## Math 6 Chapter 3 Lesson 6: Compare fractions

## 1. Summary of theory Tóm

### 1.1. Compare two fractions with the same denominator

**Of any two fractions with the same positive denominator, the fraction with the larger numerator is larger.**

**For example: **Compare the following pairs of fractions

a) \(\dfrac{-3}{4} ;\dfrac{-7}{4}\)

b) \(\dfrac{5}{-8} ;\dfrac{-7}{8}\)

**Solution**

a) Because \(-3>-7\Rightarrow \dfrac{-3}{4} >\dfrac{-7}{4}\)

b) Since the 2 fractions do not have the same positive denominator, we will transform:

\(\dfrac{5}{-8}=\dfrac{-5}{8}\) and we will compare \(\dfrac{-5}{8};\dfrac{-7}{8}\ )

Because \(-5>-7\Rightarrow \dfrac{5}{-8}=\dfrac{-5}{8}>\dfrac{-7}{8}\)

### 1.2. Compare two fractions that do not have the same denominator

**To compare two fractions that do not have the same denominator, we write them as two fractions with the same positive denominator and then compare the numerators. The fraction with the larger numerator is larger.**

**For example: **Compare the following 2 fractions: \(\dfrac{2}{-3}\) and \(\dfrac{-5}{9}\)

**Solution:**

– Return the positive sample: \(\dfrac{2}{-3}=\dfrac{-2}{3}\)

– Denominator of fractions: \(\dfrac{-2}{3}\) and \(\dfrac{-5}{9 }\)

\(\dfrac{-2}{3}=\dfrac{(-2).3}{3.3}=\dfrac{-6}{9}\); keep \(\dfrac{-5}{9}\)

Because \(-6<-5\Rightarrow \dfrac{-6}{9}<\dfrac{-5}{9}\Rightarrow \dfrac{-2}{3}<\dfrac{-5}{9} \Rightarrow \dfrac{2}{-3}<\dfrac{-5}{9}\)

## 2. Illustrated exercise

**Question 1:** Put the appropriate sign (< , >) in the box:

\(\eqalign{& {{ – 8} \over 9}\,\, \square \,{{ – 7} \over 9}\,;\,\,\,\,\,\,\,\ ,\,\,\,\,\,{{ – 1} \over 3}\,\, \square \,\,{{ – 2} \over 3} \cr & {3 \over 7}\, \, \square \,\,{{ – 6} \over 7};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,{{ – 3} \over {11}}\,\, \square \,\,{0 \over {11}} \cr} \)

\(\eqalign{& {{ – 8} \over 9}\,\, < \,{{ - 7} \over 9}\,;\,\,\,\,\,\,\,\, \,\,\,\,\,{{ - 1} \over 3}\,\, > \,\,{{ – 2} \over 3} \cr & {3 \over 7}\,\, > \,\,{{ – 6} \over 7};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{ { – 3} \over {11}}\,\, < \,\,{0 \over {11}} \cr} \)

**Verse 2: **Compare fractions:

a)\(\displaystyle \,\,{{ – 11} \over {12}};\,\,\, \,\,\,{{17} \over { – 18}}\,\,\ ,\,\)

b) \(\displaystyle \,\,{{ – 14} \over {21}};\,\,\, \,\,\,{{ – 60} \over { – 72}}\)

**Solution guide**

a) Change \(\dfrac{{17}}{{ – 18}} = \dfrac{{ – 17}}{{18}}\)

We have:

\(12 = 2^2.3\)

\(18 = 2. 3^2\)

Deduce \(BCNN(12,18) = 2^2.3^2= 36\)

\(\eqalign{& {{ – 11} \over {12}} = {{ – 11.3} \over {12.3}} = {{ – 33} \over {36}} \cr & {{17} \over { – 18}} ={{-17} \over { 18}}= {{-17.2} \over { 18.2}} = {{ – 34} \over {36}} \cr & {{ – 33} \ over {36}} > {{ – 34} \over {36}} \cr & \Rightarrow {{ – 11} \over {12}} > {{17} \over { – 18}} \cr} \)

b) \(\dfrac{{ – 14}}{{21}} = \dfrac{{ – 14:7}}{{21:7}} = \dfrac{{ – 2}}{3}\)

\(\dfrac{{ – 60}}{{ – 72}} = \dfrac{{ – 60:\left( { – 12} \right)}}{{ – 72:\left( { – 12} \right )}} = \dfrac{5}{6}\)

Let’s reduce the two fractions: \(\dfrac{{ – 2}}{3};\dfrac{5}{6}\)

The common denominator is \(BCNN(3, 6) = 6\)

Synonyms: \(\dfrac{{ – 2}}{3} = \dfrac{{ – 2.2}}{{3.2}} = \dfrac{{ – 4}}{6};\dfrac{5}{6 } = \dfrac{5}{6}\)

Compare: Since \(\dfrac{{ – 4}}{6} < \dfrac{5}{6}\) \(\dfrac{{ - 14}}{{21}} < \dfrac{{ - 60}}{{ - 72}}\)

**Question 3: **Compare the following fractions with 0: \(\dfrac{3}{5};\dfrac{{ – 2}}{{ – 3}};\dfrac{{ – 3}}{5};\dfrac{ 2}{{ – 7}}\)

**Solution guide**

Negative fractions are \(\dfrac{{ – 3}}{5};\dfrac{2}{{ – 7}}\) so \(\dfrac{{ – 3}}{5}<0;\ dfrac{2}{{ - 7}}<0\)

The positive fractions are \(\dfrac{{ 3}}{5};\dfrac{-2}{{ – 3}}\) so \(\dfrac{3}{5} > 0;\)\( \dfrac{{ – 2}}{{ – 3}} =\dfrac{{ 2}}{{ 3}}> 0\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Compare the following two fractions:

a) \(\dfrac{2}{5};\dfrac{3}{5}\)

b) \(\dfrac{2}{-9};\dfrac{-4}{9}\)

**Verse 2:** Compare the following 2 fractions: \(\dfrac{7}{12};\dfrac{9}{16}\)

**Question 3:** Find fractions whose denominator is 12 greater than \(\dfrac{-2}{3}\) and less than \(\dfrac{-1}{4}\)

**Question 4: **Arrange the following fractions in ascending order: \(\dfrac{-5}{6};\dfrac{7}{8};\dfrac{7}{24};\dfrac{16}{17} \)

### 3.2. Multiple choice exercises Bài

**Question 1: **How many of the following fractions are positive: \(\dfrac{1}{4};\dfrac{-2}{3};\dfrac{5}{-7};\dfrac{-4} {-9};\dfrac{-(-3)}{7}\)

A. 1

B. 2

C. 3

D. 4

**Verse 2: **Two people A and B travel the same distance: Knowing that the time taken by person A and person B to travel the entire distance is: \(\dfrac{5}{6}\) hours and \(\ dfrac{7}{8}\) hours.

Which of the following assertion is true?

A. A is slower than B

B. B is faster than A

C. B is slower than A

D. Unable to determine

**Question 3: **Compare 2 fractions: \(\dfrac{-3}{5}\) and \(\dfrac{-4}{5}\)

A. \(\dfrac{-3}{5}>\dfrac{-4}{5}\)

B. \(\dfrac{-3}{5}=\dfrac{-4}{5}\)

C. \(\frac{-3}{5}<\frac{-4}{5}\)

D. Incomparable

**Question 4: **Compare 2 fractions \(\dfrac{-11}{12};\dfrac{17}{-18}\)

A. \(\dfrac{-11}{12}< \dfrac{17}{-18}\)

B. \(\dfrac{-11}{12}> \dfrac{17}{-18}\)

C. \(\dfrac{-11}{12}= \dfrac{17}{-18}\)

D. Incomparable

**Question 5: **Arrange the following fractions in descending order: \(\dfrac{4}{5};\dfrac{7}{10};\dfrac{23}{25}\)

A. \(\dfrac{23}{25}>\dfrac{7}{10}>\dfrac{4}{5}\)

B. \(\dfrac{23}{25}>\dfrac{4}{5}>\dfrac{7}{10}\)

C. \(\dfrac{4}{5}>\dfrac{23}{25}>\dfrac{7}{10}\)

D. \(\dfrac{4}{5}>\dfrac{7}{10}>\dfrac{23}{25}\)

## 4. Conclusion

Through this lesson, you should know the following:

- Compare two fractions with the same denominator and two fractions with the same denominator.
- Do related math problems.

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