## Math 6 Chapter 3 Lesson 5: Reducing the denominator of many fractions

## 1. Summary of theory Tóm

### 1.1. Denominator of two fractions

To convert two fractions, we do the following:

**Step 1:** Find a common multiple of 2 denominators to use as a common denominator

**Step 2: **Find the sub-factor of each denominator (divide the common denominator by each denominator)

**Step 3: **Multiply both the numerator and denominator of each fraction by the corresponding sub-factor

**For example: **Denominator of two fractions \(\dfrac{3}{4}\) and \(\dfrac{2}{5}\)

Consider two fractions \(\dfrac{3}{4}\) and \(\dfrac{2}{5}\). We have 20 which is a common multiple of 4 and 5.

We have: \(\dfrac{3}{4}=\dfrac{3.5}{4.5}=\dfrac{15}{20}\) and \(\dfrac{2}{5}=\frac{2.4} {5.4}=\dfrac{8}{20}\).

The fractions \(\dfrac{3}{4}\) and \(\dfrac{2}{5}\) can also be denominators with other common denominators such as: 40, 60, 80, ….

For simplicity when denominating two fractions, we usually take the common denominator as the BCNN of the samples.

### 1.2. Denominator of many fractions

Since every fraction can be written as a fraction with a positive denominator, we have the rule:

To reduce the denominator of many fractions with a positive denominator, do the following:

**Step 1:** Find a common multiple of the samples (usually the BCNN) to use as a common denominator

**Step 2: **Find the sub-factor of each denominator (by dividing the common denominator by each).

**Step 3: **Multiply the numerator and denominator of each fraction with the corresponding sub-factor

**For example: **Denominator of fractions: \(\dfrac{9}{10}\), \(\dfrac{4}{15}\) and \(\dfrac{7}{6}\)

-Find state report: State report (10,15,6)=30

– Find the sub-factors:

30:10=3, 30:15=2, 30:6=5

– Multiply and denominator with respective sub-factors phụ

\(\dfrac{9}{10}=\dfrac{9.3}{10.3}=\dfrac{27}{30}\),

\(\dfrac{4}{15}=\dfrac{4.2}{15.2}=\dfrac{18}{30}\);

\(\dfrac{7}{6}=\dfrac{7.5}{6.5}=\dfrac{35}{30}\)

## 2. Illustrated exercise

**Question 1: **Convert the following two fractions: \(\dfrac{5}{6};\dfrac{6}{7}\)

**Solution guide**

We have: BCNN (6;7)=42

Candlestick:

\(\dfrac{5}{7}=\dfrac{5.7}{6.7}=\dfrac{35}{42}\)

\(\dfrac{6}{7}=\dfrac{6.6}{7.6}=\dfrac{36}{42}\)

**Verse 2: **Reduce the denominator of the following fractions: \(\dfrac{3}{4};\dfrac{7}{6};\dfrac{5}{8}\)

**Solution guide**

We have: BCNN (4; 6; 8)=24

Candlestick

\(\dfrac{3}{4}=\dfrac{3.6}{4.6}=\dfrac{18}{24}\)

\(\dfrac{7}{6}=\dfrac{7.4}{6.4}=\dfrac{28}{24}\)

\(\dfrac{5}{8}=\dfrac{5.3}{8.3}=\dfrac{15}{24}\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Convert the following two fractions to the denominator: \(\dfrac{2}{5};\dfrac{3}{7}\)

**Verse 2:** Reduce the denominator of the following fractions: \(\dfrac{7}{3};\dfrac{5}{6};\dfrac{3}{4}\)

**Question 3:** Simplify 2 expressions and concur:

\(\dfrac{2^{5}.7+2^{5}}{2^{5}.5^{2}-2^{5}.3}\) and \(\dfrac{3^ {4}.5-3^{6}}{3^{4}.13+3^{4}}\)

**Question 4:** Combine the following two expressions: \(\dfrac{a+b}{a^{2}}; \dfrac{a}{b(a+b)}; a,b \in Z; a,b,( a+b)\neq 0\)

### 3.2. Multiple choice exercises Bài

**Question 1:** Using the denominators of the fractions \(\dfrac{-3}{7};\dfrac{-5}{9};\dfrac{4}{21}\) we get the new fractions:

A. \(\dfrac{-21}{63};\dfrac{-35}{63};\dfrac{16}{63}\)

B. \(\dfrac{-27}{63};\dfrac{-35}{63};\dfrac{12}{63}\)

C. \(\dfrac{-27}{63};\dfrac{-30}{63};\dfrac{16}{63}\)

D. \(\dfrac{-21}{63};\dfrac{-30}{63};\dfrac{12}{63}\)

**Verse 2: **After denominating 2 fractions \(\dfrac{3}{5};\dfrac{7}{6}\) we get what 2 fractions?

A. \(\dfrac{18}{30};\dfrac{42}{30}\)

B. \(\dfrac{18}{30};\dfrac{35}{30}\)

C. \(\dfrac{25}{30};\dfrac{35}{30}\)

D. \(\dfrac{25}{30};\dfrac{42}{30}\)

**Question 3: **The next fraction of the sequence: \(\dfrac{9}{20};\dfrac{3}{5};\dfrac{3}{4};..\) is:

A. \(\dfrac{9}{23}\)

B. \(\dfrac{3}{9}\)

C. \(\dfrac{9}{10}\)

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

**Question 4: **After converting 2 fractions \(\dfrac{15}{120};\dfrac{2}{40}\) we get 2 new fractions:

A. \(\dfrac{60}{480};\dfrac{22}{480}\)

B. \(\dfrac{600}{4800};\dfrac{200}{4800}\)

C. \(\dfrac{15}{120};\dfrac{6}{120}\)

D. \(\dfrac{500}{4800};\dfrac{240}{4800}\)

**Question 5:** Given the sequence of fractions: \(\dfrac{1}{5};\dfrac{3}{10};\dfrac{2}{5};…\) The next fraction of the sequence is: (Write below: simplified fraction form)

A. \(\dfrac{2}{10}\)

B. \(\dfrac{4}{5}\)

C. \(\dfrac{1}{2}\)

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

## 4. Conclusion

Through this lesson, you should know the following:

- Know how to reduce the denominator of two fractions
- Equivalent to denominators of many fractions.

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