Addition and Subtraction of Fractions

The Story

"1/4 + 1/4," said Ms. Rao, "are LIKE fractions — same denominator. Just add the numerators and keep the denominator: 1+1=2, so 1/4+1/4=2/4." Kabir got this one right immediately.

"Now try 1/4 + 1/3," she said. These are UNLIKE fractions — different denominators. "You cannot add numerators directly here, because quarters and thirds are different-sized pieces." She showed the fix: convert both to the SAME denominator first, using their LCM. LCM of 4 and 3 is 12. 1/4 = 3/12 (multiply by 3/3). 1/3 = 4/12 (multiply by 4/4). NOW add: 3/12+4/12=7/12.

Anaya tried subtraction the same way: 3/5 - 1/4. LCM of 5 and 4 is 20. 3/5=12/20. 1/4=5/20. Subtract: 12/20-5/20=7/20. "Same rule as addition," she said, "convert to a common denominator first, THEN combine the numerators."

"Here's the one thing that never changes whether adding or subtracting," said Ms. Rao. "The DENOMINATOR only gets combined once both fractions describe the same-sized pieces. Get that step right, and the rest is just simple addition or subtraction of whole numbers on top."

Kabir wanted to double-check the method on a bigger example: 2/3 + 1/6. He listed multiples of 3 (3,6,9) and of 6 (6,12,18) and spotted the LCM, 6. Since 6 was already 3's neighbor in the LCM sense, he only needed to convert 2/3: multiplying by 2/2 gave 4/6. Now both fractions shared sixths: 4/6 + 1/6 = 5/6. "Sometimes only ONE fraction needs converting," he realised, "if the other's denominator is already the LCM."

Anaya tried a subtraction where the answer came out negative: 1/3 - 3/4. Converting to twelfths gave 4/12 - 9/12 = -5/12. "A subtraction can genuinely go negative," said Ms. Rao, "exactly like whole-number subtraction can — 3-8 is -5, and fractions follow the same logic once the pieces match in size." This convinced both students that fraction addition and subtraction were never a separate universe of rules — just ordinary addition and subtraction, applied carefully once the pieces being combined were the same size.