How to subtract fractional numbers with different denominators. Subtraction of common fractions: rules, examples, solutions

Has your child brought homework from school and you don’t know how to solve it? Then this mini tutorial is for you!

How to add decimals

It is more convenient to add decimal fractions in a column. To add decimal fractions, you must adhere to one simple rule:

• The digit must be below the digit, a comma below the comma.

As you can see in the example, whole units are under each other, the tenths and hundredths are under each other. Now add the numbers, ignoring the comma. What to do with the comma? The comma is transferred to the place where it was in the place of integers.

Adding fractions with equal denominators

To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction, which will be the total.

Adding fractions with different denominators by the method of finding the common multiple

The first thing to look at is the denominators. The denominators are different, are they not divisible, are prime numbers... First, you need to bring to one common denominator, for this there are several ways:

• 1/3 + 3/4 = 13/12, to solve this example we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b - LCM (a; b). In this example, the LCM (3; 4) = 12. We check: 12: 3 = 4; 12: 4 = 3.
• We multiply the factors and add the numbers obtained, we get 13/12 - an improper fraction.

• In order to convert an incorrect fraction into a correct one, divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.

Adding fractions by cross-to-cross multiplication

To add fractions with different denominators there is another way according to the “cross to cross” formula. This is a guaranteed way to level the denominators by multiplying the numerators with the denominator of one fraction and vice versa. If you're only on initial stage studying fractions, then this method is the simplest and most accurate, how to get the correct result when adding fractions with different denominators.

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

Subtraction of mixed fractions with the same denominator.

Consider an example with the condition that the reduced integer and fractional parts are greater than the subtracted whole and fractional parts, respectively. Under these conditions, the deduction takes place separately. Subtract the whole part from the whole part, and the fractional part from the fractional part.

Let's consider an example:

Subtract the 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) \)

Consider an example with the condition when the fractional part of the reduced is less, respectively, the fractional part of the subtracted. In this case, we borrow one from the whole in the decreasing one.

Let's consider an example:

Perform mixed fraction subtraction \ (6 \ frac (1) (4) \) and \ (3 \ frac (3) (4) \).

The reduced \ (6 \ frac (1) (4) \) has a fractional part less than the fractional part of the subtracted \ (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 an integer.

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

Reduced 3 does not have a fractional part, so we cannot immediately subtract it. Let's borrow one from the integer part of 3, and then perform 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) \)

Subtraction of mixed fractions with different denominators.

Consider an example with the condition if the fractional parts of the reduced and subtracted with different denominators. You need to bring to a common denominator, and then perform the subtraction.

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

The common denominator is 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) \)

Questions on the topic:
How to subtract mixed fractions? How to solve mixed fractions?
Answer: you need to decide what type the expression belongs to and, by the type of expression, apply the solution algorithm. Subtract the whole from the whole part, subtract the fractional part from the fractional part.

How to subtract a fraction from an integer? How to subtract a fraction from an integer?
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 the correct fraction from one: a) \ (1- \ frac (8) (33) \) b) \ (1- \ frac (6) (7) \)

Solution:
a) We represent the unit as a fraction with the denominator 33. We get \ (1 = \ frac (33) (33) \)

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

b) We represent the unit as a fraction with the 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 an integer: a) \ (21-10 \ frac (4) (5) \) b) \ (2-1 \ frac (1) (3) \)

Solution:
a) We borrow 21 units from an 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 borrow a unit 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 No. 4:
Subtract the correct fraction from the 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) \)

Instructions

It is customary to separate ordinary and decimal fractions, acquaintance with which begins in high school. There is currently no area of ​​expertise that does not apply this. Even in we say the first 17th century, and all at once, which means 1600-1625. You also often have to deal with elementary actions on, as well as their transformation from one type to another.

Bringing fractions to a common denominator is perhaps the most important action on. This is the basis for absolutely all calculations. So let's say there are two fractions a / b and c / d. Then, in order to bring them to a common denominator, you need to find the least common multiple (M) of the numbers b and d, and then multiply the numerator of the first fractions on (M / b), and the second numerator on (M / d).

Comparing fractions is another important task. In order to do this, give the given simple fractions to the common denominator and then compare the numerators, whose numerator is greater, that fraction and more.

To perform addition or subtraction common fractions, you need to bring them to a common denominator, and then produce the desired mathematical with these fractions. The denominator remains unchanged. Suppose you need to subtract c / d from a / b. To do this, you need to find the least common multiple M of the numbers b and d, and then subtract the other from one numerator without changing the denominator: (a * (M / b) - (c * (M / d)) / M

It is enough just to multiply one fraction by another, for this you just need to multiply their numerators and denominators:
(a / b) * (c / d) = (a * c) / (b * d) To divide one fraction by another, you need to multiply the fraction of the dividend by the inverse of the divisor. (a / b) / (c / d) = (a * d) / (b * c)
It is worth recalling that in order to get the reciprocal fraction, the numerator and denominator must be reversed.

Find the numerator and denominator. A fraction includes two numbers: the number above the line is called the numerator, and the number below the line is called the denominator. The denominator denotes the total number of parts into which a whole is divided, and the numerator is the number of such parts under consideration.

• For example, in the fraction ½, the numerator is 1 and the denominator is 2.

Determine the denominator. If two or more fractions have a common denominator, such fractions have the same number under the line, that is, in this case, some whole is divided into the same number of parts. It is very easy to add fractions with a common denominator, since the denominator of the total fraction will be the same as for the added fractions. For instance:

• Fractions 3/5 and 2/5 have a common denominator of 5.
• Fractions 3/8, 5/8, 17/8 have a common denominator of 8.

Define the numerators. To add fractions with a common denominator, add their numerators, and write the result over the denominator of the fractions to add.

• Fractions 3/5 and 2/5 have numerators 3 and 2.
• Fractions 3/8, 5/8, 17/8 have numerators 3, 5, 17.

Add up the numerators. For problem 3/5 + 2/5, add the numerators 3 + 2 = 5. For problem 3/8 + 5/8 + 17/8, add the numerators 3 + 5 + 17 = 25.

Write down the total fraction. Remember that when you add fractions with a common denominator, it remains unchanged - only the numerators are added.

• 3/5 + 2/5 = 5/5
• 3/8 + 5/8 + 17/8 = 25/8

Convert the fraction if necessary. Sometimes a fraction can be written as an integer, rather than an ordinary one, or decimal... For example, 5/5 is easy to convert to 1, since any fraction with the numerator equal to the denominator is 1. Imagine a pie cut into three pieces. If you eat all three pieces, then you will eat a whole (one) pie.

• Any fraction can be converted to decimal; to do this, divide the numerator by the denominator. For example, the fraction 5/8 can be written like this: 5 ÷ 8 = 0.625.

Simplify the fraction if possible. A simplified fraction is a fraction whose numerator and denominator do not have common factors.

• For example, consider 3/6. Here, both the numerator and the denominator have common divisor equal to 3, that is, the numerator and denominator are completely divisible by 3. Therefore, the fraction 3/6 can be written as follows: 3 ÷ 3/6 ÷ 3 = ½.

If necessary, convert the improper fraction to a mixed number (mixed number). An improper fraction has a higher numerator than the denominator, for example, 25/8 (a regular fraction has a lower numerator). An irregular fraction can be converted to a mixed fraction, which consists of an integer part (that is, an integer) and a fractional part (that is, a regular fraction). To convert an improper fraction, such as 25/8, to a mixed number, follow these steps:

• Divide the numerator of the improper fraction by its denominator; write down the incomplete quotient (whole answer). In our example: 25 ÷ 8 = 3 plus some remainder. In this case, the whole answer is the whole part of the mixed number.
• Find the remainder. In our example: 8 x 3 = 24; Subtract the result from the original numerator: 25 - 24 = 1, that is, the remainder is 1. In this case, the remainder is the numerator of the fractional part of the mixed number.
• Write down the mixed fraction. The denominator does not change (that is, it is equal to the denominator of the improper fraction), so 25/8 = 3 1/8.

Note! Before you write your final answer, see if you can reduce the fraction you received.

Subtracting fractions from the same denominators,examples:

,

,

Subtracting a correct fraction from one.

If it is necessary to subtract a fraction from the unit that is correct, the unit is transferred to the form of an incorrect fraction, its denominator is equal to the denominator of the fraction to be subtracted.

An example of subtracting a correct fraction from one:

The denominator of the subtracted fraction = 7 , i.e., we represent the unit as an irregular fraction 7/7 and subtract according to the rule of subtraction of fractions with the same denominators.

Subtracting a right fraction from an integer.

Fraction subtraction rules - correct from an integer (natural number):

• We translate the given fractions, which contain an integer part, into incorrect ones. We get normal terms (it doesn't matter if they have different denominators), which we count according to the rules given above;
• Next, we calculate the difference of the fractions that we got. As a result, we will almost find the answer;
• We carry out the inverse transformation, that is, we get rid of the incorrect fraction - we select the whole part in the fraction.

Subtract the correct fraction from the integer: represent the natural number as a mixed number. Those. we occupy a unit in natural number and convert it to the form of an irregular fraction, the denominator is the same as that of the subtracted fraction.

An example of subtracting fractions:

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

Subtraction of fractions with different denominators.

Or, to put it in other words, subtraction of different fractions.

The rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to bring these fractions to the lowest common denominator (LCN), and only after this subtract as with fractions with the same denominator.

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

Attention! If the numerator and denominator have common factors in the final fraction, then the fraction must be canceled. A bad fraction is best represented as a mixed fraction. Leaving the result of subtraction without canceling the fraction where possible is an unfinished 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 a common denominator under all the fractions;
• subtract the numerators of the 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.

Subtraction of fractions, examples:

Subtraction of mixed fractions.

At subtracting mixed fractions (numbers) separately from the whole part, subtract the whole part, and subtract the fractional part from the fractional part.

The first option is to subtract mixed fractions.

If fractional parts the same denominators and numerator of the fractional part of the subtracted (subtract from it) ≥ numerator of the fractional part of the subtracted (subtract it).

For instance:

The second option is to subtract mixed fractions.

When fractional parts have 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, and the fractional part from the fractional part.

For instance:

The third option for subtracting mixed fractions.

The fractional part of the reduced is less than the fractional part of the subtracted.

Example:

Because the 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 subtracted is less than the numerator of the fractional part of the subtracted.3 < 14. Hence, we take a unit from the whole part and bring this unit to the form of an irregular fraction with the same denominator and numerator = 18.

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