Topic: Ordinary fractions (theory and practice with test tasks). View the contents of the document "Reducing fractions to the lowest common denominator"

>>Math: Reducing fractions to a common denominator

10. Reducing fractions to a common denominator

Let's multiply the numerator and denominator of the fraction by the same number 2. We get a fraction equal to it, i.e. They say that we corrected the fraction to a new denominator 8. The fraction can be reduced to any multiple of the denominator of this fraction.

The number by which the denominator of a fraction must be multiplied to obtain a new denominator is called an additional factor.

When converting a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.

Example 1. Let's reduce the fraction to the denominator 35.
Solution. The number 35 is a multiple of 7, since 35:7 = 5. An additional factor is the number 5. Multiply the numerator and denominator of this decimals by 5, we get

Any two fractions can be reduced to the same denominator, or otherwise to a common denominator.
For example,
The common denominator of fractions can be any common multiple of their denominators (for example, the product of the denominators).

Fractions usually reduce to their lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.

Example 2. Reduce to the lowest common denominator of the fraction
Solution. The least common multiple of 4 and 6 is 12.

To convert a fraction to a denominator of 12, you must multiply the numerator and denominator of this fraction by an additional
multiplier 3 (12:4 = 3). We get
To bring a fraction to a denominator of 12, you need to multiply the numerator and denominator of this fraction by an additional factor 2 (12:6=2).

We get
So A

To reduce fractions to their lowest common denominator:

1) find the least common multiple of the denominators of these fractions, it will be their least common denominator;

2) divide the lowest common denominator by the denominators of these fractions, i.e. find an additional factor for each fraction;

3) multiply the numerator and denominator of each fraction by its additional factor.

In more complex cases, the lowest common denominator and additional factors are found using prime factorization.

Example 3. Let's reduce the fractions to their lowest common denominator.

Solution. Let's decompose the denominators of these fractions into simple factors: 60=2 2 3 5; 168 = 2 2 2 3 7. Find the lowest common denominator:

2 2 2 3 5 7 = 840.
An additional factor for a fraction is the product of 2 7, i.e. those factors that need to be added to the expansion numbers 60 to get a common denominator expansion of 840. Therefore

? What new denominator can this fraction be reduced to? Is it possible to reduce a fraction to a denominator of 35? to the denominator 25? What number is called an additional factor? How to find an additional multiplier? What number can serve as the common denominator of two fractions? How do you reduce fractions to their lowest common denominator?

TO 264. Give the fraction:

265. Express in minutes, and then in sixtieths of an hour:

266. How much does it contain:

267. Reduce fractions and then bring them to the denominator 24.

268. Is it possible to reduce the fraction to denominator 36:

269. Is it possible to represent it in the form decimal:

270. Write as a decimal fraction, giving:

271. Write as a decimal:

272. Reduce to the lowest common denominator of the fraction:

273. Calculate conventionally:

274. Find the missing numbers if x=0.8; 0.16; 0.06; 1:

275. What number should 24 be multiplied by? 8; 16; 6; 12 to get 48?

276. Using a protractor, divide one circle into 6 and the other into 3 equal arcs. Construct the polygons shown in drawing 14. Each of these polygons has equal sides and equal angles. Such polygons are called regular. Consider whether a rectangle is a regular polygon; square.

277 Abbreviate:

278. Find the greatest common divisor of the numerator and denominator and reduce the fraction:

279. At what value of x is the equality true:

280. A beetle crawls up a tree trunk (Fig. 15) at a speed of 6 cm/s. A caterpillar is crawling down the same tree. Now she is 60 cm below the beetle. At what speed does the caterpillar crawl if after 5 s the distance between it and the beetle is 100 cm?

281. The Vega-1 spacecraft was moving towards Halley's comet at a speed of 34 km/s, and the comet itself was moving towards it at a speed of 46 km/s. What was the distance between them 15 minutes before the meeting? "

282. Shorten:

284 Follow the steps and check your calculations using a micro calculator:

1) 111 - ((0,9744:0,24 +1,02) 2,5 - 2,7 5);
2) 200 - ((9,08 - 2,6828:0,38) 8,5 + 0,84).

D 285. Give the fraction:

286. Express as a decimal:

287. Reduce fractions and then bring them to the denominator 60.

288. Reduce fractions to lowest common denominator:

289. From two points, the distance between which is 40 km, a pedestrian and a cyclist set off towards each other at the same time. The speed of a cyclist is 4 times the speed of a pedestrian. Find the speeds of the pedestrian and the cyclist if it is known that they met 2.5 hours after they left.

290. From two points, the distance between which is 210 km, two electric trains left simultaneously towards each other. The speed of one of them is 5 km/h more than the speed of the other. Find the speed of each electric train if they met 2 hours after they left.

291. Follow these steps:

a) 62.3+(50.1 - 3.3 (96.96:9.6)) 1.8;
b) 51.6 + (70.2 - 4.4 (73.73:7.3)) 1.6.

N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school

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2.1 The concept of an ordinary fraction. Basic properties of fractions. Comparison of fractions.

Fractional numbers arise when one object (orange, tomato, apple, piece of paper, cake) or unit of measurement (meter, hour, kilogram) is divided into several equal parts.

Fractional numbers can be written using ordinary fractions.

Common fractions are written using two natural numbers and a fraction stroke.

The number written above the line is called numerator fractions The number written below the line is called denominator fractions

The denominator shows how many parts the whole was divided into, and the numerator shows how many such parts were taken.

Let's look at our orange. We divided it into 8 parts, that is, at first our orange was like 8/8, and when we took three slices from 8 slices, then there were 5 slices left and the orange remained as 5/8, and three slices from the orange were 3/5.

A fraction whose numerator is less than its denominator is called correct. Conversely, a fraction whose numerator is greater than or equal to the denominator is called wrong.

For example: 3/5, 1/2, 23/54 are proper fractions,
8/8, 27/3, 7/5 are improper fractions. Improper fractions are usually written as 8/8=1; 27/3=9; 7/5=1+2/5. Such numbers are read as one whole, nine whole, one whole two fifths. The number 1 2/5 is called a mixed number, the natural number 1 is called whole part of a mixed number, 2/5 fractional part.

To convert an improper fraction, the numerator of which is not completely divisible by the denominator, into a mixed number, the numerator must be divided by the denominator; write the resulting incomplete quotient as an integer part of the mixed number, and the remainder as the numerator of its fractional part.

If the numerator of an improper fraction is divisible by the denominator, then this fraction is equal to the natural number (27/3, 8/8).

To convert a mixed number into an improper fraction, you need to multiply the whole part of the number by the denominator of the fractional part and add the numerator of the fractional part to the resulting product; write this amount as the numerator of the improper fraction, and write the denominator of the fractional part of the mixed number in the denominator.

For example: 5 4/9=(5 9+4)/9=49/9.

Of two fractions with the same denominator, the one with the larger numerator is greater, and the one with the smaller numerator is smaller.

3/7>2/7; 1/8<3/8.

All proper fractions are less than one, and all improper fractions are greater than or equal to one.

Every improper fraction is greater than every proper fraction, and vice versa.

The main property of a fraction:

If the numerator and denominator of a fraction are multiplied or divided by the same number other than zero, you get a fraction equal to the given one.

If the numerator and denominator of a fraction are natural numbers, then dividing the numerator and denominator by their common divisor other than one is called reducing a fraction.

For example: 27/36=3/4, means that the fraction has been reduced by 9.

A fraction whose numerator and denominator are mutually prime numbers is called irreducible.

Using the basic property of fractions, any two fractions can be reduced to a common denominator.

To reduce fractions to LCD (lowest common denominator), you need to:

1. Find the LCM of the denominators of these fractions;
2. Find additional factors for each fraction by dividing the common denominator by the denominator of the given fractions;
3. Multiply the numerator and denominator of each fraction by its additional factor.

For example: let's bring to NOS 7/8 and 11/12.

1. We are looking for NOZ: we multiply 8 2 = 16, 8 3 = 24, then 12 3 = 24. We found NOZ=24.

We multiply the numerators of the fractions by an additional factor 7 3=21, 11 2=22.

We got equalities: 7/8=21/24 and 11/12=22/24

To compare two fractions with different denominators, you need to reduce them to the same denominator.

2.2 Arithmetic operations with ordinary fractions.

1. To add two fractions with the same denominators, you need to add the numerators of the fractions and leave the denominator unchanged.

2/5+1/5=(2+1)/5=3/5.

2. To subtract two fractions with the same denominators, you need to subtract the numerator of the other fraction from the numerator of one fraction, leaving the denominator unchanged.

2/5-1/5=(2-1)/5=1/5

1. To add or subtract fractions with unlike denominators, you must bring them to a common denominator, and then apply the rule for adding or subtracting fractions with like denominators.

To multiply one fraction by another, you must multiply the numerator of one fraction by the numerator of another and multiply the denominator of one fraction by the denominator of the other.

4/7 2/3=(4 2)/(7 3)=8/21.

Two fractions whose product is equal to 1 are called mutually inverse.

For example: 4/9 and 9/4

1. To divide one fraction by another, you need to multiply the first fraction by the inverse fraction of the second fraction (that is, the fraction that is the divisor must be turned over, that is, the numerator and denominator in the second fraction are swapped).

For example: 6/35: 2/5= 6/35 5/2=3/7.

Now that we're done with the theory of ordinary fractions, let's move on to the test.

Lesson #27. Subject: " Reducing fractions to a common denominator »

The purpose of the lesson:

subject:

develop the ability to reduce a fraction to a new denominator and lowest common denominator

meta-subject:

personal:

develop the ability to formulate your own opinion.

Planned results: The student will learn to reduce a fraction to a new denominator and a lowest common denominator.

Basic concepts: Reducing fractions to a common denominator, additional factor, common denominator of two fractions, least common denominator, rule for reducing a fraction to least common

denominator.

Lesson type : a lesson in learning new material.

Lesson equipment: blackboard, chalk, textbook, cards for independent work.

During the classes:

Organizational moment

Preparing students for work in the classroom.

The cheerful bell rang,

Are we ready to start the lesson?

Let's listen and talk

And help each other.

Hello, please sit down.

We are calm, kind and welcoming. Take a deep breath. Exhale yesterday's resentment, anger, anxiety. Breathe in the warmth of the sun's rays. I wish you a good mood. I hope you remain in a good mood until the end of the lesson.

Checking homework

Let's check your homework.

Swap notebooks with a neighbor and check that your homework is completed correctly.

What mistakes were made?

Updating knowledge

So that mistakes don’t get into the notebook,

You need to remember and know the rules.

What did we talk about in previous lessons?

What does it mean to reduce a fraction?

Can every fraction be reduced?

What is the basis for reducing fractions?

State the main property of a fraction.

1) Find the greatest common divisor and least common multiple of the numbers:

and 12; 12 and 16; 15 and 25; 3 and 4; 6 and 18; 4 and 15; 12 and 5; 6 and 20; 3 and 7.

Motivational stage

2) Compare fractions: and,

How to compare?

What are your assumptions?

Learning new material

Reduce to the same numerator 6. To do this, multiply the numerator and denominator of the first fraction by 3, and those of the second fraction by 2.

The resulting fractions are 6/9 and 6/8. The second fraction is larger.

Reduce the fractions to the same denominator 12. To do this, multiply the numerator and denominator of the first fraction by 4, and that of the other fraction by 3. We get the fractions 8/12 and 9/12. The second fraction is larger.

How can any two fractions be reduced to a common denominator? Today in class we have to learn this. And so, we write down the topic of the lesson: “Bringing fractions to a common denominator.”

For both fractions, the numerators and denominators must be multiplied by numbers such that the denominators are the same. That is, this number must be divisible by both 3 and 4. This is 12. Another way we find the LCM of these numbers. Now we are looking for the numbers by which the numerators are multiplied. For this 12: 3 = 4, this is the additional factor of the first fraction. 12: 4 = 3 – additional factor of the second fraction. Then we multiply the numerators of the fractions by additional fractions. We get the fractions 8/12 and 9/12. The second fraction is larger.

Reducing fractions to lowest common denominator (LCD)

To reduce multiple fractions to their lowest common denominator:

1) find the least common multiple of the denominators of these fractions, it will be their least common denominator;

2) divide the lowest common denominator by the denominators of these fractions, i.e. find an additional factor for each fraction;

3) multiply the numerator and denominator of each fraction by its additional factor.

Fizminutka

All the guys stood up together

And they walked on the spot.

Stretch on your toes

And they turned to each other.

We sat down like springs,

And then they sat down quietly.

Primary consolidation of new material

№236, 238, 239(1, 3, 5,7)

Reflection

Continue the statement about evaluating your work in class.

I worked in class for a grade...

I learned today...

I don't quite understand...

Homework - P.9, questions 1-3, No. 237, 240, 263

Example 1. Let's bring the fractions 1/8 and 5/6 to a common denominator. The number that is the common denominator of these fractions must be divisible by both the number 8 and the number 6, i.e. it is a common multiple of the numbers 8 and 6. And there are infinitely many common multiples of 8 and 6: 24, 48, 72, etc. LCM (8,6) = 24. This means that the lowest common denominator of the fractions 1/8 and 5/6 is the number 24.

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"Reducing fractions to their lowest common denominator"

Reducing fractions to their lowest common denominator

Mathematics teacher Kereeva Zh.T. G AKTOBE SSL No. 20

9/24, then 5/6 3/8. "width="640"

Comparing fractions with different numerators and different denominators. Example 4 Compare the fractions 5/6 and 3/8. We reduce the fractions being compared to their lowest common denominator. Thus, we equate the denominators of these fractions. LCM (6,8) = 24 5/6 = 20/24; 3/8 = 9/24 since 20/24 is 9/24, then 5/6 is 3/8.

c/d if adbc, for example, 3/72/9, since 3*97*2; 3) a/b" width="640"

The rule for comparing fractions can be reduced to the general form 1) a/b=c/d, if ad=bc, for example, 2/5=4/10, since 2*10=5*4; 2) a/bc/d, if adbc, for example, 3/72/9, since 3*97*2; 3) a/b
1/3. "width="640"

Comparison of mixed numbers Example 5 Let's compare the mixed numbers 2+5/7 and 3+1/7. Compare the integer part of mixed numbers. Since 2 2+1/3, since 5/7 1/3.