Q1: What is the sum of \(\frac{3}{5}\) and \(\frac{-2}{5}\)? (a) \(\frac{5}{5}\) (b) \(\frac{1}{5}\) (c) \(\frac{-1}{5}\) (d) \(\frac{1}{10}\)
Solution:
Ans: (b) Explanation: When adding rational numbers with the same denominator, we add the numerators and keep the denominator the same. \(\frac{3}{5} + \frac{-2}{5} = \frac{3 + (-2)}{5} = \frac{1}{5}\). Option (a) would result from adding absolute values incorrectly, option (c) reverses the sign, and option (d) incorrectly adds denominators.
Q2: Which expression represents the additive inverse of \(\frac{-7}{12}\)? (a) \(\frac{-12}{7}\) (b) \(\frac{12}{7}\) (c) \(\frac{7}{12}\) (d) \(\frac{-7}{-12}\)
Solution:
Ans: (c) Explanation: The additive inverse of a number is the number that, when added to the original number, gives zero. The additive inverse of \(\frac{-7}{12}\) is \(\frac{7}{12}\) because \(\frac{-7}{12} + \frac{7}{12} = 0\). Option (a) is the reciprocal with wrong sign, option (b) is the reciprocal, and option (d) equals the original number.
Ans: (a) Explanation: When subtracting rational numbers with the same denominator, subtract the numerators and keep the denominator: \(\frac{5}{8} - \frac{3}{8} = \frac{5-3}{8} = \frac{2}{8}\). While this can be simplified to \(\frac{1}{4}\), option (a) is the direct result. Option (b) adds instead of subtracts, option (c) is undefined, and option (d) skips the calculation step.
Q4: What is \(\frac{-3}{4} + \frac{5}{6}\)? (a) \(\frac{2}{10}\) (b) \(\frac{1}{12}\) (c) \(\frac{-1}{12}\) (d) \(\frac{1}{2}\)
Solution:
Ans: (b) Explanation: To add rational numbers with different denominators, find the LCD (Least Common Denominator). The LCD of 4 and 6 is 12. \(\frac{-3}{4} = \frac{-9}{12}\) and \(\frac{5}{6} = \frac{10}{12}\) \(\frac{-9}{12} + \frac{10}{12} = \frac{1}{12}\). Option (a) incorrectly adds denominators, option (c) has the wrong sign, and option (d) is an incorrect simplification.
Ans: (b) Explanation: Subtracting a negative number is the same as adding its positive: \(\frac{2}{3} - \frac{-1}{3} = \frac{2}{3} + \frac{1}{3} = \frac{2+1}{3} = \frac{3}{3} = 1\). Option (a) subtracts instead of adding, option (c) has the wrong sign, and option (d) incorrectly multiplies denominators.
Q6: Which of the following is equivalent to \(\frac{-5}{9} + \frac{-2}{9}\)? (a) \(\frac{-7}{18}\) (b) \(\frac{3}{9}\) (c) \(\frac{-7}{9}\) (d) \(\frac{7}{9}\)
Solution:
Ans: (c) Explanation: When adding two negative rational numbers with the same denominator, add the numerators (keeping the negative signs) and keep the denominator: \(\frac{-5}{9} + \frac{-2}{9} = \frac{-5 + (-2)}{9} = \frac{-7}{9}\). Option (a) incorrectly doubles the denominator, option (b) has the wrong operation and sign, and option (d) has the wrong sign.
Ans: (b) Explanation: Convert to a common denominator. The LCD of 10 and 5 is 10. \(\frac{3}{5} = \frac{6}{10}\) \(\frac{7}{10} - \frac{6}{10} = \frac{1}{10}\). Option (a) subtracts denominators incorrectly, option (c) uses wrong LCD, and option (d) is an incorrect simplification.
Q8: What is the value of \(\frac{-1}{2} + \frac{3}{4} - \frac{1}{4}\)? (a) \(\frac{-1}{2}\) (b) \(\frac{0}{4}\) (c) \(\frac{1}{4}\) (d) \(\frac{3}{4}\)
Solution:
Ans: (b) Explanation: Convert all fractions to common denominator 4: \(\frac{-1}{2} = \frac{-2}{4}\) \(\frac{-2}{4} + \frac{3}{4} - \frac{1}{4} = \frac{-2 + 3 - 1}{4} = \frac{0}{4} = 0\). Option (a) only combines first two terms, option (c) makes a calculation error, and option (d) ignores the negative term.
Section B: Fill in the Blanks
Q9: The sum of a rational number and its additive inverse is always __________.
Solution:
Ans: 0 (or zero) Explanation: By definition, the additive inverse of any number is the number that when added to it produces zero. For example, \(\frac{3}{5} + \frac{-3}{5} = 0\).
Q10: When adding two rational numbers with different denominators, we must first find the __________.
Solution:
Ans: Least Common Denominator (or LCD or common denominator) Explanation: To add or subtract rational numbers with different denominators, we need to convert them to equivalent fractions with the same denominator, preferably the LCD.
Q11: The result of \(\frac{4}{7} + \frac{3}{7}\) is __________.
Solution:
Ans: \(\frac{7}{7}\) or 1 Explanation: Since the denominators are the same, add the numerators: \(\frac{4}{7} + \frac{3}{7} = \frac{4+3}{7} = \frac{7}{7} = 1\). This tests understanding of adding like fractions.
Q12: Subtracting \(\frac{-3}{8}\) is the same as adding __________.
Solution:
Ans: \(\frac{3}{8}\) Explanation: Subtracting a negative number is equivalent to adding its opposite (additive inverse). Therefore, subtracting \(\frac{-3}{8}\) equals adding \(\frac{3}{8}\).
Q13: The LCD of denominators 6 and 8 is __________.
Solution:
Ans: 24 Explanation: The Least Common Denominator is the smallest number that both denominators divide into evenly. Multiples of 6: 6, 12, 18, 24... Multiples of 8: 8, 16, 24... The smallest common multiple is 24.
Q14: When both rational numbers being added are negative, the sum will be __________.
Solution:
Ans: negative Explanation: Adding two negative numbers always results in a negative sum. For example, \(\frac{-2}{5} + \frac{-3}{5} = \frac{-5}{5} = -1\), which is negative.
Section C: Word Problems
Q15: A scuba diver descends \(\frac{3}{4}\) meters below sea level, then descends another \(\frac{5}{4}\) meters. What is the diver's total depth below sea level?
Solution:
Ans: Depth is measured as negative when below sea level. First descent: \(\frac{-3}{4}\) meters Second descent: \(\frac{-5}{4}\) meters Total depth = \(\frac{-3}{4} + \frac{-5}{4} = \frac{-3 + (-5)}{4} = \frac{-8}{4} = -2\) meters Final Answer: 2 meters below sea level (or -2 meters)
Q16: A recipe requires \(\frac{2}{3}\) cup of sugar, but Maria only has \(\frac{1}{6}\) cup. How much more sugar does she need?
Solution:
Ans: Amount needed = \(\frac{2}{3}\) cup Amount available = \(\frac{1}{6}\) cup Amount more needed = \(\frac{2}{3} - \frac{1}{6}\) LCD of 3 and 6 is 6 \(\frac{2}{3} = \frac{4}{6}\) \(\frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\) Final Answer: \(\frac{1}{2}\) cup of sugar
Q17: The temperature in the morning was \(\frac{-5}{2}\)°C. By afternoon, it rose by \(\frac{9}{2}\)°C. What was the afternoon temperature?
Solution:
Ans: Morning temperature = \(\frac{-5}{2}\)°C Temperature increase = \(\frac{9}{2}\)°C Afternoon temperature = \(\frac{-5}{2} + \frac{9}{2} = \frac{-5 + 9}{2} = \frac{4}{2} = 2\)°C Final Answer: 2°C
Q18: A water tank had \(\frac{7}{8}\) of its capacity filled. After using \(\frac{3}{8}\) of the capacity for irrigation, how much water remains in the tank as a fraction of its capacity?
Solution:
Ans: Initial amount = \(\frac{7}{8}\) of capacity Amount used = \(\frac{3}{8}\) of capacity Amount remaining = \(\frac{7}{8} - \frac{3}{8} = \frac{7-3}{8} = \frac{4}{8} = \frac{1}{2}\) Final Answer: \(\frac{1}{2}\) of the tank's capacity
Q19: A submarine was at a depth of \(\frac{-15}{4}\) meters. It rose \(\frac{7}{4}\) meters and then descended \(\frac{3}{4}\) meters. What is the submarine's final depth?
Q20: Jamie walked \(\frac{3}{5}\) of a mile to school, then walked \(\frac{2}{3}\) of a mile to the library. What is the total distance Jamie walked?
Solution:
Ans: Distance to school = \(\frac{3}{5}\) mile Distance to library = \(\frac{2}{3}\) mile Total distance = \(\frac{3}{5} + \frac{2}{3}\) LCD of 5 and 3 is 15 \(\frac{3}{5} = \frac{9}{15}\) \(\frac{2}{3} = \frac{10}{15}\) \(\frac{9}{15} + \frac{10}{15} = \frac{19}{15}\) miles \(\frac{19}{15} = 1\frac{4}{15}\) miles Final Answer: \(\frac{19}{15}\) miles or \(1\frac{4}{15}\) miles
The document Worksheet (with Solutions): Rational Numbers: Addition And Subtraction is a part of the Grade 7 Course Math Grade 7.
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