In the course of secondary and high school, students passed the theme of "Fractions." However, this concept is much broader than that given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can calculate any expression, for example, multiplying fractions.
What is a fraction?
So historically, fractional numbers appeared due to the need to measure. As practice shows, examples are often found to determine the length of a segment, the volume of a rectangular box, the area of a rectangle.
Initially, students get acquainted with the concept of share. For example, if you divide a watermelon into 8 parts, then everyone will get one eighth of a watermelon. This one part of eight is called a share.
A fraction equal to ½ of a quantity is called half; ⅓ - the third; ¼ - a quarter. Records of the form 5/8, 4/5, 2/4 are called ordinary fractions. An ordinary fraction is divided into a numerator and a denominator. Between them there is a fraction line, or a fractional line. The fractional line can be drawn in the form of both a horizontal and an inclined line. In this case, it denotes a division sign.
The denominator represents how many equal shares divide the value, the subject; and the numerator - how many identical shares are taken. The numerator is written above the fractional bar, the denominator is under it.
It is most convenient to show ordinary fractions on the coordinate beam. If you divide a single segment into 4 equal shares, designate each share with a Latin letter, then as a result you can get an excellent visual aid. So, point A shows a fraction equal to 1/4 of the entire unit segment, and point B marks 2/8 of this segment.
Varieties of fractions
Fractions are ordinary, decimal, as well as mixed numbers. In addition, fractions can be divided into right and wrong. This classification is more suitable for common fractions.
The correct fraction is understood to mean a number whose numerator is less than the denominator. Accordingly, an incorrect fraction is a number whose numerator is greater than the denominator. The second kind is usually written as a mixed number. Such an expression consists of an integer and a fractional part. For example, 1½. 1 - the integer part, ½ - fractional. However, if you need to carry out any manipulations with the expression (division or multiplication of fractions, their reduction or transformation), the mixed number is converted to the wrong fraction.
The correct fractional expression is always less than one, and the incorrect fraction is greater than or equal to 1.
As for decimal fractions, this expression means a record in which any number is represented, the denominator of the fractional expression of which can be expressed in terms of a unit with several zeros. If the fraction is correct, then the integer part in the decimal notation will be equal to zero.
To write a decimal fraction, you must first write the integer part, separate it from the fractional using a comma, and then write down the fractional expression. It must be remembered that after the decimal point, the numerator must contain as many digital characters as there are zeros in the denominator.
An example . Represent the fraction 7 21/1000 in decimal notation.
Algorithm for translating the wrong fraction into a mixed number and vice versa
It is incorrect to write the wrong fraction in the answer of the problem, so it must be converted to a mixed number:
- divide the numerator by the existing denominator;
- in a specific example, an incomplete quotient is a whole;
- and the remainder is the numerator of the fractional part, and the denominator remains unchanged.
An example . Convert the wrong fraction to a mixed number: 47/5 .
Solution . 47: 5. Incomplete quotient equals 9, remainder = 2. Therefore, 47/5 = 9 2/5 .
Sometimes you need to represent a mixed number as an incorrect fraction. Then you need to use the following algorithm:
- the whole part is multiplied by the denominator of the fractional expression;
- the resulting product is added to the numerator;
- the result is written in the numerator, the denominator remains unchanged.
An example . Present the number in mixed form as an incorrect fraction: 9 8/10 .
Solution . 9 x 10 + 8 = 90 + 8 = 98 - the numerator.
Answer : 98/10 .
Multiplication of common fractions
Various fractions can be performed on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, the multiplication of fractions with different denominators does not differ from the product of fractional numbers with the same denominators.
It happens that after finding the result, you need to reduce the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, one cannot say that the wrong fraction in the answer is a mistake, but it is also difficult to call it the correct answer.
An example . Find the product of two ordinary fractions: ½ and 20/18 .
As you can see from the example, after finding the product, we got a reducible fractional record. In this case, both the numerator and the denominator are divided by 4, and the answer is 5/9.
Decimal fraction multiplication
The product of decimal fractions is quite different from the product of ordinary fractions by its principle. So, the multiplication of fractions is as follows:
- two decimal fractions should be written under each other so that the rightmost digits are one under the other;
- you need to multiply the recorded numbers, despite the commas, that is, as natural ones;
- count the number of digits after the comma in each of the numbers;
- in the result obtained after multiplication, you need to count on the right as many numeric characters as the sum in both multipliers after the decimal point, and put a separating sign;
- if the numbers in the work turned out to be less, then you need to write so many zeros in front of them to cover this number, put a comma and ascribe the integer part equal to zero.
An example . Calculate the product of two decimal fractions: 2.25 and 3.6.
Solution .
Multiplication of Fractions
To calculate the product of two mixed fractions, you need to use the fraction multiplication rule:
- translate mixed numbers into the wrong fractions;
- find the product of the numerators;
- find the product of the denominators;
- record the result;
- simplify the expression as much as possible.
An example . Find the product of 4½ and 6 2/5 .
Multiplying a number by a fraction (fractions by a number)
In addition to finding the product of two fractions, mixed numbers, there are tasks where you need to multiply a natural number by a fraction.
So, to find the product of the decimal fraction and the natural number, you need:
- write the number under the fraction so that the rightmost digits are one above the other;
- find the work, despite the comma;
- in the result, separate the integer part from the fractional one using a comma, counting to the right the number of characters that are after the comma in the fraction.
To multiply an ordinary fraction by a number, you need to find the product of the numerator and the natural factor. If the answer is a contractible fraction, it should be converted.
An example . Calculate the product of 5/8 and 12.
Solution . 5/8 * 12 = (5 * 12) / 8 = 60/8 = 30/4 = 15/2 = 7 1/2 .
Answer : 7 1/2 .
As you can see from the previous example, it was necessary to reduce the result and convert the incorrect fractional expression to a mixed number.
Also, fraction multiplication also concerns finding the product of a number in a mixed form and a natural factor. To multiply these two numbers, the integer part of the mixed factor should be multiplied by a number, the numerator should be multiplied by the same value, and the denominator should be left unchanged. If required, you need to simplify the resulting result.
An example . Find the product 9 5/6 and 9.
Solution . 9 5/6 x 9 = 9 x 9 + (5 x 9) / 6 = 81 + 45/6 = 81 + 7 3/6 = 88 1/2 .
Answer : 88 1/2 .
Multiplication by factors of 10, 100, 1000 or 0.1; 0.01; 0.001
The following rule follows from the previous paragraph. To multiply the decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma to the right by as many digits as the number of zeros in the factor after unity.
Example 1 Find the product of 0.065 and 1000.
Solution . 0.065 x 1000 = 0065 = 65.
The answer is 65.
Example 2 Find the product of 3.9 and 1000.
Solution . 3.9 x 1000 = 3.900 x 1000 = 3900.
Answer : 3900.
If you need to multiply a natural number and 0.1; 0.01; 0.001; 0.0001, etc., move the comma to the left in the resulting product by as many digits as there are zeros to one. If necessary, zeros in sufficient quantities are written before the natural number.
Example 1 Find the product of 56 and 0.01.
Solution . 56 x 0.01 = 0056 = 0.56.
Answer : 0.56.
Example 2 Find the product of 4 and 0.001.
Solution . 4 x 0.001 = 0004 = 0.004.
Answer : 0.004.
So, finding the product of various fractions should not cause difficulties, unless counting the result; in this case, you just can’t do without a calculator.