Rational Numbers

A rational number is a number that can be written in the form p/q, where p and q are integers and q is not zero. Every integer and every fraction is a rational number, so 5 = 5/1, -3/4 and 0 are all rational. A rational number is positive when the numerator and denominator have the same sign and negative when they have opposite signs. Equivalent rational numbers are formed by multiplying or dividing the numerator and denominator by the same non-zero integer, so -2/3 = -4/6 = 10/-15. A rational number is in standard form when its denominator is positive and the numerator and denominator have no common factor other than 1. In this chapter you represent rational numbers on a number line, compare them by first converting them to the same positive denominator, and find many rational numbers between any two given ones. You also add, subtract, multiply and divide rational numbers. For example, -3/4 + 5/6 = -9/12 + 10/12 = 1/12, and dividing by a rational number means multiplying by its reciprocal.

Detailed Worksheet: Rational Numbers Section A - Definitions (10 marks) 1. Define a rational number. Is 0 a rational number? Explain. (2 marks) 2. What is the standard form of a rational number? Write -18/45 in standard form. (2 marks) 3. What are equivalent rational numbers? Write two rational numbers equivalent to 3/-5. (2 marks) 4. When is a rational number positive and when negative? Classify 2/-3 and -5/-7. (2 marks) 5. What is the reciprocal of a rational number? Write the reciprocals of -4/9 and -1. (2 marks) Section B - Calculations and Applications (15 marks) 6. Write in standard form: (i) -44/72 (ii) 36/-24 (iii) -3/-15 (iv) -14/49. (3 marks) 7. Compare and put the correct sign (greater than, less than or equal): (i) -5/7 and 2/3 (ii) -4/5 and -5/7 (iii) -7/8 and 14/-16. (3 marks) 8. Find: (i) -7/11 + 2/11 (ii) 5/9 + (-3/6) (iii) -2/3 - 4/5 (iv) 3/14 - (-5/7). (3 marks) 9. Find: (i) 9/2 x (-7/4) (ii) -6/5 x 5/7 (iii) -4/5 divided by 12/-15 (iv) -2/3 divided by 1/3. (3 marks) 10. List five rational numbers between -1 and 0 and three rational numbers between -2/3 and -1/2. (3 marks) Section C - Diagrams (10 marks) 11. Draw a number line and represent 3/4, -5/8, -7/4 and 1 1/2 on it. (4 marks) 12. Draw a number line from -2 to 0 divided into thirds and mark -2/3, -1, -4/3 and -5/3. (3 marks) 13. Draw a number line and show that -1/2 and 2/-4 are represented by the same point. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Arrange in ascending order: -3/5, -2/5, -1/5, 2/-3, -5/6, 7/-12. (i) Convert to a common positive denominator. (ii) Write the order. (iii) Which is the greatest? (iv) Which is the smallest? (5 marks) 15. State whether true or false and justify: (i) Every integer is a rational number. (ii) Every rational number is an integer. (iii) 0 is neither positive nor negative. (iv) -3/4 is greater than -1/2. (v) There are infinitely many rational numbers between any two rational numbers. (5 marks) 16. The temperatures on a winter night changed by -3/4 degrees C per hour for 6 hours. (i) Find the total change. (ii) If it started at 2 degrees C, what was the final temperature? (iii) Write the final temperature in standard form. (iv) Find the average change per hour if the total change over 5 hours was -15/4 degrees C. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Always make the denominator positive before comparing or adding rational numbers.