## Math 6 Chapter 3 Lesson 2: Equal fractions

## 1. Summary of theory Tóm

Two fractions \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\) are said to be equal if the cross product \(ad=bc\).

**Example 1:**

\(\dfrac{-5}{3}=\dfrac{10}{-6}\) because \((-5).(-6)=3.10\) \((=30)\);

\(\dfrac{-4}{9} \neq \dfrac{-5}{11}\) because \(-4.11\neq 9.(-5)\)

## 2. Illustrated exercise

**Question 1:** Are the following pairs of fractions equal? Why?

a) \( \displaystyle{1 \over 4}\) and \( \displaystyle{3 \over 12}\)

b) \( \displaystyle{2 \over 3}\) and \( \displaystyle{6 \over 8}\)

c) \( \displaystyle{{ – 3} \over 5}\) and \( \displaystyle{9 \over { – 15}}\)

d) \( \displaystyle{4 \over 3}\) and \( \displaystyle{{ – 12} \over 9}\)

**Solution guide**

a) We have: \( \displaystyle1 . 12 = 12 ; 3 . 4 = 12\)

So \( \displaystyle{1 \over 4}= \displaystyle{3 \over 2}\)

b) We have: \(2 . 8 = 16 ; 3 . 6 = 18 ≠ 16\)

Derive \( \displaystyle{2 \over 3} \ne {6 \over 8}\)

c) We have: \(-3 . (-15 ) = 45 ; 9 . 5 = 45\)

Derive \( \displaystyle{{ – 3} \over 5}\)= \( \displaystyle{9 \over { – 15}}\)

d) We have: \(4 . 9 = 36 ; -12 . 3 = – 36 ≠ 36\)

Derive \( \displaystyle{4 \over 3} \ne {{ – 12} \over 9}\)

**Verse 2: **Can you immediately confirm that the following pairs of fractions are not equal, why?

a) \(\dfrac {-2}{5}\) and \(\dfrac {2}{5}\)

b) \(\dfrac {4}{-21}\) and \(\dfrac {5}{20}\)

c) \(\dfrac {-9}{-11}\) and \(\dfrac {7}{-10}\)

**Solution guide**

a) Since \(\dfrac {-2}{5}<0\) and \(\dfrac {2}{5}>0\) \(\dfrac {-2}{5}<\dfrac {2 }{5}\) or these two fractions are not equal.

b) Since \(\dfrac {4}{-21}<0\) and \(\dfrac {5}{20}>0\) \(\dfrac {4}{-21}<\dfrac {5 }{20}\) or these two fractions are not equal.

c) Since \(\dfrac {-9}{-11}>0\) and \(\dfrac {7}{-10}<0\) \(\dfrac {-9}{-11}>\ dfrac {7}{-10}\) or these two fractions are not equal.

**Question 3: **Find an integer x, knowing: \(\dfrac{x}{15}=\dfrac{3}{9}\)

**Solution guide**

Since \(\dfrac{x}{15}=\dfrac{3}{9}\) \(x.9=3.15\Rightarrow x=\dfrac{3.15}{9}=5\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Find 3 fractions that are equal to the fraction \(\dfrac{-3}{7}\)

**Verse 2:** Make equal fractions from equality: 6.7=14.3

**Question 3:** Prove that the following pairs of numbers are equal:

a) \(\dfrac{a}{-b}\) and \(\dfrac{-a}{b}\)

b) \(\dfrac{-a}{-b}\) and \(\dfrac{a}{b}\)

**Question 4: **Find the numbers x, y, z, t know: \(\dfrac{12}{-6}=\dfrac{x}{5}=\dfrac{-y}{3}=\dfrac{z}{- 7}=\dfrac{-t}{-8}\)

**Question 5: **Given 2 equal fractions \(\dfrac{a}{b}=\dfrac{c}{d}\) . Prove that \(\dfrac{a\pm b}{b}=\dfrac{c\pm d}{d}\)

### 3.2. Multiple choice exercises Bài

**Question 1: **Which of the following equations is true?

A. \(\frac{-(-a)}{(-b)}=\frac{-a}{-b}\)

B. \(\frac{(-a)}{(-b)}=\frac{-a}{-(-b)}\)

C. \(\frac{-(-a)}{(-b)}=\frac{a}{b}\)

D. \(\frac{-(-a)}{-(-b)}=\frac{a}{b}\)

**Verse 2: **Find pairs of integers x, y know: \(\frac{x}{3}=\frac{7}{y}\)

A. (3,7), (7,3)

B. (1,21), (3,7)

C. (3,7), (7,3), (1,21), (21.1)

D. (1,21), (21.1)

**Question 3: **What value of x would satisfy the following equality: \(\frac{x}{5}=\frac{6}{3}\)

A. 8

B. 9

C. 10

D. 11

**Question 4: **The following pairs of fractions are equal fractions:

A. \(\frac{-3}{5}\) and \(\frac{9}{15}\)

B. \(\frac{-3}{5}\) and \(\frac{-9}{-15}\)

C. \(\frac{-3}{5}\) and \(\frac{9}{-15}\)

D. \(\frac{-3}{5}\) and \(\frac{5}{3}\)

**Question 5: **Are there integers x,y that satisfy the following fractions equally: \(\frac{x}{11}=\frac{4}{22}\);\(\frac{13}{ y}=\frac{4}{17}\)

A. There exist both x and y

B. Neither of them exist

C. There exists only x and no y

D. There exists only y and no x

## 4. Conclusion

Through this lesson, you should know the following:

- Know the definition of two equal fractions
- Know how to solve math problems involving two equal fractions.

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