## Math 6 Chapter 3 Lesson 9: Fractional Subtraction

## 1. Summary of theory Tóm

### 1.1. The opposite number

Two numbers are said to be opposite if their sum is 0.

– The countersign of the fraction \(\dfrac{a}{b}\) is \( – \dfrac{a}{b}\).

We have:

\(\dfrac{a}{b} + \left( { – \dfrac{a}{b}} \right) = 0\)

\( – \dfrac{a}{b} = \dfrac{a}{{ – b}} = \dfrac{{ – a}}{b}\)

**For example: **

The fractions \(\dfrac{3}{4}\) and \(\dfrac{-3}{4}\) are opposite fractions because \(\dfrac{3}{4}+ \dfrac{ -3}{4}=0\)

\( – \dfrac{2}{7} = \dfrac{{ – 2}}{7} = \dfrac{2}{{ – 7}}\)

### 1.2. Fraction subtraction

To subtract a fraction from a fraction, add the subtracted number with the opposite of the minus number.

\(\dfrac{a}{b} – \dfrac{c}{d} = \dfrac{a}{b} + \left( { – \dfrac{c}{d}} \right)\)

**For example: **Calculate \(\dfrac{3}{8} – \left( {\dfrac{{ – 3}}{5}} \right)\)

**Solution**

\(\dfrac{3}{8} – \left( {\dfrac{{ – 2}}{5}} \right) = \dfrac{3}{8} + \dfrac{2}{5} = \ dfrac{{3.5 + 2.8}}{{40}} = \dfrac{{31}}{{40}}\)

**Comment: **Subtraction (fraction) is the inverse operation of addition (fraction).

## 2. Illustrated exercise

**Question 1:** Calculate

\(\dfrac{3}{5} – \dfrac{{ – 1}}{2}\);

\(\dfrac{{ – 5}}{7} – \dfrac{1}{3}\);

\(\dfrac{{ – 2}}{5} – \dfrac{{ – 3}}{4}\); \( – 5 – \dfrac{1}{6}\)

**Solution guide**

We have

\(\dfrac{3}{5} – \dfrac{{ – 1}}{2} = \dfrac{3}{5} + \dfrac{1}{2} \)

\(= \dfrac{6}{{10}} + \dfrac{5}{{10}} = \dfrac{{11}}{{10}}\)

\(\dfrac{{ – 5}}{7} – \dfrac{1}{3} = \dfrac{{ – 5}}{7} + \left( {\dfrac{{ – 1}}{3} } \right) \)

\(= \dfrac{{ – 15}}{{21}} + \dfrac{{ – 7}}{{21}} \)

\(= \dfrac{{ – 15 + \left( { – 7} \right)}}{{21}} = \dfrac{{ – 22}}{{21}}\)

\(\dfrac{{ – 2}}{5} – \dfrac{{ – 3}}{4} = \dfrac{{ – 2}}{5} + \dfrac{3}{4} \)

\(= \dfrac{{ – 8}}{{20}} + \dfrac{{15}}{{20}} = \dfrac{{ – 8 + 15}}{{20}} = \dfrac{7) }{{20}}\)

\( – 5 – \dfrac{1}{6} = – 5 + \left( {\dfrac{{ – 1}}{6}} \right) = \dfrac{{ – 30}}{6} + \ dfrac{{ – 1}}{6} \)

\(= \dfrac{{ – 30 + \left( { – 1} \right)}}{6} = \dfrac{{ – 31}}{6}\)

**Verse 2: **

a) Calculate \(1 – \dfrac{1}{2},\,\,\,\dfrac{1}{2} – \dfrac{1}{3},\,\,\,\dfrac{1 }{3} – \dfrac{1}{4},\,\,\,\dfrac{1}{4} – \dfrac{1}{5},\,\,\,\dfrac{1}{ 5} – \dfrac{1}{6}\)

b) Use the result of a) to quickly calculate the following sum:

\(\dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{{12}} + \dfrac{1}{{20}} + \dfrac{1}{{30 }}\)

**Solution guide**

a) \(\dfrac{1}{2},\,\,\dfrac{1}{6},\,\,\dfrac{1}{{12}},\,\dfrac{1}{{ 20}},\,\,\dfrac{1}{{30}}\)

b) \(\dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{{12}} + \dfrac{1}{{20}} + \dfrac{1}{ {30}} \)

\(= \left( {1 – \dfrac{1}{2}} \right) + \left( {\dfrac{1}{2} – \dfrac{1}{3}} \right) + \left ( {\dfrac{1}{3} – \dfrac{1}{4}} \right) + \left( {\dfrac{1}{4} – \dfrac{1}{5}} \right) + \left( {\dfrac{1}{5} – \dfrac{1}{6}} \right)\)

\( = 1 + \left( {\dfrac{{ – 1}}{2} + \dfrac{1}{2}} \right) + \left( {\dfrac{{ – 1}}{3} + \dfrac{1}{3}} \right) + \left( {\dfrac{{ – 1}}{4} + \dfrac{1}{4}} \right) + \left( {\dfrac{{ – 1}}{5} + \dfrac{1}{5}} \right) + \dfrac{{ – 1}}{6}\)

\(= \dfrac{5}{6}\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Quick calculation

\(A = \dfrac{1}{6} + \dfrac{1}{{12}} + \dfrac{1}{{20}} + \dfrac{1}{{30}} + \dfrac{1 }{{42}} + \dfrac{1}{{56}}\)

**Verse 2: **

a) Show that for \(n \in \mathbb{N},n \ne 0\) then:

\(\dfrac{1}{{n(n + 1)}} = \dfrac{1}{n} – \dfrac{1}{{n + 1}}\)

b) Apply the results in a) to calculate:

\(A = \dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + … + \dfrac{1}{{9.10} }\)

**Question 3: **Cuong’s 1-day time is distributed as follows:

– Sleep \(\dfrac{1}{3}\) days

– Studying at school: \(\dfrac{1}{6}\) days

– Playing sports: \(\dfrac{1}{{12}}\) days

– Study and practice at home: \(\dfrac{1}{8}\) days

– Help the family with chores: \(\dfrac{1}{{24}}\) days)

How much free time does Cuong have?

**Question 4: **Prove that: \(D = \dfrac{1}{{{2^2}}} + \dfrac{1}{{{3^2}}} + \dfrac{1}{{{4^2}) }} + …. + \dfrac{1}{{{{10}^2}}} < 1\)

### 3.2. Multiple choice exercises Bài

**Question 1:** The argument of \( – \left( { – \dfrac{{27}}{{11}}} \right)\) is

A. \({ – \dfrac{{27}}{{11}}}\)

B. \({ – \dfrac{{11}}{{27}}}\)

C. \({ \dfrac{{27}}{{11}}}\)

D. \( – \left( { – \dfrac{{27}}{{11}}} \right)\)

**Verse 2:** Do the calculation \(\dfrac{{ – 1}}{6} – \dfrac{{ – 4}}{9}\)

A. \(\dfrac{5}{{18}}\)

B. \(\dfrac{5}{{36}}\)

C. -\(\dfrac{11}{{18}}\)

D. \(\dfrac{5}{{6}}\)

**Question 3: **Find x knowing \(x + \dfrac{1}{{14}} = \dfrac{5}{7}\)

A. \(\dfrac{9}{{14}}\)

B. \(\dfrac{1}{{14}}\)

C. \(\dfrac{11}{{14}}\)

D. \(\dfrac{7}{{14}}\)

**Question 4:** The value of x satisfies \(\dfrac{{15}}{{20}} – x = \dfrac{7}{{16}}\)

A. \( – \dfrac{{5}}{{16}}\)

B. \(\dfrac{{5}}{{16}}\)

C. \(\dfrac{{19}}{{16}}\)

D. \(\dfrac{{-19}}{{16}}\)

**Question 5: **Fill in the blanks with the appropriate number \(\dfrac{1}{3} + \dfrac{{…}}{{24}} = \dfrac{3}{8}\)

A. 2

B. 1

C. -1

D. 5

## 4. Conclusion

Through this lesson, you should know the following:

- Know how to find the opposite of a number.
- Know how to subtract fractions.

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