## Math 6 Chapter 3 Lesson 12: Fraction division

## 1. Summary of theory Tóm

### 1.1. Inverse number

Two numbers are said to be inverses of each other if their product is equal to 1.

**For example: **

\(\frac{{ – 3}}{5}\) and \(\frac{{ 5}}{-3}\) are inverses because their Product is \(\frac{{ – 3} }{5}.\frac{{ 5}}{-3}=1\)

4 and \(\dfrac{-1}{4}\) are inverses because \(4.\dfrac{-1}{4}=1\)

### 1.2. Fraction division

**Rule: **To divide a fraction or an integer by a fraction, multiply the divisor by the reciprocal of the divisor.

\(\frac{a}{b}\,\,:\,\,\frac{c}{d} = \frac{a}{b}.\frac{c}{d} = \frac{{ a\,\,.\,\,d}}{{b\,.\,\,c}}\)

\(a:\frac{c}{d} = a.\frac{d}{c} = \frac{{ad}}{c}\,\,(c \ne 0)\)

**For example: **

\(\begin{array}{l}

\frac{2}{3}:\frac{{ – 4}}{5} = \frac{2}{3}.\frac{5}{{ – 4}} = \frac{{10}}{ { – 12}} = \frac{{ – 5}}{6}\\

3:\frac{2}{{ – 5}} = 3.\frac{{ – 5}}{2} = \frac{{ – 15}}{2}\\

\frac{{ – 9}}{4}:3 = \frac{{ – 9}}{4}.\frac{1}{3} = \frac{{ – 9}}{{4.3}} = \ frac{{ – 9}}{{12}} = \frac{{ – 3}}{4}

\end{array}\)

**Comment:** To divide a fraction by an integer (non-zero), we keep the atom of the fraction and multiply the denominator by the integer.

\(\frac{a}{b}:c = \frac{a}{{b\,\,.\,\,\,c}}\)

## 2. Illustrated exercise

**Question 1: **Find the reciprocals of the following fractions:

\( \displaystyle {1 \over 7};\,\, – 5;\,\,{{ – 11} \over {10}};\,\,{a \over b}\) \((a , b ∈ Z, a 0, b 0)\)

**Solution guide**

– The inverse of \( \displaystyle {1 \over 7}\) is \( \displaystyle {7 \over 1}\)

– The reciprocal of -5 is \( \displaystyle {{ – 1} \over 5}\)

– The inverse of \( \displaystyle {{ – 11} \over {10}}\) is \( \displaystyle {{10} \over { – 11}}\)

– The inverse of \( \displaystyle {a \over b}\) is \( \displaystyle {b \over a}\)

**Cau 2: **Let’s calculate and compare:

\(\dfrac{2}{7}:\dfrac{3}{4}\) and \(\dfrac{2}{7}.\dfrac{4}{3}\)

**Solution guide**

\(\dfrac{2}{7}:\dfrac{3}{4} = \dfrac{2}{7}.\dfrac{4}{3} \)\(= \dfrac{{2.4}}{ {7.3}} = \dfrac{8}{{21}}\)

\(\dfrac{2}{7}.\dfrac{4}{3} = \dfrac{{2.4}}{{7.3}} = \dfrac{8}{{21}}\)

Derive \(\dfrac{2}{7}:\dfrac{3}{4}=\dfrac{2}{7}.\dfrac{4}{3}\)

**Question 3:** Complete the following calculations:

\(\begin{array}{l}

a)\,\dfrac{2}{3}:\dfrac{1}{2} = \dfrac{2}{3}.\dfrac{{…}}{1} = …\\

b)\,\,\dfrac{{ – 4}}{5}:\dfrac{3}{4} = \dfrac{{…}}{{…}}.\dfrac{4}{ 3} = …\\

c)\,\, – 2:\dfrac{4}{7} = \dfrac{{ – 2}}{1}.\dfrac{{…}}{{…}} = …

\end{array}\)

**Solution guide**

\(\begin{array}{*{20}{l}}

{a){\mkern 1mu} \dfrac{2}{3}:\dfrac{1}{2} = \dfrac{2}{3}.\dfrac{2}{1} = \dfrac{4}{ 3}}\\

{b){\mkern 1mu} {\mkern 1mu} \dfrac{{ – 4}}{5}:\dfrac{3}{4} = \dfrac{{ – 4}}{5}.\dfrac{4 }{3} = \dfrac{{ – 16}}{{15}}}\\

{c){\mkern 1mu} {\mkern 1mu} – 2:\dfrac{4}{7} = \dfrac{{ – 2}}{1}.\dfrac{7}{4} = \dfrac{{ – 7}}{2}}

\end{array}\)

**Question 4: **Do the math:

\(a)\,\,\dfrac{5}{6}:\dfrac{{ – 7}}{{12}}\) \(b) \,\,-7:\dfrac{{14}} {3}\) \(c)\,\,\dfrac{{ – 3}}{7}:9\)

**Solution guide**

\(\begin{array}{*{20}{l}}

{a){\mkern 1mu} {\mkern 1mu} \dfrac{5}{6}:\dfrac{{ – 7}}{{12}} = \dfrac{5}{6}.\dfrac{{12 }}{{ – 7}} = \dfrac{{60}}{{ – 42}} = \dfrac{{ – 10}}{7}}\\

{b){\mkern 1mu} {\mkern 1mu} – 7:\dfrac{{14}}{3} = \dfrac{{ – 7}}{1}.\dfrac{3}{{14}} = \dfrac{{ – 21}}{{14}} = \dfrac{{ – 3}}{2}}\\

{c){\mkern 1mu} {\mkern 1mu} \dfrac{{ – 3}}{7}:9 = \dfrac{{ – 3}}{{7.9}} = \dfrac{{ – 1}}{ {21}}}

\end{array}\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Calculate the following quotients and arrange them in ascending order:

\(\frac{3}{2}:\frac{9}{4};\,\,\,\frac{{48}}{{55}}:\frac{{12}}{{11} };\,\,\frac{7}{{10}}:\frac{7}{5};\,\,\frac{6}{7}:\frac{8}{7}\)

**Verse 2: **Write the fraction \(\frac{{14}}{{15}}\) as the quotient of two fractions whose numerator and denominator are single-digit positive integers.

**Question 3: **Calculate the value of the expression

\(A = \frac{{\frac{2}{3} + \frac{2}{5} – \frac{2}{9}}}{{\frac{4}{3} + \frac{ 4}{5} – \frac{4}{9}}}.\)

**Question 4: **Given two fractions \(\frac{8}{{15}}\) and \(\frac{{18}}{{35}}.\) Find the largest number such that when each of these fractions is divided by the number then we get an integer result.

**Question 5: **Find two numbers, knowing that \(\frac{9}{{11}}\) of one is equal to \(\frac{6}{7}\) of the other and that the sum of the two is 258.

### 3.2. Multiple choice exercises Bài

**Question 1: **Which of the following values of x satisfy \(\left( { – \frac{3}{5}} \right).x = \frac{4}{{15}}\)

A. \( – \frac{1}{{10}}\)

B. \( – \frac{4}{{9}}\)

C. \( – \frac{4}{{3}}\)

D. -4

**Verse 2: **Calculate \(\frac{2}{3}:\frac{7}{{12}}:\frac{4}{{18}}\)

A. \(\frac{7}{{18}}\)

B. \(\frac{9}{{14}}\)

C. \(\frac{36}{{7}}\)

D. \(\frac{`8}{{7}}\)

**Question 3: **Calculate \(\frac{2}{3}:\frac{1}{2}\)

A. 3

B. 1

C. \(\frac{1}{3}\)

D. \(\frac{4}{3}\)

**Question 4: **The result of the calculation \(\frac{{\left( { – 7} \right)}}{6}:\left( { – \frac{{14}}{3}} \right)\) has the value value is

A. \(\frac{1}{4}\)

B. \(\frac{1}{2}\)

C. \(\frac{-1}{2}\)

D. 1

**Question 5: **Find x knowing \(\frac{{13}}{{25}}:x = \frac{5}{6}\)

A. \(\frac{2}{5}\)

B. \(\frac{338}{125}\)

C. \(\frac{5}{2}\)

D. \(\frac{125}{338}\)

**Question 6: **The expression value \(M = \frac{5}{6}:{\left( {\frac{5}{2}} \right)^2} + \frac{7}{{15}}\) is a fraction of the form \(\frac{a}{b}\) where a > 0. Calculate b + a

A. 8

B. 9/5

C. 3/5

D. 2

## 4. Conclusion

Through this lesson, you should know the following:

- Determine the reciprocal of a number.
- Can perform fractional division.

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