1. Which of the following rational numbers are equal?
(i) [tex]$\frac{-7}{21}$[/tex] and [tex]$\frac{3}{-9}$[/tex]
(ii) [tex]$\frac{-8}{-14}$[/tex] and [tex]$\frac{4}{7}$[/tex]
2. Which pairs represent the same rational number?
(i) [tex]$\frac{-11}{-13}$[/tex] and [tex]$\frac{33}{39}$[/tex]
(ii) [tex]$\frac{7}{13}$[/tex] and [tex]$\frac{35}{-65}$[/tex]
Answer (2)
Pairs 1, 2, and 3 are equal. โ
Explanation
Problem Analysis We need to determine which pairs of rational numbers are equal. To do this, we will simplify each fraction to its lowest terms and then compare them.
Comparing -7/21 and 3/-9 First, let's analyze the pair 21 โ 7 โ and โ 9 3 โ .
Simplify 21 โ 7 โ by dividing both the numerator and the denominator by 7: 21 รท 7 โ 7 รท 7 โ = 3 โ 1 โ
Simplify โ 9 3 โ by dividing both the numerator and the denominator by 3: โ 9 รท 3 3 รท 3 โ = โ 3 1 โ = 3 โ 1 โ
Since 3 โ 1 โ = 3 โ 1 โ , the rational numbers 21 โ 7 โ and โ 9 3 โ are equal.
Comparing -8/-14 and 4/7 Next, let's analyze the pair โ 14 โ 8 โ and 7 4 โ .
Simplify โ 14 โ 8 โ by dividing both the numerator and the denominator by -2: โ 14 รท โ 2 โ 8 รท โ 2 โ = 7 4 โ
Since 7 4 โ = 7 4 โ , the rational numbers โ 14 โ 8 โ and 7 4 โ are equal.
Comparing -11/-13 and 33/39 Now, let's analyze the pair โ 13 โ 11 โ and 39 33 โ .
Simplify โ 13 โ 11 โ by canceling the negative signs: โ 13 โ 11 โ = 13 11 โ
Simplify 39 33 โ by dividing both the numerator and the denominator by 3: 39 รท 3 33 รท 3 โ = 13 11 โ
Since 13 11 โ = 13 11 โ , the rational numbers โ 13 โ 11 โ and 39 33 โ are equal.
Comparing 7/13 and 35/-65 Finally, let's analyze the pair 13 7 โ and โ 65 35 โ .
The fraction 13 7 โ is already in simplest form.
Simplify โ 65 35 โ by dividing both the numerator and the denominator by 5: โ 65 รท 5 35 รท 5 โ = โ 13 7 โ = โ 13 7 โ
Since 13 7 โ ๎ = โ 13 7 โ , the rational numbers 13 7 โ and โ 65 35 โ are not equal.
Conclusion In summary:
21 โ 7 โ and โ 9 3 โ are equal.
โ 14 โ 8 โ and 7 4 โ are equal.
โ 13 โ 11 โ and 39 33 โ are equal.
13 7 โ and โ 65 35 โ are not equal.
Examples
Rational numbers are used in everyday life to represent parts of a whole or ratios. For example, when you divide a pizza into slices, each slice represents a rational number of the whole pizza. If you cut a pizza into 8 slices, each slice is 8 1 โ of the pizza. Comparing rational numbers helps you determine which slice is bigger or if two slices are the same size. Similarly, in cooking, recipes often use rational numbers to specify the amount of ingredients needed, such as 2 1 โ cup of flour or 4 3 โ teaspoon of salt.
The rational numbers 21 โ 7 โ and โ 9 3 โ are equal, as are โ 14 โ 8 โ and 7 4 โ , and โ 13 โ 11 โ and 39 33 โ . However, 13 7 โ and โ 65 35 โ are not equal.
;
