Actions with fractions. In this article, we will analyze examples, everything is detailed with explanations. We will consider ordinary fractions. In the future, we will analyze decimals. I recommend to watch the whole and study sequentially.
1. Sum of fractions, difference of fractions.
Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be is equal to the sum fraction numerators.
Rule: when calculating the difference of fractions with the same denominators, we get a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.
Formal notation of the sum and difference of fractions with equal denominators:
Examples (1):
It is clear that when ordinary fractions are given, then everything is simple, but if they are mixed? Nothing complicated...
Option 1- you can convert them into ordinary ones and then calculate them.
Option 2- you can separately "work" with the integer and fractional parts.
Examples (2):
Yet:
And if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? It can also be done in two ways.
Examples (3):
* Translated into ordinary fractions, calculated the difference, converted the resulting improper fraction into a mixed one.
* Divided into integer and fractional parts, got three, then presented 3 as the sum of 2 and 1, with the unit presented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it as a fraction with the denominator we need, then from this fraction we can already subtract another.
Another example:
Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted into improper ones, then perform the necessary action. After that, if as a result we get an improper fraction, we translate it into a mixed one.
Above, we looked at examples with fractions that have equal denominators. What if the denominators differ? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the main property of the fraction is used.
Consider simple examples:
In these examples, we immediately see how one of the fractions can be converted to get equal denominators.
If we designate ways to reduce fractions to one denominator, then this one will be called METHOD ONE.
That is, immediately when “evaluating” the fraction, you need to figure out whether such an approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.
Now look at these examples:
This approach does not apply to them. There are other ways to reduce fractions to common denominator Let's take a look at them.
Method SECOND.
Multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:
*In fact, we bring fractions to the form when the denominators become equal. Next, we use the rule of adding timid with equal denominators.
Example:
*This method can be called universal, and it always works. The only negative is that after the calculations, a fraction may turn out that will need to be further reduced.
Consider an example:
It can be seen that the numerator and denominator are divisible by 5:
Method THIRD.
Find the least common multiple (LCM) of the denominators. This will be the common denominator. What is this number? This is the smallest natural number that is divisible by each of the numbers.
Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, 30, 60, 90 are divisible by them .... Least 30. Question - how to determine this least common multiple?
There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15), no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, such as 51 and 119.
Algorithm. In order to determine the least common multiple of several numbers, you must:
- decompose each of the numbers into SIMPLE factors
- write out the decomposition of the BIGGER of them
- multiply it by the MISSING factors of other numbers
Consider examples:
50 and 60 50 = 2∙5∙5 60 = 2∙2∙3∙5
in decomposition more missing one five
=> LCM(50,60) = 2∙2∙3∙5∙5 = 300
48 and 72 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3
in the expansion of a larger number, two and three are missing
=> LCM(48,72) = 2∙2∙2∙2∙3∙3 = 144
* The least common multiple of two prime numbers is equal to their product
Question! And why is it useful to find the least common multiple, because you can use the second method and simply reduce the resulting fraction? Yes, you can, but it's not always convenient. See what the denominator will be for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. Agree that it is more pleasant to work with smaller numbers.
Consider examples:
*51 = 3∙17 119 = 7∙17
in the expansion of a larger number, a triple is missing
=> LCM(51,119) = 3∙7∙17
And now we apply the first method:
* Look at the difference in the calculations, in the first case there is a minimum of them, and in the second you need to work separately on a piece of paper, and even the fraction that you got needs to be reduced. Finding the LCM simplifies the work considerably.
More examples:
* In the second example, it is already clear that the smallest number that is divisible by 40 and 60 is 120.
TOTAL! GENERAL CALCULATION ALGORITHM!
- we bring fractions to ordinary ones, if there is an integer part.
- we bring the fractions to a common denominator (first we look to see if one denominator is divisible by another, if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).
- having received fractions with equal denominators, we perform actions (addition, subtraction).
- if necessary, we reduce the result.
- if necessary, select the whole part.
2. Product of fractions.
The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:
Examples:
This section deals with operations with ordinary fractions. If it is necessary to perform a mathematical operation with mixed numbers, then it is enough to translate mixed fraction into extraordinary, carry out the necessary operations and, if necessary, present the final result again as a mixed number. This operation will be described below.
Fraction reduction
mathematical operation. Fraction reduction
To reduce the fraction \frac(m)(n) you need to find the greatest common divisor of its numerator and denominator: gcd(m,n), then divide the numerator and denominator of the fraction by this number. If gcd(m,n)=1, then the fraction cannot be reduced. Example: \frac(20)(80)=\frac(20:20)(80:20)=\frac(1)(4)
Usually, immediately finding the greatest common divisor is a difficult task, and in practice the fraction is reduced in several stages, step by step highlighting obvious common factors from the numerator and denominator. \frac(140)(315)=\frac(28\cdot5)(63\cdot5)=\frac(4\cdot7\cdot5)(9\cdot7\cdot5)=\frac(4)(9)
Bringing fractions to a common denominator
mathematical operation. Bringing fractions to a common denominator
To reduce two fractions \frac(a)(b) and \frac(c)(d) to a common denominator, you need:
- find the least common multiple of the denominators: M=LCM(b,d);
- multiply the numerator and denominator of the first fraction by M / b (after which the denominator of the fraction becomes equal to the number M);
- multiply the numerator and denominator of the second fraction by M/d (after which the denominator of the fraction becomes equal to the number M).
Thus, we convert the original fractions to fractions with the same denominators (which will be equal to the number M).
For example, the fractions \frac(5)(6) and \frac(4)(9) have LCM(6,9) = 18. Then: \frac(5)(6)=\frac(5\cdot3)(6 \cdot3)=\frac(15)(18);\quad\frac(4)(9)=\frac(4\cdot2)(9\cdot2)=\frac(8)(18) . Thus, the resulting fractions have a common denominator.
In practice, finding the least common multiple (LCM) of denominators is not always an easy task. Therefore, a number equal to the product of the denominators of the original fractions is chosen as a common denominator. For example, the fractions \frac(5)(6) and \frac(4)(9) are reduced to a common denominator N=6\cdot9:
\frac(5)(6)=\frac(5\cdot9)(6\cdot9)=\frac(45)(54);\quad\frac(4)(9)=\frac(4\cdot6)( 9\cdot6)=\frac(24)(54)
Fraction Comparison
mathematical operation. Fraction Comparison
To compare two common fractions:
- compare the numerators of the resulting fractions; a fraction with a larger numerator will be larger.
When comparing fractions, there are several special cases:
- From two fractions with the same denominators the greater is the fraction whose numerator is greater. For example \frac(3)(15)
- From two fractions with the same numerators the larger is the fraction whose denominator is smaller. For example, \frac(4)(11)>\frac(4)(13)
- That fraction, which at the same time larger numerator and smaller denominator, more. For example, \frac(11)(3)>\frac(10)(8)
Attention! Rule 1 applies to any fractions if their common denominator is a positive number. Rules 2 and 3 apply to positive fractions (which have both numerator and denominator greater than zero).
Addition and subtraction of fractions
mathematical operation. Addition and subtraction of fractions
To add two fractions, you need:
- bring them to a common denominator;
- add their numerators and leave the denominator unchanged.
Example: \frac(7)(9)+\frac(4)(7)=\frac(7\cdot7)(9\cdot7)+\frac(4\cdot9)(7\cdot9)=\frac(49 )(63)+\frac(36)(63)=\frac(49+36)(63)=\frac(85)(63)
To subtract another fraction from one, you need:
- bring fractions to a common denominator;
- subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged.
Example: \frac(4)(15)-\frac(3)(5)=\frac(4)(15)-\frac(3\cdot3)(5\cdot3)=\frac(4)(15) -\frac(9)(15)=\frac(4-9)(15)=\frac(-5)(15)=-\frac(5)(3\cdot5)=-\frac(1)( 3)
If the original fractions initially have a common denominator, then point 1 (reduction to a common denominator) is skipped.
Converting a mixed number to an improper fraction and vice versa
mathematical operation. Converting a mixed number to an improper fraction and vice versa
To convert a mixed fraction to an improper one, it is enough to sum the whole part of the mixed fraction with the fractional part. The result of such a sum will be an improper fraction, the numerator of which is equal to the sum of the product of the integer part and the denominator of the fraction with the numerator of the mixed fraction, and the denominator remains the same. For example, 2\frac(6)(11)=2+\frac(6)(11)=\frac(2\cdot11)(11)+\frac(6)(11)=\frac(2\cdot11+ 6)(11)=\frac(28)(11)
To convert an improper fraction to a mixed number:
- divide the numerator of a fraction by its denominator;
- write the remainder of the division into the numerator, and leave the denominator the same;
- write the result of the division as an integer part.
For example, the fraction \frac(23)(4) . When dividing 23:4=5.75, that is, the integer part is 5, the remainder of the division is 23-5*4=3. Then the mixed number will be written: 5\frac(3)(4) . \frac(23)(4)=\frac(5\cdot4+3)(4)=5\frac(3)(4)
Converting a Decimal to a Common Fraction
mathematical operation. Converting a Decimal to a Common Fraction
In order to turn decimal in the ordinary, you need:
- take the n-th power of ten as a denominator (here n is the number of decimal places);
- as a numerator, take the number after the decimal point (if the integer part of the original number is not equal to zero, then take all leading zeros as well);
- the non-zero integer part is written in the numerator at the very beginning; the zero integer part is omitted.
Example 1: 0.0089=\frac(89)(10000) (4 decimal places, so the denominator 10 4 =10000, since the integer part is 0, the numerator is the number after the decimal point without leading zeros)
Example 2: 31.0109=\frac(310109)(10000) (in the numerator we write the number after the decimal point with all zeros: "0109", and then we add the integer part of the original number "31" before it)
If the integer part of a decimal fraction is different from zero, then it can be converted to a mixed fraction. To do this, we translate the number into common fraction as if the integer part is equal to zero (points 1 and 2), and the integer part is simply rewritten before the fraction - this will be the integer part of the mixed number. Example:
3.014=3\frac(14)(100)
To convert an ordinary fraction to a decimal, it is enough to simply divide the numerator by the denominator. Sometimes you get an infinite decimal. In this case, it is necessary to round to the desired decimal place. Examples:
\frac(401)(5)=80.2;\quad \frac(2)(3)\approx0.6667
Multiplication and division of fractions
mathematical operation. Multiplication and division of fractions
To multiply two common fractions, you need to multiply the numerators and denominators of the fractions.
\frac(5)(9)\cdot\frac(7)(2)=\frac(5\cdot7)(9\cdot2)=\frac(35)(18)
To divide one common fraction by another, you need to multiply the first fraction by the reciprocal of the second ( reciprocal is a fraction in which the numerator and denominator are reversed.
\frac(5)(9):\frac(7)(2)=\frac(5)(9)\cdot\frac(2)(7)=\frac(5\cdot2)(9\cdot7)= \frac(10)(63)
If one of the fractions is a natural number, then the above multiplication and division rules remain in force. Just keep in mind that an integer is the same fraction, the denominator of which equal to one. For example: 3:\frac(3)(7)=\frac(3)(1):\frac(3)(7)=\frac(3)(1)\cdot\frac(7)(3)= \frac(3\cdot7)(1\cdot3)=\frac(7)(1)=7
496. To find X, if:
497. 1) If you add 10 1/2 to 3/10 of an unknown number, you get 13 1/2. Find an unknown number.
2) If you subtract 10 1/2 from 7/10 of an unknown number, you get 15 2/5. Find an unknown number.
498 *. If you subtract 10 from 3 / 4 of an unknown number and multiply the resulting difference by 5, you get 100. Find the number.
499 *. If an unknown number is increased by 2/3 of it, you get 60. What is this number?
500 *. If we add the same amount to an unknown number, and even 20 1/3, then we get 105 2/5. Find an unknown number.
501. 1) The yield of potatoes with a square-nest planting is on average 150 centners per 1 ha, and with a normal planting 3/5 of this amount. How many more potatoes can be harvested from an area of 15 hectares if potatoes are planted in a square-nest way?
2) An experienced worker made 18 parts in 1 hour, and an inexperienced worker 2/3 of this amount. How many more parts can an experienced worker produce in a 7-hour working day?
502. 1) Pioneers assembled within three days 56 kg of different seeds. On the first day, 3/14 of the total amount was collected, on the second, one and a half times more, and on the third day, the rest of the grain. How many kilograms of seeds did the pioneers collect on the third day?
2) When grinding wheat, it turned out: flour 4/5 of the total amount of wheat, semolina - 40 times less than flour, and the rest is bran. How much flour, semolina and bran separately did you get when grinding 3 tons of wheat?
503. 1) Three garages fit 460 cars. The number of cars that fit in the first garage is 3/4 of the number of cars that fit in the second, and in the third garage there are 1 1/2 times as many cars as in the first. How many cars fit in each garage?
2) The plant, which has three workshops, employs 6,000 workers. The number of workers in the second workshop is 1 1/2 times less than in the first, and the number of workers in the third workshop is 5/6 of the number of workers in the second workshop. How many workers are in each shop?
504. 1) First, 2/5 was poured from the tank with kerosene, then 1/3 of the total kerosene, and after that 8 tons of kerosene remained in the tank. How much kerosene was in the tank originally?
2) The cyclists raced for three days. On the first day they covered 4/15 of the entire journey, on the second day they covered 2/5, and on the third day the remaining 100 km. How far did the cyclists travel in three days?
505. 1) The icebreaker made its way through the ice field for three days. On the first day he covered 1/2 of the total distance, on the second day 3/5 of the remaining distance, and on the third day the remaining 24 km. Find the distance traveled by the icebreaker in three days.
2) Three detachments of schoolchildren planted trees for landscaping the village. The first detachment planted 7/20 of all the trees, the second 5/8 of the remaining trees, and the third the remaining 195 trees. How many trees did the three teams plant in total?
506. 1) A combine harvester harvested wheat from one plot in three days. On the first day he harvested from 5/18 of the total area of the plot, on the second day from 7/13 of the remaining area, and on the third day from the remaining area of 30 1/2 hectares. On average, 20 centners of wheat were harvested from each hectare. How much wheat was harvested in the entire plot?
2) On the first day, the participants of the rally covered 3/11 of the entire path, on the second day 7/20 of the remaining path, on the third day 5/13 of the new remainder, and on the fourth day, the remaining 320 km. How long is the rally route?
507. 1) On the first day, the car covered 3/8 of the entire distance, on the second day 15/17 of what it passed on the first, and on the third day the remaining 200 km. How much gasoline was consumed if the car consumes 1 3/5 kg of gasoline for 10 km of travel?
2) The city consists of four districts. And in the first district live 4/13 of all the inhabitants of the city, in the second 5/6 of the inhabitants of the first district, in the third 4/11 of the inhabitants of the first; two districts combined, and the fourth district is home to 18,000 people. How much bread does the entire population of the city need for 3 days, if on average one person consumes 500 g per day?
508. 1) The tourist walked on the first day 10/31 of the entire path, on the second 9/10 of what he walked on the first day, and on the third the rest of the path, and on the third day he walked 12 km more than on the second day. How many kilometers did the tourist walk on each of the three days?
2) The car traveled all the way from city A to city B in three days. On the first day, the car covered 7/20 of the entire distance, on the second day, 8/13 of the remaining distance, and on the third day, the car covered 72 km less than on the first day. What is the distance between cities A and B?
509. 1) The executive committee allotted land to the workers of three factories for garden plots. The first plant was assigned 9/25 of the total number of plots, the second plant 5/9 of the number of plots allocated for the first, and the third - the rest of the plots. How many plots were allotted to the workers of three factories if the first plant was given 50 fewer plots than the third?
2) The plane delivered a shift of winterers to the polar station from Moscow in three days. On the first day he flew 2/5 of the entire path, on the second - 5/6 of the path he traveled on the first day, and on the third day he flew 500 km less than on the second day. How far did the plane fly in three days?
510. 1) The plant had three workshops. The number of workers in the first workshop is 2/5 of all factory workers; in the second workshop there are 1 1/2 times fewer workers than in the first, and in the third workshop there are 100 more workers than in the second. How many workers are in the factory?
2) The collective farm includes residents of three neighboring villages. The number of families in the first village is 3/10 of all the families of the collective farm; in the second village the number of families is 1 1/2 times greater than in the first, and in the third village the number of families is 420 fewer than in the second. How many families are on the collective farm?
511. 1) The Artel spent in the first week 1/3 of its stock of raw materials, and in the second 1/3 of the remainder. How much raw material is left in the artel if in the first week the consumption of raw materials was 3/5 tons more than in the second week?
2) Of the imported coal for heating the house in the first month, 1/6 of it was spent, and in the second month - 3/8 of the remainder. How much coal is left for heating the house if 1 3/4 more was used in the second month than in the first month?
512. 3/5 of the entire land of the collective farm is allocated for sowing grain, 13/36 of the rest is occupied by vegetable gardens and meadows, the rest of the land is forested, and the sown area of the collective farm is 217 hectares more than the forest area, 1/3 of the land allotted for sowing grain is sown with rye, and the rest is wheat. How many hectares of land did the collective farm sow with wheat and how many with rye?
513. 1) The tram route is 14 3/8 km long. During this route, the tram makes 18 stops, spending on average up to 1 1/6 minutes per stop. The average tram speed along the entire route is 12 1/2 km per hour. How long does it take for a tram to make one trip?
2) Bus route 16 km. During this route, the bus makes 36 stops of 3/4 min. each on average. The average bus speed is 30 km per hour. How long does it take for a bus to make one route?
514*. 1) It is now 6 o'clock. evenings. What part is the remaining part of the day from the past and what part of the day is left?
2) A steamboat travels downstream between two cities in 3 days. and back the same distance in 4 days. How many days will the rafts float from one city to another?
515. 1) How many boards will be used to lay the floor in a room whose length is 6 2/3 m, width h 5 1/4 m, if the length of each board is 6 2/3 m, and its width is 3/80 of the length?
2) A rectangular platform has a length of 45 1/2 m, and its width is 5/13 of the length. This area is bordered by a path 4/5 m wide. Find the area of the path.
516. Find the mean arithmetic numbers:
517. 1) The arithmetic mean of two numbers 6 1 / 6 . One of the numbers 3 3 / 4 . Find another number.
2) The arithmetic mean of two numbers is 14 1 / 4 . One of these numbers is 15 5 / 6 . Find another number.
518. 1) The freight train was on the road for three hours. In the first hour he walked 36 1/2 km, in the second 40 km, and in the third 39 3/4 km. Find the average speed of the train.
2) The car traveled 81 1/2 km in the first two hours, and 95 km in the next 2 1/2 hours. How many kilometers did he walk on average per hour?
519. 1) The tractor driver completed the task of plowing the land in three days. On the first day he plowed 12 1/2 ha, on the second day 15 3/4 ha, and on the third day 14 1/2 ha. How many hectares of land did a tractor driver plow on average per day?
2) A detachment of schoolchildren, making a three-day tourist trip, was on the way on the first day 6 1 / 3 hours, on the second 7 hours. and on the third day, 4 2/3 hours. How many hours on average were students on the road every day?
520. 1) Three families live in the house. The first family for lighting the apartment has 3 light bulbs, the second 4 and the third 5 bulbs. How much should each family pay for electricity if all the lamps were the same and the total electricity bill (for the whole house) was 7 1/5 rubles?
2) The polisher rubbed the floors in the apartment where three families lived. The first family had a living area of 36 1/2 sq. m, the second in 24 1/2 sq. m, and the third - in 43 sq. m. For all the work was paid 2 rubles. 08 kop. How much did each family pay?
521. 1) In the garden plot, potatoes were harvested from 50 bushes, 1 1/10 kg from one bush, from 70 bushes, 4/5 kg from one bush, from 80 bushes, 9/10 kg from one bush. How many kilograms of potatoes are harvested on average from each bush?
2) A field-growing team on an area of 300 ha received a harvest of 20 1/2 centners of winter wheat per 1 ha, from 80 hectares 24 centners per 1 ha, and from 20 hectares - 28 1/2 centners per 1 ha. What is the average yield in a brigade from 1 hectare?
522. 1) The sum of two numbers is 7 1 / 2 . One number is greater than another by 4 4 / 5 . Find these numbers.
2) If we add the numbers expressing the width of the Tatar and Kerch Straits together, we get 11 7 / 10 km. The Tatar Strait is 3 1/10 km wider than the Kerch Strait. What is the width of each strait?
523. 1) The sum of three numbers is 35 2 / 3 . The first number is 5 1/3 greater than the second and 3 5/6 greater than the third. Find these numbers.
2) Islands New Earth, Sakhalin and Severnaya Zemlya together occupy an area of 196 7/10 thousand square meters. km. The area of Novaya Zemlya is 44 1/10 thousand square meters. km more area Severnaya Zemlya and 5 1/5 thousand square meters. km larger than the area of Sakhalin. What is the area of each of the listed islands?
524. 1) The apartment consists of three rooms. The area of the first room is 24 3/8 sq. m and is 13/36 of the entire area of the apartment. The area of the second room is 8 1/8 sq. m more than the area of the third. What is the area of the second room?
2) The cyclist during the three-day competition on the first day traveled 3 1/4 hours, which was 13/43 of the total travel time. On the second day he rode 1 1/2 hours more than on the third day. How many hours did the cyclist travel on the second day of the competition?
525. Three pieces of iron weigh together 17 1/4 kg. If the weight of the first piece is reduced by 1 1/2 kg, the weight of the second by 2 1/4 kg, then all three pieces will have the same weight. How much did each piece of iron weigh?
526. 1) The sum of two numbers is 15 1 / 5 . If the first number is reduced by 3 1/10 and the second is increased by 3 1/10, then these numbers will be equal. What is each number equal to?
2) There were 38 1/4 kg of cereal in two boxes. If 4 3/4 kg of cereals are poured from one box into another, then in both boxes there will be equal amounts of cereals. How many cereals are in each box?
527 . 1) The sum of two numbers is 17 17 / 30 . If you subtract 5 1/2 from the first number and add to the second, then the first will still be more than the second by 2 17/30. Find both numbers.
2) Two boxes contain 24 1/4 kg of apples. If 3 1/2 kg are transferred from the first box to the second, then in the first there will still be 3/5 kg more apples than in the second. How many kilograms of apples are in each box?
528 *. 1) The sum of two numbers is 8 11/14, and their difference is 2 3/7. Find these numbers.
2) The boat was moving along the river at a speed of 15 1/2 km per hour, and against the current 8 1/4 km per hour. What is the speed of the river?
529. 1) There are 110 cars in two garages, and in one of them there are 1 1/5 times more than in the other. How many cars are in each garage?
2) The living area of an apartment consisting of two rooms is 47 1/2 sq. m. The area of one room is 8/11 of the area of the other. Find the area of each room.
530. 1) An alloy consisting of copper and silver weighs 330 g. The weight of copper in this alloy is 5/28 of the weight of silver. How much silver and how much copper is in the alloy?
2) The sum of two numbers is 6 3 / 4 , and the quotient is 3 1 / 2 . Find these numbers.
531. The sum of three numbers is 22 1 / 2 . The second number is 3 1/2 times and the third is 2 1/4 times the first. Find these numbers.
532. 1) The difference of two numbers is 7; the quotient of dividing the larger number by the smaller is 5 2 / 3 . Find these numbers.
2) The difference of two numbers is 29 3/8, and their multiple ratio is 8 5/6. Find these numbers.
533. In a class, the number of absent students is 3/13 of the number of those present. How many students are in the class according to the list, if there are 20 more people present than absent?
534. 1) The difference of two numbers is 3 1 / 5 . One number is 5/7 of another. Find these numbers.
2) Father older than son for 24 years. The number of the son's years is 5/13 of the father's years. How old is the father and how old is the son?
535. The denominator of a fraction is 11 more than its numerator. What is a fraction equal to if its denominator is 3 3/4 times the numerator?
No. 536 - 537 orally.
536. 1) The first number is 1/2 of the second. How many times greater is the second number than the first?
2) The first number is 3/2 of the second. What part of the first number is the second number?
537. 1) 1/2 of the first number is equal to 1/3 of the second number. What part of the first number is the second number?
2) 2/3 of the first number is equal to 3/4 of the second number. What part of the first number is the second number? What part of the second number is the first?
538. 1) The sum of two numbers is 16. Find these numbers if 1/3 of the second number is equal to 1/5 of the first.
2) The sum of two numbers is 38. Find these numbers if 2/3 of the first number is equal to 3/5 of the second.
539 *. 1) Two boys picked 100 mushrooms together. 3/8 of the number of mushrooms picked by the first boy is numerically equal to 1/4 of the number of mushrooms picked by the second boy. How many mushrooms did each boy collect?
2) The institution employs 27 people. How many men and how many women work if 2/5 of all men are equal to 3/5 of all women?
540 *. Three boys bought a volleyball. Determine the contribution of each boy, knowing that 1/2 of the contribution of the first boy is equal to 1/3 of the contribution of the second, or 1/4 of the contribution of the third, and that the contribution of the third boy is 64 kopecks more than the contribution of the first.
541 *. 1) One number is 6 greater than another. Find these numbers if 2/5 of one number is equal to 2/3 of another.
2) The difference of two numbers is 35. Find these numbers if 1/3 of the first number is equal to 3/4 of the second number.
542. 1) The first brigade can complete some work in 36 days, and the second in 45 days. How many days will it take both teams working together to complete this task?
2) A passenger train travels the distance between two cities in 10 hours, and a freight train travels this distance in 15 hours. Both trains left these cities at the same time towards each other. In how many hours will they meet?
543. 1) A fast train travels the distance between two cities in 6 1/4 hours, and a passenger train in 7 1/2 hours. In how many hours will these trains meet if they leave both cities at the same time towards each other? (Round answer to the nearest 1 hour.)
2) Two motorcyclists left two cities at the same time towards each other. One motorcyclist can travel the entire distance between these cities in 6 hours, and another in 5 hours. How many hours after the departure will the motorcyclists meet? (Round answer to the nearest 1 hour.)
544. 1) Three cars of different carrying capacity can carry some cargo, working separately: the first in 10 hours, the second in 12 hours. and the third in 15 hours In how many hours can they move the same cargo by working together?
2) Two trains leave two stations simultaneously towards each other: the first train covers the distance between these stations in 12 1/2 hours, and the second in 18 3/4 hours. How many hours after leaving will the trains meet?
545. 1) There are two taps connected to the bath. Through one of them, the bath can be filled in 12 minutes, through the other 1 1/2 times faster. How many minutes will it take to fill 5/6 of the entire bath if both taps are opened at once?
2) Two typists must retype the manuscript. The first woman can do this job in 3 1/3 days, and the second one 1 1/2 times faster. In how many days will both typists complete the work if they work at the same time?
546. 1) The pool is filled with the first pipe in 5 hours, and through the second pipe it can be emptied in 6 hours After how many hours will the entire pool be filled if both pipes are opened at the same time?
Instruction. In an hour, the pool is filled to (1 / 5 - 1 / 6 of its capacity.)
2) Two tractors plowed the field in 6 hours. The first tractor, working alone, could plow this field in 15 hours How many hours would it take the second tractor to plow this field, working alone?
547 *. Two trains leave two stations at the same time towards each other and meet after 18 hours. after its release. How long does it take the second train to travel the distance between stations if the first train travels this distance in 1 day and 21 hours?
548 *. The pool is filled with two pipes. First, the first pipe was opened, and then after 3 3/4 hours, when half the pool was full, the second pipe was opened. After 2 1/2 hours of working together, the pool filled up. Determine the capacity of the pool if 200 buckets of water per hour were poured through the second pipe.
549. 1) A courier train left Leningrad for Moscow, which travels 1 km in 3/4 minutes. 1/2 hour after the departure of this train, a fast train left Moscow for Leningrad, the speed of which was equal to 3/4 of the speed of the courier. How far will the trains be from each other 2 1/2 hours after the departure of the courier train, if the distance between Moscow and Leningrad is 650 km?
2) From the collective farm to the city 24 km. A truck has left the collective farm and travels 1 km in 2 1/2 minutes. After 15 min. after the departure of this car from the city, a cyclist left the collective farm, at a speed half that of a truck. How long will it take for the cyclist to meet the truck after leaving?
550. 1) A pedestrian came out of one village. 4 1/2 hours after the pedestrian left, a cyclist left in the same direction, whose speed is 2 1/2 times the speed of the pedestrian. In how many hours after the pedestrian leaves, the cyclist will overtake him?
2) A fast train travels 187 1/2 km in 3 hours, and a freight train 288 km in 6 hours. 7 1/4 hours after the departure of the freight train, an ambulance leaves in the same direction. How long will it take for the fast train to overtake the freight train?
551. 1) From two collective farms, through which the road to the district center passes, two collective farmers left at the same time to the district on horseback. The first of them traveled 8 3/4 km per hour, and the second 1 1/7 times the first. The second collective farmer overtook the first in 3 4/5 hours. Determine the distance between collective farms.
2) 26 1/3 hours after the departure of the Moscow-Vladivostok train, the average speed of which is 60 km per hour, the TU-104 aircraft took off in the same direction, at a speed 14 1/6 times the speed of the train. How many hours after the flight will the plane overtake the train?
552. 1) The distance between cities along the river is 264 km. This distance the steamer traveled downstream in 18 hours, spending 1/12 of this time on stops. The speed of the river is 1 1/2 km per hour. How long would it take a steamer to travel 87 km without stopping? standing water?
2) The motorboat traveled 207 km downstream in 13 1/2 hours, spending 1/9 of that time on stops. The speed of the river is 1 3/4 km per hour. How many miles can this boat travel in still water in 2 1/2 hours?
553. The boat on the reservoir covered a distance of 52 km without stopping in 3 hours and 15 minutes. Further, going along the river against the current, the speed of which is 1 3 / 4 km per hour, this boat traveled 28 1 / 2 km in 2 1 / 4 hours, making 3 equal stops in the process. How many minutes did the boat stop at each stop?
554. From Leningrad to Kronstadt at 12 noon. the next day a steamboat set out and covered the entire distance between these cities in 1 1/2 hours. On the way, he met another steamer that left Kronstadt for Leningrad at 12:18. and walking at a speed 1 1/4 times greater than the first. At what time did the two ships meet?
555. The train had to cover a distance of 630 km in 14 hours. Having covered 2/3 of this distance, he was delayed for 1 hour and 10 minutes. At what speed must he continue his journey in order to arrive at his destination without delay?
556. At 4 o'clock 20 min. in the morning a freight train left Kyiv for Odessa average speed 31 1/5 km per hour. After some time, a mail train left Odessa to meet it, the speed of which is 1 17/39 times the speed of the freight train, and met with the freight train 6 1/2 hours after its departure. At what time did the postal train leave Odessa if the distance between Kiev and Odessa is 663 km?
557*. The clock shows noon. How long does it take for the hour and minute hands to coincide?
558. 1) The factory has three workshops. The number of workers in the first workshop is 9/20 of all the workers of the plant, in the second workshop there are 1 1/2 times fewer workers than in the first, and in the third workshop there are 300 workers less than in the second. How many workers are in the factory?
2) There are three secondary schools in the city. The number of students in the first school is 3/10 of all students in these three schools; in the second school there are 1 1/2 times more students than in the first, and in the third school there are 420 students less than in the second. How many students are in the three schools?
559. 1) Two combine operators worked at the same site. After one combiner harvested 9/16 of the entire area, and the second 3/8 of the same area, it turned out that the first combiner harvested 97 1/2 hectares more than the second. On average, 32 1/2 centners of grain were threshed from each hectare. How many quintals of grain did each combine thresh?
2) Two brothers bought a camera. One had 5/8, and the second had 4/7 of the cost of the camera, and the first had 2 rubles. 25 kop. more than the second. Each paid half the cost of the apparatus. How much money does each have?
560. 1) From city A to city B, the distance between them is 215 km, a car left at a speed of 50 km per hour. At the same time, a truck left city B for city A. How many kilometers did the car travel before meeting the truck if the speed of the truck per hour was 18/25 of the speed of the car?
2) Between cities A and B 210 km. A car left town A for town B. At the same time, a truck left city B for city A. How many kilometers did the truck travel before meeting with the car if the car was moving at a speed of 48 km per hour, and the speed of the truck per hour was 3/4 of the speed of the car?
561. The collective farm harvested wheat and rye. Wheat was sown 20 hectares more than rye. The total harvest of rye amounted to 5/6 of the total harvest of wheat with a yield of 20 centners per 1 ha for both wheat and rye. The collective farm sold 7/11 of the entire harvest of wheat and rye to the state, and left the rest of the grain to meet its needs. How many trips did the two-ton trucks need to make to take out the grain sold to the state?
562. Rye and wheat flour was brought to the bakery. The weight of wheat flour was 3/5 of the weight of rye flour, and rye flour was brought 4 tons more than wheat. How much wheat and how much rye bread will be baked by the bakery from this flour, if the baked goods are 2/5 of all flour?
563. Within three days, a team of workers completed 3/4 of the entire work to repair the highway between the two collective farms. On the first day, 2 2/5 km of this highway was repaired, on the second day 1 1/2 times more than on the first, and on the third day 5/8 of what was repaired in the first two days together. Find the length of the highway between collective farms.
564. Fill in the empty spaces in the table, where S is the area of the rectangle, but- the base of the rectangle, a h-height (width) of the rectangle.
565. 1) The length of a rectangular plot of land is 120 m, and the width of the plot is 2/5 of its length. Find the perimeter and area of the plot.
2) The width of the rectangular section is 250 m, and its length is 1 1/2 times the width. Find the perimeter and area of the plot.
566. 1) The perimeter of a rectangle is 6 1/2 dm, its base is 1/4 dm more than the height. Find the area of this rectangle.
2) The perimeter of a rectangle is 18 cm, its height is 2 1/2 cm less than the base. Find the area of the rectangle.
567. Calculate the areas of the figures shown in Figure 30, dividing them into rectangles and finding the dimensions of the rectangle by measuring.
568. 1) How many sheets of dry plaster will be required to upholster the ceiling of a room whose length is 4 1/2 m and the width is 4 m, if the dimensions of the plaster sheet are 2 m x l 1/2 m?
2) How many boards 4 1/2 l long and 1/4 m wide will be required to lay a floor that is 4 1/2 m long and 3 1/2 m wide?
569. 1) A rectangular plot 560 m long and 3/4 of its length wide was sown with beans. How many seeds were required to sow the plot if 1 centner was sown per 1 hectare?
2) A wheat crop was harvested from a rectangular field at 25 centners per 1 ha. How much wheat was harvested from the whole field if the field is 800 m long and 3/8 of its length wide?
570 . 1) A rectangular plot of land, having a length of 78 3/4 m and a width of 56 4/5 m, is built up so that 4/5 of its area is occupied by buildings. Determine the area of land under the buildings.
2) On a rectangular plot of land, the length of which is 9/20 km, and the width is 4/9 of its length, the collective farm proposes to plant a garden. How many trees will be planted in this garden if, on average, an area of 36 square meters is required for each tree?
571. 1) For normal daylight illumination of the room, it is necessary that the area of \u200b\u200ball windows be at least 1/5 of the floor area. Determine if there is enough light in a room that is 5 1/2 m long and 4 m wide. Does the room have one window measuring 1 1/2 m x 2 m?
2) Using the condition of the previous problem, find out if there is enough light in your classroom.
572. 1) The barn measures 5 1/2 m x 4 1/2 m x 2 1/2 m. m of hay weighs 82 kg?
2) The woodpile has the shape of a rectangular parallelepiped, the dimensions of which are 2 1/2 m x 3 1/2 m x 1 1/2 m. What is the weight of the woodpile if 1 cu. m of firewood weighs 600 kg?
573. 1) A rectangular aquarium is filled with water up to 3/5 of the height. The length of the aquarium is 1 1/2 m, the width is 4/5 m, the height is 3/4 m. How many liters of water are poured into the aquarium?
2) The pool, having the shape of a rectangular parallelepiped, has a length of 6 1/2 m, a width of 4 m and a height of 2 m. The pool is filled with water up to 3/4 of its height. Calculate the amount of water poured into the pool.
574. A fence is to be built around a rectangular piece of land 75 m long and 45 m wide. How many cubic meters of boards should go to his device if the thickness of the board is 2 1/2 cm, and the height of the fence should be 2 1/4 m?
575. 1) What is the angle between the minute hand and the hour hand at 13:00? at 15 o'clock? at 17 o'clock? at 21 o'clock? at 23:30?
2) By how many degrees will the hour hand turn in 2 hours? 5 o'clock? 8 o'clock? 30 minutes.?
3) How many degrees does an arc equal to half a circle contain? 1/4 circle? 1/24 circle? 5 / 24 circles?
576. 1) Draw with a protractor: a) a right angle; b) an angle of 30°; c) an angle of 60°; d) an angle of 150°; e) an angle of 55°.
2) Measure the angles of the figure with a protractor and find the sum of all the angles of each figure (Fig. 31).
577. Run actions:
578. 1) A semicircle is divided into two arcs, one of which is 100° larger than the other. Find the magnitude of each arc.
2) A semicircle is divided into two arcs, one of which is 15° less than the other. Find the magnitude of each arc.
3) The semicircle is divided into two arcs, of which one is twice the other. Find the magnitude of each arc.
4) The semicircle is divided into two arcs, of which one is 5 times smaller than the other. Find the magnitude of each arc.
579. 1) The chart "Literacy of the population in the USSR" (Fig. 32) shows the number of literate per hundred people of the population. According to the diagram and its scale, determine the number of literate men and women for each of the indicated years.
Record the results in a table:
2) Using the data of the diagram "Soviet envoys to space" (Fig. 33), make up tasks.
580. 1) According to the sector diagram "Daily routine for a student of grade V" (Fig. 34), fill in the table and answer the questions: what part of the day is devoted to sleep? for homework? to school?
2) Build a pie chart about the mode of your day.
1º. Integers are the numbers used in counting. The set of all natural numbers is denoted by N, i.e. N=(1, 2, 3, …).
Shot is called a number consisting of several fractions of one. Common fraction is called a number of the form where the natural number n shows how many equal parts a unit is divided, and a natural number m shows how many such equal parts are taken. Numbers m And n are called respectively numerator And denominator fractions.
If the numerator is less than the denominator, then the fraction is called correct; If the numerator is equal to or greater than the denominator, then the fraction is called wrong. A number that consists of an integer and a fractional part is called mixed number.
For example, 
- regular fractions 
- improper ordinary fractions, 1 - mixed number.
2º. When performing operations on ordinary fractions, remember the following rules:
1)Basic property of a fraction. If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then a fraction equal to the given one will be obtained.
For example, a) 
; b) 
.
Dividing the numerator and denominator of a fraction by their common divisor, which is different from one, is called fraction reduction.
2) In order to represent a mixed number as an improper fraction, you need to multiply its integer part by the denominator of the fractional part and add the numerator of the fractional part to the resulting product, write the resulting amount as the numerator of the fraction, and leave the denominator the same.
Similarly, any natural number can be written as an improper fraction with any denominator.
For example, a) 
, because 
; b) 
etc.
3) In order to write an improper fraction as a mixed number (i.e., select an integer part from an improper fraction), you need to divide the numerator by the denominator, take the quotient as the integer part, the remainder as the numerator, leave the denominator the same.
For example, a) 
, since 200: 7 = 28 (remaining 4); b) 
, since 20: 5 = 4 (remaining 0).
4) To bring fractions to the lowest common denominator, you need to find the least common multiple (LCM) of the denominators of these fractions (it will be their least common denominator), divide the least common denominator by the denominators of these fractions (i.e. find additional factors for fractions) , multiply the numerator and denominator of each fraction by its additional factor.
For example, let's take fractions 
to the lowest common denominator:
,
,
;
630: 18 = 35, 630: 10 = 63, 630: 21 = 30.
Means, 
;
;
.
5) Rules for arithmetic operations on ordinary fractions:
a) Addition and subtraction of fractions with the same denominators is performed according to the rule:
.
b) Addition and subtraction of fractions with different denominators is carried out according to the rule a), having previously reduced the fractions to the lowest common denominator.
c) When adding and subtracting mixed numbers, you can convert them to improper fractions, and then follow the steps under the rules a) and b),
d) When multiplying fractions, use the rule:
.
e) To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor:
.
f) When multiplying and dividing mixed numbers, first convert them to improper fractions, and then use the rules d) and e).
3º. When solving examples for all actions with fractions, remember that the actions in brackets are performed first. Both inside and outside of parentheses, multiplication and division are performed first, followed by addition and subtraction.
Consider the implementation of the above rules with an example.
Example 1 Calculate: 
.
1) 
;
2) 
;
5) 
. Answer: 3.
1. The rule for adding fractions with the same denominators:
Example 1:
Example 2:
Rule for adding fractions with different denominators:
Example 1:
Example 2:
Here the denominators were not multiplied, but the least common factor a2 was taken.
(The denominator is the highest power of 2.)
Additional multiplier for the first fraction 1, for the second a.
2. The rule for subtracting fractions with the same denominators:
Rule for subtracting fractions with different denominators:
3. The rule for multiplying ordinary fractions:
4. The rule for dividing fractions:
Example:
Ordinary (simple) fraction. The numerator and denominator of a fraction.
Proper and improper fraction. Mixed number.
Incomplete quotient. Integer and fractional part. Reverse fractions. A part of a unit or several of its parts is called an ordinary or simple fraction. The number of equal parts into which the unit is divided is called the denominator, and the number of parts taken is called the numerator. The fraction is written as:
Here 3 is the numerator, 7 is the denominator.
If the numerator is less than the denominator, then the fraction is less than 1 and is called proper fraction
. If the numerator is equal to the denominator, then the fraction is 1. If the numerator is greater than the denominator, then the fraction is greater than 1. In both the latter cases, the fraction is called improper. If the numerator is divisible by the denominator, then this fraction is equal to the quotient: 63 / 7 = 9. If the division is performed with a remainder, then this improper fraction can be represented mixed number:
Here 9 - incomplete quotient(the integer part of the mixed number), 2 is the remainder (the numerator of the fractional part), 7 is the denominator.
Often it is necessary to solve the inverse problem - reverse a mixed number into a fraction. To do this, multiply the integer part of the mixed number by the denominator and add the numerator of the fractional part. This will be the numerator of an ordinary fraction, and the denominator remains the same. 
Reciprocals are two fractions whose product is 1. For example, 3/7 and 7/3; 15/1 and 1/15 etc.
Fraction expansion. Fraction reduction. Fraction comparison.
Reduction to a common denominator. Addition and subtraction fractions.
Multiplication of fractions. Division of fractions
Fraction expansion.The value of a fraction does not change if its numerator and denominator are multiplied by the same non-zero number by expanding the fraction. For example,
Fraction reduction. The value of a fraction does not change if its numerator and denominator are divided by the same non-zero number.. This transformation is calledfraction reduction. For example, 
Fraction comparison.Of two fractions with the same numerator, the larger one is the one with the smaller denominator:
Of two fractions with the same denominators, the one with the larger numerator is greater:
To compare fractions that have different numerators and denominators, you need to expand them to bring them to a common denominator.
EXAMPLE Compare two fractions:
The transformation used here is called reducing fractions to a common denominator.
Addition and subtraction of fractions.If the denominators of fractions are the same, then in order to add the fractions, you need to add their numerators, and in order to subtract the fractions, you need to subtract their numerators (in the same order). The resulting sum or difference will be the numerator of the result; the denominator will remain the same. If the denominators of the fractions are different, you must first reduce the fractions to a common denominator. When adding mixed numbers, their integer and fractional parts are added separately. When subtracting mixed numbers, we recommend that you first convert them to the form improper fractions, then subtract from one another, and after that again reduce the result, if necessary, to the form of a mixed number.
EXAMPLE
Multiplication of fractions.To multiply a number by a fraction means to multiply it by the numerator and divide the product by the denominator. Hence we have general rule multiplying fractions:to multiply fractions, you need to multiply their numerators and denominators separately and divide the first product by the second.
EXAMPLE 
Division of fractions. In order to divide a certain number by a fraction, it is necessary to multiply this number by its reciprocal. This rule follows from the definition of division (see the Arithmetic Operations section).
EXAMPLE 
Decimal. Whole part. Decimal point.
Decimals. Properties of decimal fractions.
Periodic decimal. Period
Decimalis the result of dividing one by ten, one hundred, one thousand, etc. parts. These fractions are very convenient for calculations, since they are based on the same positional system on which counting and notation of integers are built. Due to this, the notation and rules for decimals are actually the same as for integers. When writing decimal fractions, there is no need to mark the denominator, this is determined by the place that the corresponding figure occupies. First spelled whole part numbers, then put on the rightdecimal point. The first digit after the decimal point means the number of tenths, the second - the number of hundredths, the third - the number of thousandths, etc. The numbers after the decimal point are calleddecimal places.
EXAMPLE 
One of the advantages of decimal fractions is that they are easily reduced to the form of ordinary ones: the number after the decimal point (in our case 5047) is the numerator; the denominator is equal n -th degree 10, where n - the number of decimal places (in our case n = 4): 
If the decimal fraction does not contain an integer part, then a zero is placed before the decimal point:
Properties of decimal fractions.
1. Decimal does not change if zeros are added to the right:
2.
The decimal fraction does not change if you remove the zeros located
at the end of the decimal:
0.00123000 = 0.00123 .
Attention! You can not delete zeros located not at the end decimal!br />
These properties allow you to quickly multiply and divide decimals by 10, 100, 1000, and so on.
Periodic decimal contains an infinitely repeating group of digits called a period. The period is written in brackets. For example, 0.12345123451234512345… = 0.(12345).
EXAMPLE If we divide 47 by 11, we get 4.27272727… = 4.(27).
Multiplying decimals.
Division of decimals.
Addition and subtraction of decimal fractions. These operations are performed in the same way as addition and subtraction of integers. It is only necessary to write the corresponding decimal places one under the other.
EXAMPLE
Multiplying decimals. In the first step, we multiply the decimal fractions as whole numbers, without taking into account the decimal point. The following rule is then applied: the number of decimal places in the product is equal to the sum of the decimal places in all factors.
Remark : before putting the decimal point inYou can't drop zeros at the end of the product!
EXAMPLE
The sum of the numbers of decimal places in the factors is: 3 + 4 = 7. The sum of the digits in the product is 6. Therefore, you need to add one zero to the left: 0197056 and put a decimal point in front of it: 0.0197056.
Decimal division
Divide a decimal by an integer
If dividend is less than divisor, we write zero in the integer part of the quotient and put a decimal point after it. Then, without taking into account the decimal point of the dividend, we add the next digit of the fractional part to its integer part and again compare the resulting integer part of the dividend with the divisor. If the new number is again less than the divisor, put one more zero after the decimal point in the quotient and add the next digit of its fractional part to the integer part of the dividend. This process is repeated until the resulting dividend becomes greater than the divisor. After that, division is performed as for integers. If dividend is greater than or equal to the divisor, first we divide its integer part, write the result of division in private and put a decimal point. After that, the division continues, as in the case of integers.
EXAMPLE Divide 1.328 by 64.
Solution: 
Division of one decimal fraction by another.
First, we transfer the decimal points in the dividend and divisor by the number of decimal places in the divisor, that is, we make the divisor an integer. Now we perform the division, as in the previous case.
EXAMPLE Divide 0.04569 by 0.0006.
Solution: Move the decimal points 4 places to the right and divide 456.9 by 6:
In order to convert a decimal to a common fraction, you need to take the number after the decimal point as the numerator, and take the nth power of ten as the denominator (here n is the number of decimal places). The non-zero integer part is preserved in the common fraction; the zero integer part is omitted. For example: 
In order to convert an ordinary fraction to a decimal, it is necessary to divide the numerator by the denominator in accordance with the rules of division.
EXAMPLE Convert 5 / 8 to a decimal.
Solution: Dividing 5 by 8 gives 0.625. (Check, please!).
In most cases, this process can continue indefinitely. Then it is impossible to accurately convert an ordinary fraction to a decimal. But in practice this is never required. The division is aborted if the decimal places of interest have already been received.
EXAMPLE Convert 1/3 to a decimal.
Solution: Dividing 1 by 3 will be infinite: 1:3 = 0.3333… .
Check it out please!
