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)
Simplify each rational number to its lowest terms.
Compare the simplified forms to determine if they are equal.
21 โ 7 โ and โ 9 3 โ simplify to 3 โ 1 โ , so they are equal.
โ 14 โ 8 โ simplifies to 7 4 โ , so โ 14 โ 8 โ and 7 4 โ are equal.
โ 13 โ 11 โ and 39 33 โ simplify to 13 11 โ , so they are equal.
13 7 โ and โ 65 35 โ simplify to 13 7 โ and 13 โ 7 โ respectively, so they are not equal.
The equal pairs are: (i) 21 โ 7 โ and โ 9 3 โ , (ii) โ 14 โ 8 โ and 7 4 โ , (iii) โ 13 โ 11 โ and 39 33 โ .
Explanation
Problem Analysis We are asked to determine which pairs of rational numbers are equal. To do this, we need to 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 โ 7 โ = 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 3 โ = 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 โ 8 โ = 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 cancelling 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 33 โ = 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 its simplest form.
Simplify โ 65 35 โ by dividing both the numerator and the denominator by 5: โ 65 35 โ = โ 65 รท 5 35 รท 5 โ = โ 13 7 โ = 13 โ 7 โ
Since 13 7 โ ๎ = 13 โ 7 โ , the rational numbers 13 7 โ and โ 65 35 โ are not equal.
Final Answer Therefore, the pairs of equal rational numbers are:
21 โ 7 โ and โ 9 3 โ
โ 14 โ 8 โ and 7 4 โ
โ 13 โ 11 โ and 39 33 โ
Examples
Understanding equal rational numbers is crucial in various real-life scenarios. For instance, when comparing discounts at different stores, you might find one store offering 100 25 โ off and another offering 4 1 โ off. Recognizing that these are equivalent rational numbers helps you quickly determine that both stores are offering the same discount of 25%. This skill is also useful in cooking, where recipes might use fractions to represent ingredient quantities, and in construction, where precise measurements are essential.
The equal pairs of rational numbers are: (1) 21 โ 7 โ and โ 9 3 โ , (2) โ 14 โ 8 โ and 7 4 โ , and (3) โ 13 โ 11 โ and 39 33 โ . The pair 13 7 โ and โ 65 35 โ are not equal.
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