## Math 6 Chapter 3 Lesson 3: Basic properties of fractions

## 1. Summary of theory Tóm

– If we multiply both the numerator and denominator of a fraction by the same non-zero integer, we get a fraction equal to the given fraction.

\(\dfrac{a}{b}=\dfrac{am}{bm}, m \in Z, m\neq0\)

**For example: **\(\dfrac{2}{5} = \dfrac{{2.2}}{{5.2}} = \dfrac{4}{{10}};\,\,\,\dfrac{9}{4} = \dfrac{{9.3}}{{4.3}} = \dfrac{{27}}{{12}}\)

– If we divide both the numerator and denominator of a fraction by the same common divisor, we get a fraction equal to the given fraction.

\(\dfrac{a}{b}=\dfrac{a:n}{b:n}, n \in\) UC(a,b)

**For example:** \(\dfrac{9}{{24}} = \dfrac{{9:3}}{{24:3}} = \dfrac{3}{8};\,\,\,\dfrac{8} {6} = \dfrac{{8:2}}{{6:2}} = \dfrac{4}{3}\)

– From the properties of the above fraction, we can write any fraction with a negative denominator into a fraction with a positive and equal it by multiplying both the numerator and denominator of the fraction by (-1).

**For example: **\(\frac{3}{-8}=\frac{3.(-1)}{(-8).(-1)}=\frac{-3}{8}\)

– Each fraction will have an infinite number of fractions equal to it.

**For example:** \(\frac{1}{3}=\frac{2}{6}=\frac{3}{9}=\frac{4}{12}=…\)

## 2. Illustrated exercise

**Question 1: **Fill in the appropriate number in the box:

**Solution guide**

\(\eqalign{& {{ – 1} \over 2} = {{ – 1.( – 3)} \over {2.( – 3)}} = {3 \over { – 6}} \cr & {5 \over { – 10}} = {{5 \div ( – 5)} \over { – 10 \div ( – 5)}} = {{ – 1} \over 2} \cr} \)

**Verse 2:** Write each of the following fractions as a fraction that is equal to it and has a positive denominator:

\(a)\dfrac{5}{{ – 17}}\) \(b)\dfrac{{ – 4}}{{ – 11}}\) \(c)\dfrac{a}{b}\ ,\,\left( {a,b \in ,b < 0} \right)\)

**Solution guide**

We have

\(\begin{array}{l}

\dfrac{5}{{ – 17}} = \dfrac{{5.\left( { – 1} \right)}}{{\left( { – 17} \right).\left( { – 1} \right)}} = \dfrac{{ – 5}}{{17}}\\

\dfrac{{ – 4}}{{ – 11}} = \dfrac{{\left( { – 4} \right).\left( { – 1} \right)}}{{\left( { – 11 } \right).\left( { – 1} \right)}} = \dfrac{4}{{11}}\\

\dfrac{a}{b} = \dfrac{{a.\left( { – 1} \right)}}{{b.\left( { – 1} \right)}} = \dfrac{{ – a }}{{ – b}}

\end{array}\)

\(\left( {do\,\,b < 0\,\,should\,\, - b > 0} \right)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1:** Write a new fraction equal to the fraction \(\dfrac{4}{11}\) such that the denominator of the new fraction is 5 times the denominator of the old fraction

**Verse 2:** Convert the following fractions to fractions with positive denominators: \(\dfrac{6}{-13};\dfrac{-3}{-5}\)

**Question 3:** Find the numbers x, y, z that satisfy: \(\dfrac{4}{5}=\dfrac{12}{x}=\dfrac{y}{20}=\dfrac{8(yx)}{z} \)

**Question 4: **Prove that: \(\dfrac{-22}{55}=\dfrac{-26}{65}\)

### 3.2. Multiple choice exercises Bài

**Question 1: **The value of x satisfying the equality \(\dfrac{x}{11}=\dfrac{-22}{121}\) is:

A. 2

B. -2

C. 12

D. -12

**Verse 2: **Which of the following positive denominator fractions is equal to the fraction \(\dfrac{5}{-8}\):

A. \(\dfrac{15}{24}\)

B. \(\dfrac{-20}{32}\)

C. \(\dfrac{10}{16}\)

D. \(\dfrac{25}{-40}\)

**Question 3: **Which of the following fractions is equal to the fraction \(\dfrac{1}{4}\):

A. \(\dfrac{7}{24}\)

B. \(\dfrac{6}{18}\)

C. \(\dfrac{4}{15}\)

D. \(\dfrac{5}{20}\)

**Question 4: **Which of the following sequences of equal fractions is correct?

A. \(\dfrac{5}{3}=\dfrac{10}{6}=\dfrac{15}{12}\)

B. \(\dfrac{2}{3}=\dfrac{4}{6}=\dfrac{8}{12}\)

C. \(\dfrac{5}{9}=\dfrac{10}{18}=\dfrac{15}{36}\)

D. \(\dfrac{7}{9}=\dfrac{14}{18}=\dfrac{26}{36}\)

**Question 5: **40 minutes make up how many parts of an hour?

A. \(\dfrac{4}{10}\)

B. \(\dfrac{2}{6}\)

C. \(\dfrac{2}{3}\)

D. \(\dfrac{40}{100}\)

## 4. Conclusion

Through this lesson, you should know the following:

- Master the basic properties of fractions
- Apply to solve related problems.

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