Arrange the rational numbers in ascending order:-3/15,-3/-10,-3/6
Answers
Answer:
We will learn how to arrange the rational numbers in ascending order.
General method to arrange from smallest to largest rational numbers (increasing):
Step 1: Express the given rational numbers with positive denominator.
Step 2: Take the least common multiple (L.C.M.) of these positive denominator.
Step 3: Express each rational number (obtained in step 1) with this least common multiple (LCM) as the common denominator.
Step 4: The number having the smaller numerator is smaller.
1. Arrange the rational numbers −710, 5−8 and 2−3 in ascending order:
Solution:
We first write the given rational numbers so that their denominators are positive.
We have,
5−8 = 5×(−1)(−8)×(−1) = −58 and 2−3 = 2×(−1)(−3)×(−1) = −23
Thus, the given rational numbers with positive denominators are
−710, −58, −23
Now, LCM of the denominators 10, 8 and 3 is 2 × 2 × 2 × 3 × 5 = 120
We now write the numerators so that they have a common denominator 120 as follows:
−710 = (−7)×1210×12 = −84120,
−58 = (−5)×158×15 = −75120 and
−23 = (−2)×403×40 = −80120.
Comparing the numerators of these numbers, we get,
- 84 < -80 < -75
Therefore, −84120 < −80120 < −75120 ⇒ −710 < −23 < −58 ⇒ −710 < 2−3 < 5−8
Hence, the given numbers when arranged in ascending order are:
−710, 2−3, 5−8
2. Arrange the rational numbers 58, 5−6, 7−4 and 35 in ascending order.
Solution:
First we write each one of the given rational numbers with positive denominator.
Clearly, denominators of 58 and 35 are positive.
The denominators of 5−6 and 7−4 are negative.
So, we express 5−6 and 7−4 with positive denominator as follows:
5−6 = 5×(−1)(−6)×(−1) = −56 and 7−4 = 7×(−1)(−4)×(−1) = −74
Thus, the given rational numbers with positive denominators are
58, −56, −74 and 35
Now, LCM of the denominators 8, 6, 4 and 5 is 2 × 2 × 2 × 3 × 5 = 120
Now we convert each of the rational numbers to their equivalent rational number with common denominator 120 as follows:
58 = 5×158×15, [Multiplying the numerator and denominator by 120 ÷ 8 = 15]
⇒ 58 = 75120
−56 = (−5)×206×20, [Multiplying the numerator and denominator by 120 ÷ 6 = 20]
⇒ −56 = −100120
−74 = (−7)×304×30, [Multiplying the numerator and denominator by 120 ÷ 4 = 30]
⇒ −74 = −210120 and
35 = 3×245×24, [Multiplying the numerator and denominator by 120 ÷ 5 = 24]
⇒ 35 = 72120
Comparing the numerators of these numbers, we get,
-210 < -100 < 72 < 75
Therefore, −210120 < −100120 < 72120 < 75120 ⇒ −74 < −56 < 35 < 5/8 ⇒ 7−4 < 5−6 < 35 < 58
Hence, the given numbers when arranged in ascending order are:
7−4, 5−6, 35, 58.
Step-by-step explanation: