This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Welcome back nishakumari81320412 — been a while! Let's pick up where you left off. 1. Identify whether the following rational numbers will make terminating decimals or not, without writing the decimal form. A rational number (p)/(q) (in simplest form) has a terminating decimal expansion if the prime factors of its denominator q are only 2s and/or 5s. Otherwise, it is a non-terminating repeating decimal. a) (1)/(4) Step 1: Simplify the fraction (already in simplest form). Step 2: Find the prime factors of the denominator. 4 = 2^2 The only prime factor is 2. Answer: Terminating b) (22)/(7) Step 1: Simplify the fraction (already in simplest form). Step 2: Find the prime factors of the denominator. 7 = 7^1 The prime factor is 7, which is not 2 or 5. Answer: Non-terminating c) (-3)/(10) Step 1: Simplify the fraction (already in simplest form). Step 2: Find the prime factors of the denominator. 10 = 2 × 5 The prime factors are 2 and 5. Answer: Terminating d) (4)/(15) Step 1: Simplify the fraction (already in simplest form). Step 2: Find the prime factors of the denominator. 15 = 3 × 5 The prime factors include 3, which is not 2 or 5. Answer: Non-terminating e) (-5)/(22) Step 1: Simplify the fraction (already in simplest form). Step 2: Find the prime factors of the denominator. 22 = 2 × 11 The prime factors include 11, which is not 2 or 5. Answer: Non-terminating f) (-11)/(16) Step 1: Simplify the fraction (already in simplest form). Step 2: Find the prime factors of the denominator. 16 = 2^4 The only prime factor is 2. Answer: Terminating 2. Express the following rational numbers as terminating decimals. a) (-10)/(4) Step 1: Simplify the fraction. (-10)/(4) = (-5)/(2) Step 2: Perform the division. (-5)/(2) = -2.5 Answer: -2.5 b) (5)/(16) Step 1: Perform the division. (5)/(16) = 0.3125 Answer: 0.3125 c) (31)/(25) Step 1: Perform the division. (31)/(25) = 1.24 Answer: 1.24 d) (-21)/(15) Step 1: Simplify the fraction. (-21)/(15) = (-7)/(5) Step 2: Perform the division. (-7)/(5) = -1.4 Answer: -1.4 3. Express the following rational numbers as non-terminating decimals. Mark the repeating decimal digits with a bar over them. a) (2)/(3) Step 1: Perform the division. (2)/(3) = 0.666... Step 2: Mark the repeating digit with a bar. 0.6 Answer: 0.6 b) (-13)/(6) Step 1: Perform the division. (-13)/(6) = -2.1666... Step 2: Mark the repeating digit with a bar. -2.16 Answer: -2.16 c) (1)/(9) Step 1: Perform the division. (1)/(9) = 0.111... Step 2: Mark the repeating digit with a bar. 0.1 Answer: 0.1 d) (3)/(7) Step 1: Perform the division. (3)/(7) = 0.428571428571... Step 2: Mark the repeating block of digits with a bar. 0.428571 Answer: 0.428571 4. Write a terminating rational number and a non-terminating rational number between (-1)/(3) and (-3)/(4). Step 1: Convert the given fractions to decimals to understand the range. (-1)/(3) ≈ -0.333... (-3)/(4) = -0.75 We need numbers between -0.75 and -0.333.... Step 2: Find a common denominator for the given fractions to identify fractions in between. The least common multiple of 3 and 4 is 12. (-1)/(3) = (-1 × 4)/(3 × 4) = (-4)/(12) (-3)/(4) = (-3 × 3)/(4 × 3) = (-9)/(12) So we are looking for numbers between (-9)/(12) and (-4)/(12). Possible fractions include (-8)/(12), (-7)/(12), (-6)/(12), (-5)/(12). Step 3: Choose a terminating rational number. A terminating rational number has a denominator whose prime factors are only 2s and/or 5s. Let's choose (-6)/(12). (-6)/(12) = (-1)/(2) The denominator is 2, so it's terminating. Its decimal value is -0.5. -0.75 < -0.5 < -0.333... Terminating rational number: (-1)/(2) (or -0.5) Step 4: Choose a non-terminating rational number. A non-terminating rational number has a denominator with prime factors other than 2s and 5s. Let's choose (-7)/(12). The denominator is 12 = 2^2 × 3. Since it has a prime factor of 3, it is non-terminating. Its decimal value is (-7)/(12) ≈ -0.5833.... -0.75 < -0.5833... < -0.333... Non-terminating rational number: (-7)/(12) (or -0.583) Drop the next question 📸