In the course of the middle and high school students took the theme of "Fractions".However, this concept is much broader than that given in the learning process.Today the concept of fractions is not uncommon, and not everyone can perform calculations of an expression, for example, multiplication of fractions.
What is a fraction?
Historically, fractional numbers that have appeared because of the need to measure.As practice shows, are common examples of the determination of the length of the segment, the volume of a cuboid, the area of a rectangle.
Initially, students become familiar with the concept of how to share.For example, if divided into 8 parts watermelon, then each will get one eighth watermelon.Here is one part of the eight called lobes.
share, equal to ½ of a value called a half;⅓ - third;¼ - quarter.Entries of the form 5/8, 4/5, 2/4 are called common fractions.Common fractions divided into the numerator and denominator.Between them is the devil shot or stroke.You can draw a slash in the form of both horizontal and sloping line.In this case it refers to the division sign.
denominator is, as far as the same proportion of share value, the subject;and the numerator - the same number of shares taken.The numerator of the fraction is written below, the denominator - underneath.
preferred to show ordinary fractions on the coordinate ray.If the unit interval divided into 4 equal parts, indicate the share of each Latin letter, as a result, you can get excellent visual aid.For example, point A shows the share equal to 1/4 of the total length of the unit and point B marks the 2/8 of this segment.
Species fractions
Fractions are common, decimal and mixed numbers.Further, fractions can be divided into right and wrong.This classification is more suitable for ordinary fractions.
Under proper fraction understand the numbers, the numerator of which is less than the denominator.Accordingly, the improper fraction - a number which has more than the numerator denominator.The second type is usually written as a mixed number.This is an expression of integer and fractional parts.For example, 1½.1 - the whole part, ½ - fractional.However, if you need to carry out any manipulation of the expression (division or multiplication of fractions, their reduction or conversion), mixed number is translated into improper fractions.
correct fractional expression is always less than unity, and wrong - more or equal to 1.
As for decimals, then by this expression understand the record, which is represented by any number, the denominator of the fractional expression of which can be expressed in a unit with a few zeros.If the roll is correct, then the whole part in decimal notation is equal to zero.
To write a decimal fraction, you must first write a whole piece, to separate it from the fraction with a comma, and then write the fractional expression.It must be remembered that after the point numerator must contain the same number of numeric characters, how many zeros in the denominator.
example .Present 721/1000 fraction in decimal notation.
translation algorithm improper fractions to mixed numbers and vice versa
Record in response task improper fractions incorrectly, so it needs to be converted to a mixed number:
- divide the numerator by the denominator is available;
- in particular example partial quotient - whole;
- and balance - the numerator of the fractional part, and the denominator remains unchanged.
example .Translate improper fractions into mixed numbers: 47/5.
decision .47: 5. Incomplete quotient is 9, the remainder = 2. So, 47/5 = 92/5.
Sometimes you need to submit a mixed number as an improper fraction.Then you need to use the following algorithm:
- integer part is multiplied by the denominator of the fractional expression;
- resulting product is added to the numerator;
- result is written in the numerator, denominator remains unchanged.
example .Present numbers in mixed form as improper fractions: 98/10.
decision .9 x 10 + 8 = 90 + 8 = 98 - the numerator.
answer : 98/10.
Multiply fractions ordinary
over ordinary fractions can perform various algebraic operations.To multiply the 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 not differ from the product of fractional numbers with the same denominator.
happens that after finding the results need to reduce a fraction.The imperative to simplify the resulting expression.Of course, we can not say that the improper fraction in the answer - it's a mistake, but the correct answer and call it too difficult.
example .Find the product of two common fractions: ½, and 20/18.
As you can see, after finding the fractional product turned cancellative record.And the numerator and denominator in this case is divided by 4, and the result appears the answer 5/9.
Multiplication of decimal fractions
product decimals is quite different from ordinary works by its principle.Thus, multiplication of fractions is as follows:
- two decimals to be written under each other so that the extreme right-wing figures were one above the other;
- need to multiply the number of recorded despite commas, that is as natural;
- count the number of digits after the decimal point in each of the numbers;
- to get after multiplying the result of the need to count on the right as numeric characters, as is contained in the amount of both multipliers after the decimal point, and put the sign separates;
- if the numbers in the product proved less time in front of them to write as many zeros to cover this amount, a comma, and attributed the integer part is zero.
example .Calculate the product of two decimal numbers: 2.25 and 3.6.
decision .
Multiplication mixed fractions
To calculate the product of two mixed fractions, you need to use the rule of multiplication of fractions:
- translate numbers in mixed form in the wrong places;
- numerators find work;
- find a product of the denominators;
- record the result obtained;
- simplify expression.
example .Find the product of 4½ and 62/5.
Multiply the number by a fraction (a fraction by a number)
addition to finding the product of two fractions, mixed numbers, there are jobs where you need to multiply an integer by a fraction.
So, to find work and a decimal integer, you need:
- record the number under the shot so that the extreme right-wing figures were one above the other;
- find work, despite a comma;
- in a result separate the integer part from the decimal by a comma, count the number of characters to the right, which is after the decimal point in fraction.
To multiply the number of fractions, you should find a natural product of the numerator and the factor.If the answer is cancellative fraction, it should be converted.
example .Calculate the product of 5/8 and 12.
decision .* 12 = 5/8 (5 * 12) / 8 = 60/8 = 30/4 = 15/2 = 71/2.
answer : 71/2.
As can be seen from the previous example, it was necessary to reduce the result obtained and transform the wrong fractional expression in the mixed number.
multiplying fractions also applies to finding the product of the number in mixed form factor and natural.To multiply the two numbers, should be the integral part of the mixed factor multiplied by the number, the numerator is multiplied by the same value as the denominator left unchanged.If you want, you need to simplify the result.
example .Find work 95/6 and 9.
decision .95/6 = 9 x 9 x 9 + (5 x 9) / 6 = 81 = 81 + 45/6 + 73/6 = 881/2.
answer : 881/2.
multiplication by factors of 10, 100, 1,000 or 0.1;0.01;0,001
from the preceding paragraph implies the following rule.For multiplying a decimal fraction 10, 100, 1000, 10000, and so on. G. To be moved to the right by as many comma character digits as zeros in the multiplier unit after.
Example 1 .Find the product of 0.065 and 1000.
decision .0.065 x 1000 = 0065 = 65.
answer : 65.
Example 2 .Find the product of 3.9 and 1000.
decision .3.9 x 1000 = 3,900 x 1000 = 3900.
answer : 3900.
If you want to multiply an integer and 0.1;0.01;0.001;0.0001 and t. E., Should be moved to the left in the resulting product of the comma in the many characters of numbers, how many zeros are to one.If necessary, before the natural number recorded zeros in sufficient quantity.
Example 1 .Find the product of 56 and 0.01.
decision .56 x 0.01 = 0056 = 0.56.
answer : 0,56.
Example 2 .Find the product of 4 and 0,001.
decision .0.001 x 4 = 0004 = 0.004.
answer : 0,004.
So finding the works of various fractions should not cause difficulties, except that counting result;in this case without a calculator just will not do.
