtract one from 1065 and subtract the desired fraction from it: 1065 - 13 62 = (1064 + 1) - 13 62
Now we need to find the answer. Using the properties of subtraction, the resulting expression can be written as 1064 + 1 - 13 62. Let's calculate the difference in brackets. To do this, let's imagine unit as a fraction 1 1.
It turns out that 1 - 13 62 = 1 1 - 13 62 = 62 62 - 13 62 = 49 62.
Now let's remember about 1064 and formulate the answer: 1064 49 62.
We use the old method to prove that it is less convenient. These are the calculations we would come up with:
1065 - 13 62 = 1065 1 - 13 62 = 1065 62 1 62 - 13 62 = 66030 62 - 13 62 = = 66030 - 13 62 = 66017 62 = 1064 4 6
The answer is the same, but the calculations are obviously more cumbersome.
We looked at the case where we need to subtract a proper fraction. If it is incorrect, we replace it with a mixed number and subtract according to familiar rules.
Example 8
Calculate the difference 644 - 73 5.
Solution
The second fraction is an improper fraction, and the whole part must be separated from it.
Now we calculate similarly to the previous example: 630 - 3 5 = (629 + 1) - 3 5 = 629 + 1 - 3 5 = 629 + 2 5 = 629 2 5
Properties of subtraction when working with fractions
The properties that subtraction of natural numbers have also apply to cases of subtraction of ordinary fractions. Let's look at how to use them when solving examples.
Example 9
Find the difference 24 4 - 3 2 - 5 6.
Solution
We have already solved similar examples when we looked at subtracting a sum from a number, so we follow the already known algorithm. First, let's calculate the difference 25 4 - 3 2, and then subtract the last fraction from it:
25 4 - 3 2 = 24 4 - 6 4 = 19 4 19 4 - 5 6 = 57 12 - 10 12 = 47 12
Let's transform the answer by separating the whole part from it. Result - 3 11 12.
A short summary of the entire solution:
25 4 - 3 2 - 5 6 = 25 4 - 3 2 - 5 6 = 25 4 - 6 4 - 5 6 = = 19 4 - 5 6 = 57 12 - 10 12 = 47 12 = 3 11 12
If the expression contains both fractions and natural numbers, it is recommended to group them by type when calculating.
Example 10
Find the difference 98 + 17 20 - 5 + 3 5.
Solution
Knowing the basic properties of subtraction and addition, we can group numbers as follows: 98 + 17 20 - 5 + 3 5 = 98 + 17 20 - 5 - 3 5 = 98 - 5 + 17 20 - 3 5
Let's complete the calculations: 98 - 5 + 17 20 - 3 5 = 93 + 17 20 - 12 20 = 93 + 5 20 = 93 + 1 4 = 93 1 4
If you notice an error in the text, please highlight it and press Ctrl+Enter
The common denominator of several fractions is the LCM (least common multiple) of the natural numbers that are the denominators of the given fractions.
You need to add additional factors to the numerators of the given fractions, equal to the ratio LOC and the corresponding denominator.
The numerators of given fractions are multiplied by their additional factors, resulting in numerators of fractions with a single common denominator. Action signs (“+” or “-”) in the recording of fractions reduced to a common denominator are stored before each fraction. For fractions with a common denominator, the action signs are preserved before each reduced numerator.
Only now can you add or subtract the numerators and sign the common denominator under the result.
Attention! If in the resulting fraction the numerator and denominator have common factors, then the fraction must be reduced. It is advisable to convert an improper fraction into a mixed fraction. Leaving the result of an addition or subtraction without canceling the fraction where possible is an incomplete solution to the example!
Adding and subtracting fractions with different denominators. Rule. To add or subtract fractions with different denominators, you must first reduce them to the lowest common denominator, and then perform addition or subtraction as with fractions with the same denominators.
Procedure for adding and subtracting fractions with different denominators
- find the LCM of all denominators;
- add additional factors to each fraction;
- multiply each numerator by an additional factor;
- take the resulting products as numerators, signing the common denominator under each fraction;
- add or subtract the numerators of fractions by signing the common denominator under the sum or difference.
Fractions can also be added and subtracted if there are letters in the numerator.
You can perform various operations with fractions, for example, adding fractions. Addition of fractions can be divided into several types. Each type of addition of fractions has its own rules and algorithm of actions. Let's look at each type of addition in detail.
Adding fractions with like denominators.
Let's look at an example of how to add fractions with a common denominator.
The tourists went on a hike from point A to point E. On the first day, they walked from point A to B or \(\frac(1)(5)\) the entire way. On the second day they walked from point B to D or \(\frac(2)(5)\) the whole way. How far did they travel from the beginning of the journey to point D?
To find the distance from point A to point D, you need to add the fractions \(\frac(1)(5) + \frac(2)(5)\).
Adding fractions with like denominators means that you need to add the numerators of these fractions, but the denominator will remain the same.
\(\frac(1)(5) + \frac(2)(5) = \frac(1 + 2)(5) = \frac(3)(5)\)
In literal form, the sum of fractions with the same denominators will look like this:
\(\bf \frac(a)(c) + \frac(b)(c) = \frac(a + b)(c)\)
Answer: the tourists walked \(\frac(3)(5)\) the entire way.
Adding fractions with different denominators.
Let's look at an example:
You need to add two fractions \(\frac(3)(4)\) and \(\frac(2)(7)\).
To add fractions with different denominators, you must first find, and then use the rule for adding fractions with like denominators.
For denominators 4 and 7, the common denominator will be the number 28. The first fraction \(\frac(3)(4)\) must be multiplied by 7. The second fraction \(\frac(2)(7)\) must be multiplied by 4.
\(\frac(3)(4) + \frac(2)(7) = \frac(3 \times \color(red) (7) + 2 \times \color(red) (4))(4 \ times \color(red) (7)) = \frac(21 + 8)(28) = \frac(29)(28) = 1\frac(1)(28)\)
In literal form we get the following formula:
\(\bf \frac(a)(b) + \frac(c)(d) = \frac(a \times d + c \times b)(b \times d)\)
Adding mixed numbers or mixed fractions.
Addition occurs according to the law of addition.
For mixed fractions, we add the whole parts with the whole parts and the fractional parts with the fractions.
If fractional parts mixed numbers have the same denominators, then we add the numerators, but the denominator remains the same.
Let's add the mixed numbers \(3\frac(6)(11)\) and \(1\frac(3)(11)\).
\(3\frac(6)(11) + 1\frac(3)(11) = (\color(red) (3) + \color(blue) (\frac(6)(11))) + ( \color(red) (1) + \color(blue) (\frac(3)(11))) = (\color(red) (3) + \color(red) (1)) + (\color( blue) (\frac(6)(11)) + \color(blue) (\frac(3)(11))) = \color(red)(4) + (\color(blue) (\frac(6 + 3)(11))) = \color(red)(4) + \color(blue) (\frac(9)(11)) = \color(red)(4) \color(blue) (\frac (9)(11))\)
If the fractional parts of mixed numbers have different denominators, then we find the common denominator.
Let's perform the addition of mixed numbers \(7\frac(1)(8)\) and \(2\frac(1)(6)\).
The denominator is different, so we need to find the common denominator, it is equal to 24. Multiply the first fraction \(7\frac(1)(8)\) by an additional factor of 3, and the second fraction \(2\frac(1)(6)\) by 4.
\(7\frac(1)(8) + 2\frac(1)(6) = 7\frac(1 \times \color(red) (3))(8 \times \color(red) (3) ) = 2\frac(1\times \color(red) (4))(6\times \color(red) (4)) =7\frac(3)(24) + 2\frac(4)(24 ) = 9\frac(7)(24)\)
Related questions:
How to add fractions?
Answer: first you need to decide what type of expression it is: fractions have the same denominators, different denominators or mixed fractions. Depending on the type of expression, we proceed to the solution algorithm.
How to solve fractions with different denominators?
Answer: you need to find the common denominator, and then follow the rule of adding fractions with the same denominators.
How to solve mixed fractions?
Answer: we add integer parts with integers and fractional parts with fractions.
Example #1:
Can the sum of two result in a proper fraction? Improper fraction? Give examples.
\(\frac(2)(7) + \frac(3)(7) = \frac(2 + 3)(7) = \frac(5)(7)\)
The fraction \(\frac(5)(7)\) is a proper fraction, it is the result of the sum of two proper fractions \(\frac(2)(7)\) and \(\frac(3)(7)\).
\(\frac(2)(5) + \frac(8)(9) = \frac(2 \times 9 + 8 \times 5)(5 \times 9) =\frac(18 + 40)(45) = \frac(58)(45)\)
The fraction \(\frac(58)(45)\) is an improper fraction, it is the result of the sum of the proper fractions \(\frac(2)(5)\) and \(\frac(8)(9)\).
Answer: The answer to both questions is yes.
Example #2:
Add the fractions: a) \(\frac(3)(11) + \frac(5)(11)\) b) \(\frac(1)(3) + \frac(2)(9)\).
a) \(\frac(3)(11) + \frac(5)(11) = \frac(3 + 5)(11) = \frac(8)(11)\)
b) \(\frac(1)(3) + \frac(2)(9) = \frac(1 \times \color(red) (3))(3 \times \color(red) (3)) + \frac(2)(9) = \frac(3)(9) + \frac(2)(9) = \frac(5)(9)\)
Example #3:
Write the mixed fraction as the sum of a natural number and a proper fraction: a) \(1\frac(9)(47)\) b) \(5\frac(1)(3)\)
a) \(1\frac(9)(47) = 1 + \frac(9)(47)\)
b) \(5\frac(1)(3) = 5 + \frac(1)(3)\)
Example #4:
Calculate the sum: a) \(8\frac(5)(7) + 2\frac(1)(7)\) b) \(2\frac(9)(13) + \frac(2)(13) \) c) \(7\frac(2)(5) + 3\frac(4)(15)\)
a) \(8\frac(5)(7) + 2\frac(1)(7) = (8 + 2) + (\frac(5)(7) + \frac(1)(7)) = 10 + \frac(6)(7) = 10\frac(6)(7)\)
b) \(2\frac(9)(13) + \frac(2)(13) = 2 + (\frac(9)(13) + \frac(2)(13)) = 2\frac(11 )(13) \)
c) \(7\frac(2)(5) + 3\frac(4)(15) = 7\frac(2\times 3)(5\times 3) + 3\frac(4)(15) = 7\frac(6)(15) + 3\frac(4)(15) = (7 + 3)+(\frac(6)(15) + \frac(4)(15)) = 10 + \frac (10)(15) = 10\frac(10)(15) = 10\frac(2)(3)\)
Task #1:
At lunch we ate \(\frac(8)(11)\) from the cake, and in the evening at dinner we ate \(\frac(3)(11)\). Do you think the cake was completely eaten or not?
Solution:
The denominator of the fraction is 11, it indicates how many parts the cake was divided into. At lunch we ate 8 pieces of cake out of 11. At dinner we ate 3 pieces of cake out of 11. Let’s add 8 + 3 = 11, we ate pieces of cake out of 11, that is, the whole cake.
\(\frac(8)(11) + \frac(3)(11) = \frac(11)(11) = 1\)
Answer: the whole cake was eaten.
