## Math 6 Chapter 3 Lesson 7: Addition of Fractions

## 1. Summary of theory Tóm

**Add two fractions with the same denominator:** To add two fractions with the same denominator, add the numerators and keep the denominator the same.

\(\dfrac{a}{m} + \dfrac{b}{m} = \dfrac{{a + b}}{m}\)

**For example: **\(\dfrac{1}{2} + \dfrac{{ – 5}}{2} = \dfrac{{1 + ( – 5)}}{2} = \dfrac{{ – 4}}{2} = – 2\)

**Add two fractions with different denominators:** To add two fractions that do not have the same denominator, we write them as two fractions with the same denominator and then add the numerators and keep the common denominator.

\(\dfrac{a}{m} + \dfrac{b}{n} = \dfrac{{an}}{{mn}} + \dfrac{{bm}}{{mn}} = \dfrac{{ an + bm}}{{mn}}\)

**For example: **\(\dfrac{{ – 3}}{2} + \dfrac{5}{3} = \dfrac{{ – 9}}{6} + \dfrac{{10}}{6} = \dfrac{{ – 9 + 10}}{6} = \dfrac{1}{6}\)

## 2. Illustrated exercise

**Question 1: **Add the following fractions:

a) \(\displaystyle \,\,{3 \over 8} + {5 \over 8}\,\,\,\,\)

b) \(\displaystyle\,\,{1 \over 7} + {{ – 4} \over 7}\,\,\,\,\)

c) \(\displaystyle\,\,{6 \over {18}} + {{ – 14} \over {21}}\)

**Solution guide**

a) \(\eqalign{{3 \over 8} + {5 \over 8} = {{3 + 5} \over 8} = {8 \over 8} = 1 \cr}\)

b) \(\eqalign{{1 \over 7} + {{ – 4} \over 7} = {{1 + ( – 4)} \over 7}\, = {{ – 3} \over 7}\ cr}\)

c) \(\displaystyle {6 \over {18}} + {{ – 14} \over {21}} = {{6:6} \over {18:6}} + {{ – 14:7} \ over {21:7}}\)

\(\displaystyle = {1 \over 3} + {{ – 2} \over 3} = {{1 + ( – 2)} \over 3} = {{ – 1} \over 3} \)

**Verse 2: **Why can we say: Adding two integers is a special case of adding two fractions? For example.

**Solution guide**

We can say: Adding two integers is a special case of adding two fractions because every integer can be written as a fraction.

**For example: **\(4 + 3 = \dfrac{4}{1} + \dfrac{3}{1} = \dfrac{{4 + 3}}{1} = \dfrac{7}{1} = 7\)

**Question 3:** Add the following fractions:

a) \(\displaystyle {{ – 2} \over 3} + {4 \over {15}};\)

b)\(\displaystyle {{11} \over {15}} + {9 \over { – 10}};\)

c) \(\displaystyle {1 \over { – 7}} + 3\)

**Solution guide**

\(\displaystyle a)\,\,{{ – 2} \over 3} + {4 \over {15}} = {{ – 2.5} \over {3.5}} + {4 \over {15}} \ )

\(\displaystyle = {{ – 10} \over {15}} + {4 \over {15}} = {{ – 10 + 4} \over {15}} \)\(\displaystyle = {{ – 6 } \over {15}}= {{ -2} \over {5}}\)

\(\displaystyle b)\,\,{{11} \over {15}} + {9 \over { – 10}} \)

\(\displaystyle = {{11.2} \over {15.2}} + {{9.( – 3)} \over { – 10.( – 3)}} \)

\(\displaystyle = {{22} \over {30}} + {{ – 27} \over {30}} = {{22 + ( – 27)} \over {30}} \)

\(\displaystyle = {{ – 5} \over {30}}= {{ – 1} \over {6}}\)

\(\displaystyle c)\,\,{1 \over { – 7}} + 3 = {1 \over { – 7}} + {{ – 21} \over { – 7}} = {{1 + ( – 21)} \over { – 7}} \)

\(\displaystyle = {{ – 20} \over { – 7}} = {{20} \over 7} \)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **

a) Write the fraction \(\dfrac{7}{{15}}\) as the sum of two simple fractions with different denominators.

b) Write the fraction \(\dfrac{1}{8}\) as the sum of two positive fractions with different numerators and denominators.

c) Write fractions with \(\dfrac{{15}}{{17}}\) whose denominator is a two-digit even natural number.

**Verse 2:** Demonstrate:

\(\dfrac{1}{{1001}} + \dfrac{1}{{1002}} + \dfrac{1}{{1003}} + …. + \dfrac{1}{{1250}} > \dfrac{1}{5}\)

**Question 3:** Given \(a,{\rm{ }}b,{\rm{ }}c \in \,{\mathbb{N}^*}\) and \(A = \dfrac{a}{{a + b }} + \dfrac{b}{{b + c}} + \dfrac{c}{{a + c}}.\) Prove that 1 < A < 2.

**Question 4: **Demonstrate:

\(\dfrac{1}{{10}} + \dfrac{1}{{15}} + \dfrac{1}{{21}} + \dfrac{1}{{28}} + \dfrac{1 }{{36}} + \dfrac{1}{{45}} = \dfrac{3}{{10}}.\)

**Question 5:** Calculate \(A = \dfrac{{11}}{{1.3}} + \dfrac{{11}}{{3.5}} + … + \dfrac{{11}}{{97.99}}\)

### 3.2. Multiple choice exercises Bài

**Question 1: **Rule for adding two fractions with the same denominator:

A. To add two fractions with the same denominator, add the numerator and the numerator and the denominator with the denominator

B. To add two fractions with the same denominator, add the numerator and numerator and keep the denominator the same

C. To add two fractions with the same denominator, multiply the numerator by the numerator and keep the denominator the same

D. To work out two fractions with the same denominator, add the denominator and the denominator and keep the atom

**Verse 2:** Tim x knows: \(x – \dfrac{3}{5} = \dfrac{{ – 6}}{7}\)

A. \(\dfrac{{ – 9}}{{30}}\)

B. \(\dfrac{{ – 9}}{{35}}\)

C. \(\dfrac{{ 9}}{{35}}\)

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

**Question 3: **The result of the calculation \(1 + \dfrac{{12}}{{21}} + \dfrac{{ – 3}}{7}\) is equal to:

A. \(\dfrac{8}{7}\)

B. \(\dfrac{7}{8}\)

C. \(\dfrac{24}{7}\)

D. \(\dfrac{3}{21}\)

**Question 4:** Logic expression \(\dfrac{{ – 9}}{7} + \dfrac{{13}}{4} + \dfrac{{ – 1}}{5} + \dfrac{{ – 5}} {7} + \dfrac{3}{4}\) we get the result:

A. \(\dfrac{9}{5}\)

B. \(\dfrac{11}{5}\)

C. \(\dfrac{-11}{5}\)

D. \(\dfrac{-1}{5}\)

**Question 5:** Find an integer x known \(\dfrac{{ – 1}}{2} \le \dfrac{x}{4} < \dfrac{3}{2}\)

A. \(x \in \left\{ { – 1;0;1;2;3;4;5} \right\}\)

B. \(x \in \left\{ { – 2; – 1;1;2;3;4;5} \right\}\)

C. \(x \in \left\{ { – 2; – 1;0;1;2;3;4;5;6} \right\}\)

D. \(x \in \left\{ { – 2; – 1;0;1;2;3;4;5} \right\}\)

## 4. Conclusion

Through this lesson, you should know the following:

- Can perform addition of fractions
- Do related math problems.

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