Irina 25 proper and improper fractions. Proper and improper fractions

They are divided into correct and incorrect.

Proper Fractions

Proper fraction is an ordinary fraction in which the numerator is less than the denominator.

To find out whether a fraction is proper, you need to compare its terms with each other. Fraction terms are compared in accordance with the rule for comparing natural numbers.

Example. Consider the fraction:

Example:

8	= 1	1
7	= 1	7

Translation rules and additional examples can be found in the topic Converting an improper fraction to a mixed number. You can also use an online calculator to convert an improper fraction to a mixed number.

Comparing proper and improper fractions

Any improper ordinary fraction is greater than a proper fraction, since a proper fraction is always less than one, and an improper fraction is greater than or equal to one.

Example:

3	>	99
2	>	100

Comparison rules and additional examples can be found in the topic Comparison of ordinary fractions. Also, to compare fractions or check comparisons, you can use

Common fractions are divided into \textit (proper) and \textit (improper) fractions. This division is based on a comparison of the numerator and denominator.

Proper Fractions

Proper fraction An ordinary fraction $\frac(m)(n)$ is called, in which the numerator is less than the denominator, i.e. $m

Example 1

For example, the fractions $\frac(1)(3)$, $\frac(9)(123)$, $\frac(77)(78)$, $\frac(378567)(456298)$ are correct, so how in each of them the numerator is less than the denominator, which meets the definition of a proper fraction.

There is a definition of a proper fraction, which is based on comparing the fraction with one.

correct, if it is less than one:

Example 2

For example, the common fraction $\frac(6)(13)$ is proper because condition $\frac(6)(13) is satisfied

Improper fractions

Improper fraction An ordinary fraction $\frac(m)(n)$ is called, in which the numerator is greater than or equal to the denominator, i.e. $m\ge n$.

Example 3

For example, the fractions $\frac(5)(5)$, $\frac(24)(3)$, $\frac(567)(113)$, $\frac(100001)(100000)$ are irregular, so how in each of them the numerator is greater than or equal to the denominator, which meets the definition of an improper fraction.

Let us give a definition of an improper fraction, which is based on its comparison with one.

The common fraction $\frac(m)(n)$ is wrong, if it is equal to or greater than one:

\[\frac(m)(n)\ge 1\]

Example 4

For example, the common fraction $\frac(21)(4)$ is improper because the condition $\frac(21)(4) >1$ is satisfied;

the common fraction $\frac(8)(8)$ is improper because the condition $\frac(8)(8)=1$ is satisfied.

Let's take a closer look at the concept of an improper fraction.

Let's take the improper fraction $\frac(7)(7)$ as an example. The meaning of this fraction is to take seven shares of an object, which is divided into seven equal shares. Thus, from the seven shares that are available, the entire object can be composed. Those. the improper fraction $\frac(7)(7)$ describes the whole object and $\frac(7)(7)=1$. So, improper fractions, in which the numerator is equal to the denominator, describe one whole object and such a fraction can be replaced by the natural number $1$.

$\frac(5)(2)$ -- it is quite obvious that from these five second parts you can make up $2$ whole objects (one whole object will be made up of $2$ parts, and to compose two whole objects you need $2+2=4$ shares) and one second share remains. That is, the improper fraction $\frac(5)(2)$ describes $2$ of an object and $\frac(1)(2)$ the share of this object.

$\frac(21)(7)$ -- from twenty-one-sevenths parts you can make $3$ whole objects ($3$ objects with $7$ shares in each). Those. the fraction $\frac(21)(7)$ describes $3$ whole objects.

From the examples considered, we can draw the following conclusion: an improper fraction can be replaced by a natural number if the numerator is divisible by the denominator (for example, $\frac(7)(7)=1$ and $\frac(21)(7)=3$) , or the sum of a natural number and a proper fraction, if the numerator is not completely divisible by the denominator (for example, $\ \frac(5)(2)=2+\frac(1)(2)$). That's why such fractions are called wrong.

Definition 1

The process of representing an improper fraction as the sum of a natural number and a proper fraction (for example, $\frac(5)(2)=2+\frac(1)(2)$) is called separating the whole part from an improper fraction.

When working with improper fractions, there is a close connection between them and mixed numbers.

An improper fraction is often written as a mixed number - a number that consists of a whole number and a fraction part.

To write an improper fraction as a mixed number, you must divide the numerator by the denominator with a remainder. The quotient will be the integer part of the mixed number, the remainder will be the numerator of the fractional part, and the divisor will be the denominator of the fractional part.

Example 5

Write the improper fraction $\frac(37)(12)$ as a mixed number.

Solution.

Divide the numerator by the denominator with a remainder:

\[\frac(37)(12)=37:12=3\ (remainder\ 1)\] \[\frac(37)(12)=3\frac(1)(12)\]

Answer.$\frac(37)(12)=3\frac(1)(12)$.

To write a mixed number as an improper fraction, you need to multiply the denominator by the whole part of the number, add the numerator of the fractional part to the resulting product, and write the resulting amount into the numerator of the fraction. The denominator of the improper fraction will be equal to the denominator of the fractional part of the mixed number.

Example 6

Write the mixed number $5\frac(3)(7)$ as an improper fraction.

Solution.

Answer.$5\frac(3)(7)=\frac(38)(7)$.

Adding mixed numbers and proper fractions

Mixed Number Addition$a\frac(b)(c)$ and proper fraction$\frac(d)(e)$ is performed by adding to a given fraction the fractional part of a given mixed number:

Example 7

Add the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$.

Solution.

Let's use the formula for adding a mixed number and a proper fraction:

\[\frac(4)(15)+3\frac(2)(5)=3+\left(\frac(2)(5)+\frac(4)(15)\right)=3+\ left(\frac(2\cdot 3)(5\cdot 3)+\frac(4)(15)\right)=3+\frac(6+4)(15)=3+\frac(10)( 15)\]

By dividing by the number \textit(5) we can determine that the fraction $\frac(10)(15)$ is reducible. Let's perform the reduction and find the result of the addition:

So, the result of adding the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$ is $3\frac(2)(3)$.

Answer:$3\frac(2)(3)$

Adding mixed numbers and improper fractions

Adding improper fractions and mixed numbers reduces to the addition of two mixed numbers, for which it is enough to isolate the whole part from the improper fraction.

Example 8

Calculate the sum of the mixed number $6\frac(2)(15)$ and the improper fraction $\frac(13)(5)$.

Solution.

First, let's extract the whole part from the improper fraction $\frac(13)(5)$:

Answer:$8\frac(11)(15)$.

The pie was cut into 8 equal parts (Fig. 122, a) and 3 parts were placed on a plate.

There was a pie on it (Fig. 122, b). If you put all 8 parts, then there will be a pie on the plate, that is, the whole pie (Fig. 122, c).

Rice. 122

So = 1.

Let's take another similar pie and cut it into 8 equal parts (Fig. 123, a). If you put, for example, 11 pieces on a plate, then there will be a pie (Fig. 123, b).

Rice. 123

In a fraction, the numerator is less than the denominator. Such fractions are called proper. In a fraction, the numerator is equal to the denominator, and in a fraction, the numerator is greater than the denominator. Such fractions are called improper.

Rice. 124

For example,< 1, = 1, > 1.

Self-test questions

What fraction is called proper?
What fraction is called an improper fraction?
Can a proper fraction be greater than 1?
Is an improper fraction always greater than 1?
Which fraction is greater if one is regular and the other is improper?

Do the exercises

974. The length of segment AB is 8 cm. Draw a segment whose length is equal to:

975. Mark points on the ray with coordinates:

Take the length of 12 notebook cells as a single segment.

976. Write:

a) all proper fractions with a denominator of 6;
b) all improper fractions with numerator 5.

977. At what values is a fraction:

978. A machine can dig a ditch 1 m long in 6 minutes. How long a ditch can a machine dig in 1 minute; 5 minutes; 7 min; 11 min?

979. One kilogram of paint can cover 5 m2 of surface. How much paint will be needed to paint 3 m2; 6 m2; 13 m2 surface?

980. The construction team built the farm in 48 days. According to the plan, this time was required. How many days were allotted to build the farm according to plan?

981. The turner turned 135 parts on a lathe in 3 hours, fulfilling the daily quota. How many parts should he have turned in a working day (8 hours) according to the norm? How many parts will he turn out in a working day if he works at the same productivity?

982. The turner turned 135 parts on a lathe, fulfilling the daily quota. What is his daily requirement?

983. The concert of young musicians lasted this time instead of the planned 3 hours, as the audience asked to repeat some of their favorite performances. How long did the concert last? How many minutes did the encore last?

984. Calculate orally:

985. How many minutes in an hour? What part of an hour is 1 minute? 7 min; 15 minutes?

986. How many times is a quintal greater than a kilogram? What part of a hundredweight is a kilogram? How many hundredweight is greater than a kilogram?

987. How many minutes

988. Add the numbers 40 and the numbers 60. From the number 72, subtract the numbers 81.

989. Half of the number is 18. Find this number. A third of the number is 27. Find this number. Three quarters of a number is 60. Find this number.

990. What part of the quadrilateral ABCD (Fig. 125) is shaded? What part was left unpainted?

Rice. 125

991. Express in grams:

a) 3 kg 400 g;
b) 2 kg 30 g;
c) 15 kg.

992. Arrange the fractions in ascending order:

Arrange the same fractions in descending order.

993. Name four fractions that are less than

994. Name 5 fractions that are greater than .

995. Draw a square with a side of 4 cm. Show in the drawing: square, square. Find the areas of these parts of the square and explain the result.

996. On the first day, the team collected 5 tons of 400 kg of potatoes, and on the second - 1 ton 200 kg less than on the first. On the third day, the team collected 2 times more potatoes than on the second. How many potatoes did the brigade collect during these three days?

997. Create a problem using the equation:

a) (y+ 6) - 2 = 15;
b) 2(a - 5) = 24;
c) 3(25 + b) + 15 = 135.

998. There were a people in the first carriage, and b people in the second. At the stop, about a person got out of the first car, and d people got out of the second. What is the meaning of the following expressions:

a + b;
a - c;
c + d;
b - d;
(a + b) - (c + d);
(a - c) + (b - d)?

Explain why

(a + b) - (c + d) = (a - c) + (b - d)

for a > c, b > d.

Check this equality with a = 45, b = 39, c = 14, d = 12.

Using the resulting equality, calculate the value of the expression:

a) (548 + 897) - (148 + 227);
b) (391 + 199) - (181 + 79).

999. Come up with five fractions whose numerator is 3 less than the denominator. Write down five fractions whose numerator is 3 times the denominator.

1000. At what values of x will the fraction be improper?

1001. The farmer planned to collect 12 tons of vegetables from the field, but he collected this amount. How many tons of vegetables did the farmer harvest?

1002. The tourist walked 18 km on the first day, which is the distance he must cover on the second day. How many kilometers should a tourist walk in these two days?

1003. A freight train left St. Petersburg for Moscow at a speed of 48 km/h, and an hour after that a fast train left Moscow for St. Petersburg at a speed of 82 km/h. Find the distance between trains:

a) 1 hour after the departure of the fast train;
b) 3 hours after the freight train leaves;
c) 5 hours after the fast train leaves.

The distance from Moscow to St. Petersburg is 650 km.

1004. Find the meaning of the expression:

a) 8060 -45 - 45 150: 75 105;
b) (2 254 175 + 94 447) : 414 - 1329;
c) (123 - 93) : (12 - 9);
d) (62 + Z2)2.