## Math 6 Chapter 3 Lesson 4: Simplify fractions

## 1. Summary of theory Tóm

### 1.1. How to shorten fractions

**Rule: **

To reduce a fraction, we divide both the numerator and denominator of the fraction by a common divisor (other than 1 and (-1)) of them.

**For example: **Simplify fractions \(\dfrac{14}{6}\)

We have UC(14, 6)=2 so we have: \(\dfrac{14}{6}=\dfrac{14:2}{6:2}=\dfrac{7}{3}\)

### 1.2. Minimal fractions

A simple fraction (or a fraction that cannot be reduced anymore) is a fraction in which the numerator and denominator have only common divisors of 1 and -1.

**For example: **The fraction \(\dfrac{4}{9}\) is a simple fraction because 4 and 9 have only common divisors of 1 and -1.

**Comment: **

To quickly reduce a given fraction to a minimal fraction, we just need to divide the numerator and denominator of the fraction by their GCLN.

**For example:** UCLN(18,30)=6 so we have: \(\dfrac{18}{30}=\dfrac{18:6}{30:6}=\dfrac{3}{5}\)

**Attention:**

– The fraction \(\dfrac{a}{b}\) is minimal if \(\left | a \right |,\left | b \right |\) is co-prime.

– To reduce a fraction with a minus sign, we can reduce an unsigned fraction and then add a sign to the result

**For example:** Simplify the fraction \(\dfrac{-25}{20}\). We have GCLN (25,20)=5 so we have: \(\dfrac{25}{20}=\dfrac{25:5}{20:5}=\dfrac{5}{4}\Rightarrow \dfrac {-25}{20}=\dfrac{-5}{4}\)

– When reducing a fraction, we often reduce that fraction to the minimum

## 2. Illustrated exercise

**Question 1: **Simplify the following fractions:

\( \displaystyle a)\,\,{{ – 5} \over {10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ ,\,\,\,\,\,\,\,\,\,\,\,b)\,\,{{18} \over { – 33}}\)

\( \displaystyle c)\,\,{{19} \over {57}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,d)\,\,{{ – 36} \over { – 12}}\)

__Solution guide__

a) \( \displaystyle{{ – 5} \over {10}} = {{ – 5 : 5} \over {10 : 5}} = {{ – 1} \over 2}\)

b) \( \displaystyle{{18} \over {33}} = {{18 : 3} \over { – 33 : 3}} = {6 \over { – 11}}\)

c) \( \displaystyle{{19} \over {57}} = {{19 : 19} \over {57 : 19}} = {1 \over 3}\)

d) \( \displaystyle{{ – 36} \over {-12}} = {{ – 36 : 12} \over { – 12 : 12}} = {{ – 3} \over { – 1}} = { 3 \over 1}\)

**Verse 2: **Find the simplest fractions in the following fractions:

\(\displaystyle {3 \over 6};\,\,\,\,\,{{ – 1} \over 4};\,\,\,\,\,{{ – 4} \over {12 }};\)

\(\displaystyle{ 9 \over {16}};\,\,\,\,\,{{14} \over {63}}\)

__Solution guide__

The simplest fractions are \(\displaystyle {{ – 1} \over 4};\,\,\,\,\,\,\,\,{9 \over {16}}\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Simplify the following fractions: \(\dfrac{44}{55};\dfrac{-72}{81}\)

**Verse 2:** Simplify the following expressions: \(\dfrac{3.7}{6.14};\dfrac{8.7-8.5}{16}\)

**Question 3:** Simplify the following expression: \(\dfrac{2^{4}.5^{2}.11^{2}.7}{2^{3}.5^{3}.7^{2}. 11}\)

**Question 4:** Simplify the following expression: \(\dfrac{5^{11}.7^{12}+5^{11}.7^{11}}{5^{12}.7^{12}+9.5^ {11}.7^{11}}\)

### 3.2. Multiple choice exercises Bài

**Question 1: **Convert \(550 cm^{2}\) =? \(m^{2}\) (Write as a fraction in minimal form)

A. \(\dfrac{11}{100}\)

B. \(\dfrac{55}{1000}\)

C. \(\dfrac{11}{20}\)

D. \(\dfrac{11}{200}\)

**Verse 2: **35 minutes=? hours (Written as a fraction in the simplest form):

A. \(\dfrac{25}{45}\)

B. \(\dfrac{5}{30}\)

C. \(\dfrac{7}{12}\)

D. \(\dfrac{5}{10}\)

**Question 3: **After reducing the expression \(\dfrac{3^{10}.(-5)^{21}}{(-5)^{20}.3^{12}}\) what is the value ?

A. \(\dfrac{5}{-3}\)

B. \(\dfrac{-5}{9}\)

C. \(\dfrac{9}{-5}\)

D. \(\dfrac{3}{-5}\)

**Question 4: **After simplifying, the fraction \(\dfrac{4}{16}\) is equal to the fraction:

A. \(\dfrac{2}{8}\)

B. \(\dfrac{4}{8}\)

C. \(\dfrac{1}{4}\)

D. \(\dfrac{1}{8}\)

**Question 5: **Which of the following fractions is the simplest?

A. \(\dfrac{3}{42}\)

B. \(\dfrac{17}{34}\)

C. \(\dfrac{3}{17}\)

D. \(\dfrac{4}{48}\)

## 4. Conclusion

Through this lesson, you should know the following:

- Reduce fractions to simplest fractions
- Apply to solve related problems.

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