Rational Numbers

3/5

Step 1: Notice that 3/5 is a common factor in both terms. Use the Distributive Property: a×b + a×c = a×(b+c). Step 2: (3/5)×(2/7) + (3/5)×(5/7) = (3/5) × (2/7 + 5/7). Step 3: Add the fractions inside the bracket: 2/7 + 5/7 = 7/7 = 1. Step 4: (3/5) × 1 = 3/5. The answer is 3/5. Using distributivity makes the calculation much simpler!

Rational numbers are closed under division by non-zero rational numbers

Step 1: Let us check each option. Closure means the result of an operation on two numbers from the set stays within the set. Step 2: For multiplication: (p/q) × (r/s) = pr/qs, which is always a rational number. So rational numbers ARE closed under multiplication. Step 3: For subtraction: (p/q) - (r/s) = (ps-rq)/qs, which is always rational. So they ARE closed under subtraction too. Step 4: For division: dividing by zero is undefined. But if we divide any rational number by a non-zero rational number, the result is always a rational number. So Option A is correct.

7/11

Step 1: The additive inverse of a number is the number that when added to it gives 0. Step 2: We need a number x such that (-7/11) + x = 0. Step 3: Solving: x = 0 - (-7/11) = 7/11. Step 4: Check: (-7/11) + (7/11) = 0/11 = 0. Correct! The additive inverse is 7/11. Common mistake: Students confuse additive inverse with multiplicative inverse (reciprocal). Multiplicative inverse of -7/11 would be -11/7.

1/16

Step 1: First solve the expression inside the brackets. Find LCM of 4 and 8, which is 8. Step 2: Convert -1/4 to -2/8. Now add: -2/8 + 3/8 = 1/8. Step 3: Now multiply: (1/2) × (1/8) = 1/16. Step 4: The answer is 1/16. Always follow BODMAS - solve brackets first, then multiplication. Common mistake: Students sometimes multiply before solving the bracket.