Rational Number — Important Questions

23/28

Step 1: Let the number to be subtracted be x. Then (-3/7) - x = -5/4. Step 2: Rearranging: x = (-3/7) - (-5/4) = -3/7 + 5/4. Step 3: Find LCM of 7 and 4 = 28. Step 4: -3/7 = -12/28 and 5/4 = 35/28. Step 5: x = -12/28 + 35/28 = 23/28. Verification: (-3/7) - (23/28) = -12/28 - 23/28 = -35/28 = -5/4. Correct! Students often confuse 'what is subtracted' with 'what is added', making sign errors.

Distributive property of multiplication over addition

Step 1: Examine the structure of the equation: a × (b + c) = a×b + a×c. Step 2: Here a = 2/3, b = -5/7, c = 3/4. Step 3: The left side shows multiplication of 2/3 with a sum of two rational numbers. Step 4: The right side shows the product distributed — 2/3 multiplied separately with each term. Step 5: This exactly matches the distributive property: a × (b+c) = a×b + a×c. Commutative property would be a×b = b×a. Associative would involve regrouping of three numbers in multiplication only. Closure means result is still rational.

-19/12

Step 1: First simplify the expression algebraically: (x-y) - (y-x) = x - y - y + x = 2x - 2y = 2(x-y). Step 2: Find x - y = (-3/8) - (5/12). LCM of 8 and 12 = 24. Step 3: x - y = -9/24 - 10/24 = -19/24. Step 4: Therefore 2(x-y) = 2 × (-19/24) = -38/24 = -19/12. Final: Answer is -19/12. Common mistake: Students compute (x-y) and (y-x) separately without simplifying, and may make sign errors when subtracting (y-x).

25/36, 26/36, 27/36, 28/36, 29/36

Step 1: Find LCM of denominators 3 and 6. LCM = 6. Step 2: Convert: 2/3 = 4/6 and 5/6 = 5/6. Step 3: Since 5 rational numbers are needed, multiply numerator and denominator of each by (5+1) = 6: 4/6 = 24/36 and 5/6 = 30/36. Step 4: Rational numbers between 24/36 and 30/36 with denominator 36 are: 25/36, 26/36, 27/36, 28/36, 29/36. Step 5: These are exactly 5 numbers between the given rational numbers. Option B uses denominator 18 which is incorrect. Options C and D are wrong ranges. This method always works reliably.