Addition with like denominators. Adding and subtracting fractions with like denominators. Mixed fractions

Mixed fractions, just like simple fractions, can be subtracted. To subtract mixed numbers of fractions you need to know several subtraction rules. Let's study these rules with examples.

Subtracting mixed fractions with like denominators.

Let's consider an example with the condition that the integer and fractional parts being reduced are greater than the integer and fractional parts being subtracted, respectively. Under such conditions, subtraction occurs separately. We subtract the integer part from the whole part, and the fractional part from the fractional part.

Let's look at an example:

Subtract mixed fractions \(5\frac(3)(7)\) and \(1\frac(1)(7)\).

\(5\frac(3)(7)-1\frac(1)(7) = (5-1) + (\frac(3)(7)-\frac(1)(7)) = 4\ frac(2)(7)\)

The correctness of the subtraction is checked by addition. Let's check the subtraction:

\(4\frac(2)(7)+1\frac(1)(7) = (4 + 1) + (\frac(2)(7) + \frac(1)(7)) = 5\ frac(3)(7)\)

Let's consider an example with the condition when the fractional part of the minuend is less than the corresponding fractional part of the subtrahend. In this case, we borrow one from the whole in the minuend.

Let's look at an example:

Subtract mixed fractions \(6\frac(1)(4)\) and \(3\frac(3)(4)\).

The minuend \(6\frac(1)(4)\) has a smaller fractional part than the fractional part of the subtrahend \(3\frac(3)(4)\). That is, \(\frac(1)(4)< \frac{1}{3}\), поэтому сразу отнять мы не сможем. Займем у целой части у 6 единицу, а потом выполним вычитание. Единицу мы запишем как \(\frac{4}{4} = 1\)

\(\begin(align)&6\frac(1)(4)-3\frac(3)(4) = (6 + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (1) + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (\frac(4)(4)) + \frac(1)(4))-3\frac(3)(4) = (5 + \frac(5)(4))-3\frac(3)(4) = \\\\ &= 5\frac(5)(4)-3\frac(3)(4) = 2\frac(2)(4) = 2\frac(1)(4)\\\\ \end(align)\)

Next example:

\(7\frac(8)(19)-3 = 4\frac(8)(19)\)

Subtracting a mixed fraction from a whole number.

Example: \(3-1\frac(2)(5)\)

The minuend 3 does not have a fractional part, so we cannot immediately subtract. Let's borrow one from the whole part of 3, and then do the subtraction. We will write the unit as \(3 = 2 + 1 = 2 + \frac(5)(5) = 2\frac(5)(5)\)

\(3-1\frac(2)(5)= (2 + \color(red) (1))-1\frac(2)(5) = (2 + \color(red) (\frac(5 )(5)))-1\frac(2)(5) = 2\frac(5)(5)-1\frac(2)(5) = 1\frac(3)(5)\)

Subtracting mixed fractions with unlike denominators.

Let's consider an example with the condition that the fractional parts of the minuend and subtrahend have different denominators. You need to bring it to a common denominator, and then perform subtraction.

Subtract two mixed fractions with different denominators \(2\frac(2)(3)\) and \(1\frac(1)(4)\).

The common denominator will be the number 12.

\(2\frac(2)(3)-1\frac(1)(4) = 2\frac(2 \times \color(red) (4))(3 \times \color(red) (4) )-1\frac(1 \times \color(red) (3))(4 \times \color(red) (3)) = 2\frac(8)(12)-1\frac(3)(12 ) = 1\frac(5)(12)\)

Related questions:
How to subtract mixed fractions? How to solve mixed fractions?
Answer: you need to decide what type the expression belongs to and apply the solution algorithm based on the type of expression. From the integer part we subtract the integer, from the fractional part we subtract the fractional part.

How to subtract a fraction from a whole number? How to subtract a fraction from a whole number?
Answer: you need to take a unit from an integer and write this unit as a fraction

\(4 = 3 + 1 = 3 + \frac(7)(7) = 3\frac(7)(7)\),

and then subtract the whole from the whole, subtract the fractional part from the fractional part. Example:

\(4-2\frac(3)(7) = (3 + \color(red) (1))-2\frac(3)(7) = (3 + \color(red) (\frac(7 )(7)))-2\frac(3)(7) = 3\frac(7)(7)-2\frac(3)(7) = 1\frac(4)(7)\)

Example #1:
Subtract a proper fraction from one: a) \(1-\frac(8)(33)\) b) \(1-\frac(6)(7)\)

Solution:
a) Let's imagine one as a fraction with a denominator 33. We get \(1 = \frac(33)(33)\)

\(1-\frac(8)(33) = \frac(33)(33)-\frac(8)(33) = \frac(25)(33)\)

b) Let's imagine one as a fraction with a denominator 7. We get \(1 = \frac(7)(7)\)

\(1-\frac(6)(7) = \frac(7)(7)-\frac(6)(7) = \frac(7-6)(7) = \frac(1)(7) \)

Example #2:
Subtract a mixed fraction from a whole number: a) \(21-10\frac(4)(5)\) b) \(2-1\frac(1)(3)\)

Solution:
a) Let's borrow 21 units from the integer and write it like this \(21 = 20 + 1 = 20 + \frac(5)(5) = 20\frac(5)(5)\)

\(21-10\frac(4)(5) = (20 + 1)-10\frac(4)(5) = (20 + \frac(5)(5))-10\frac(4)( 5) = 20\frac(5)(5)-10\frac(4)(5) = 10\frac(1)(5)\\\\\)

b) Let's take one from the integer 2 and write it like this \(2 = 1 + 1 = 1 + \frac(3)(3) = 1\frac(3)(3)\)

\(2-1\frac(1)(3) = (1 + 1)-1\frac(1)(3) = (1 + \frac(3)(3))-1\frac(1)( 3) = 1\frac(3)(3)-1\frac(1)(3) = \frac(2)(3)\\\\\)

Example #3:
Subtract an integer from a mixed fraction: a) \(15\frac(6)(17)-4\) b) \(23\frac(1)(2)-12\)

a) \(15\frac(6)(17)-4 = 11\frac(6)(17)\)

b) \(23\frac(1)(2)-12 = 11\frac(1)(2)\)

Example #4:
Subtract a proper fraction from a mixed fraction: a) \(1\frac(4)(5)-\frac(4)(5)\)

\(1\frac(4)(5)-\frac(4)(5) = 1\\\\\)

Example #5:
Calculate \(5\frac(5)(16)-3\frac(3)(8)\)

\(\begin(align)&5\frac(5)(16)-3\frac(3)(8) = 5\frac(5)(16)-3\frac(3 \times \color(red) ( 2))(8 \times \color(red) (2)) = 5\frac(5)(16)-3\frac(6)(16) = (5 + \frac(5)(16))- 3\frac(6)(16) = (4 + \color(red) (1) + \frac(5)(16))-3\frac(6)(16) = \\\\ &= (4 + \color(red) (\frac(16)(16)) + \frac(5)(16))-3\frac(6)(16) = (4 + \color(red) (\frac(21 )(16)))-3\frac(3)(8) = 4\frac(21)(16)-3\frac(6)(16) = 1\frac(15)(16)\\\\ \end(align)\)

Adding and subtracting fractions with like denominators
Adding and subtracting fractions with different denominators
Concept of NOC
Reducing fractions to the same denominator
How to add a whole number and a fraction

1 Adding and subtracting fractions with like denominators

To add fractions with the same denominators, you need to add their numerators, but leave the denominator the same, for example:

To subtract fractions with the same denominators, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:

To add mixed fractions, you need to separately add their whole parts, and then add their fractional parts, and write the result as a mixed fraction,

If, when adding fractional parts, you get an improper fraction, select the whole part from it and add it to the whole part, for example:

2 Adding and subtracting fractions with different denominators

In order to add or subtract fractions with different denominators, you must first reduce them to the same denominator, and then proceed as indicated at the beginning of this article. The common denominator of several fractions is the LCM (least common multiple). For the numerator of each fraction, additional factors are found by dividing the LCM by the denominator of this fraction. We will look at an example later, after we understand what an NOC is.

3 Least common multiple (LCM)

The least common multiple of two numbers (LCM) is the smallest natural number that is divisible by both numbers without leaving a remainder. Sometimes the LCM can be found orally, but more often, especially when working with large numbers, you have to find the LCM in writing, using the following algorithm:

In order to find the LCM of several numbers, you need:

Factor these numbers into prime factors
Take the largest expansion and write these numbers as a product
Select numbers in other decompositions that do not appear in the largest decomposition (or occur fewer times in it), and add them to the product.
Multiply all the numbers in the product, this will be the LCM.

For example, let's find the LCM of the numbers 28 and 21:

4Reducing fractions to the same denominator

Let's return to adding fractions with different denominators.

When we reduce fractions to the same denominator, which is equal to the LCM of both denominators, we must multiply the numerators of these fractions by additional multipliers. You can find them by dividing the LCM by the denominator of the corresponding fraction, for example:

Thus, to reduce fractions to the same exponent, you must first find the LCM (that is, the smallest number that is divisible by both denominators) of the denominators of these fractions, then put additional factors to the numerators of the fractions. You can find them by dividing the common denominator (CLD) by the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction by an additional factor, and put the LCM as the denominator.

5How to add a whole number and a fraction

In order to add a whole number and a fraction, you just need to add this number before the fraction, which will result in a mixed fraction, for example.

Adding and subtracting fractions with like denominators
Adding and subtracting fractions with different denominators
Concept of NOC
Reducing fractions to the same denominator
How to add a whole number and a fraction

1 Adding and subtracting fractions with like denominators

To add fractions with the same denominators, you need to add their numerators, but leave the denominator the same, for example:

To subtract fractions with the same denominators, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:

To add mixed fractions, you need to separately add their whole parts, and then add their fractional parts, and write the result as a mixed fraction,

Example 1:

Example 2:

If, when adding fractional parts, you get an improper fraction, select the whole part from it and add it to the whole part, for example:

2 Adding and subtracting fractions with different denominators.

3 Least common multiple (LCM)

In order to find the LCM of several numbers, you need:

Factor these numbers into prime factors
Take the largest expansion and write these numbers as a product
Select numbers in other decompositions that do not appear in the largest decomposition (or occur fewer times in it), and add them to the product.
Multiply all the numbers in the product, this will be the LCM.

For example, let's find the LCM of the numbers 28 and 21:

4 Reducing fractions to the same denominator

Let's return to adding fractions with different denominators.

5 How to add a whole number and a fraction

To add a whole number and a fraction, you simply add that number before the fraction to create a mixed fraction, for example:

If we add a whole number and a mixed fraction, we add that number to the whole number part of the fraction, for example:

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Adding and subtracting fractions with like denominators.

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This test tests your ability to add fractions with like denominators. In this case, two rules must be observed:

If the result is an improper fraction, you need to convert it to a mixed number.
If a fraction can be shortened, be sure to shorten it, otherwise an incorrect answer will be counted.

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Subtracting fractions with like denominators, examples:

Subtracting a proper fraction from one.

If it is necessary to subtract a fraction from a unit that is proper, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.

An example of subtracting a proper fraction from one:

Denominator of the fraction to be subtracted = 7 , i.e., we represent one as an improper fraction 7/7 and subtract it according to the rule for subtracting fractions with like denominators.

Subtracting a proper fraction from a whole number.

Rules for subtracting fractions - correct from a whole number (natural number):

We convert given fractions that contain an integer part into improper ones. We obtain normal terms (it doesn’t matter if they have different denominators), which we calculate according to the rules given above;
Next, we calculate the difference between the fractions that we received. As a result, we will almost find the answer;
We perform the reverse transformation, that is, we get rid of the improper fraction - we select the whole part in the fraction.

Subtract a proper fraction from a whole number: represent the natural number as a mixed number. Those. We take a unit in a natural number and convert it to the form of an improper fraction, the denominator being the same as that of the subtracted fraction.

Example of subtracting fractions:

In the example, we replaced one with the improper fraction 7/7 and instead of 3 we wrote down a mixed number and subtracted a fraction from the fractional part.

Subtracting fractions with different denominators.

Or, to put it another way, subtracting different fractions.

Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to reduce these fractions to the lowest common denominator (LCD), and only after this, perform the subtraction as with fractions with the same denominators.

The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of these fractions.

Attention! If in the final fraction the numerator and denominator have common factors, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the subtraction result without reducing the fraction where possible is an incomplete solution to the example!

Procedure for subtracting fractions with different denominators.

find the LCM for all denominators;
put additional factors for all fractions;
multiply all numerators by an additional factor;
We write the resulting products into the numerator, signing the common denominator under all fractions;
subtract the numerators of fractions, signing the common denominator under the difference.

In the same way, addition and subtraction of fractions is carried out if there are letters in the numerator.

Subtracting fractions, examples:

Subtracting mixed fractions.

At subtracting mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.

The first option for subtracting mixed fractions.

If the fractional parts identical denominators and numerator of the fractional part of the minuend (we subtract it from it) ≥ numerator of the fractional part of the subtrahend (we subtract it).

For example:

The second option for subtracting mixed fractions.

When fractional parts different denominators. To begin with, we bring the fractional parts to a common denominator, and after that we subtract the whole part from the whole part, and the fractional part from the fractional part.

For example:

The third option for subtracting mixed fractions.

The fractional part of the minuend is less than the fractional part of the subtrahend.

Example:

Because Fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.

The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. This means we take a unit from the whole part and reduce this unit to the form of an improper fraction with the same denominator and numerator = 18.

In the numerator on the right side we write the sum of the numerators, then we open the brackets in the numerator on the right side, that is, we multiply everything and give similar ones. We do not open the parentheses in the denominator. It is customary to leave the product in the denominators. We get:

The following rules apply to proper and improper fractions (a mixed fraction can always be converted to an improper fraction) with the same denominators.

Rule. To add fractions with the same denominators, you need to add their numerators and leave the same denominator.

For example:

Rule. To subtract fractions with the same denominators, you need to subtract the numerator of the second fraction from the numerator of the first fraction and leave the same denominator.

For example:

The following rules apply to mixed fractions with like denominators.

Rule. To add mixed fractions, you need to separately add their whole and fractional parts and write down the sum of the whole parts and the sum of the fractional parts as a mixed fraction.

If the total fractional part turns out to be an improper fraction, then they should be converted into a mixed fraction, and the whole part separated from the improper fraction should be added to the sum of the whole parts. Write the final sum of the whole and fractional parts as a mixed fraction.

For example, adding fractions:

Rule: To subtract mixed fractions, you must separately subtract their whole parts and separately their fractional parts and write down the sum of the resulting differences as a mixed fraction.

If the fractional part of the minuend is less than the fractional part of the subtrahend, then we “borrow” 1 from the integer part of the minuend, which we represent as a fraction with the same denominator as the fractional part of mixed fractions, and with a numerator equal to this denominator. The borrowed 1, expressed as an improper fraction with the same numerator and denominator, is summed with the fractional part of the minuend. After this, we carry out calculations according to the rule for subtracting mixed fractions.