Unit 4 · Topic 04 · Fractions
Kabir needed half of three-quarters of a chocolate bar, and had no idea where to start.
Anaya said the answer was smaller than either fraction alone — which felt backwards.
Multiplying fractions makes things SMALLER, not bigger — the opposite of what whole-number multiplication ever does.
Ms. Rao held up a chocolate bar already split into quarters, 3 of the 4 pieces shaded. "This shaded part is 3/4 of the bar. Kabir wants HALF of that shaded part. How much of the WHOLE bar does he get?"
Kabir guessed. "More than 3/4, since we're combining two fractions?"
"Let's check with the picture instead of guessing," said Ms. Rao. She split each quarter piece in half, turning the whole bar into eighths. The shaded 3/4 was now 6 of the 8 tiny pieces. Half of those 6 pieces is 3 pieces — 3 out of 8.
"So half of three-quarters is three-eighths?" said Kabir. "That's SMALLER than three-quarters. Multiplying made it smaller, not bigger."
"Exactly the twist beginners always trip on," said Ms. Rao. "Multiplying by a fraction less than 1 always shrinks the amount, because you're taking only a PART of a part. The rule matches the picture: multiply numerator by numerator, denominator by denominator. 1/2 × 3/4 = (1×3)/(2×4) = 3/8."
Anaya tried one herself: 2/3 × 3/5. "Numerators: 2×3=6. Denominators: 3×5=15. So 6/15." She noticed 6 and 15 share a common factor of 3. "Simplified, that's 2/5."
"Always simplify your final answer," said Ms. Rao. "Now — division is the opposite trick. To divide by a fraction, flip that fraction upside down and multiply instead. Dividing 1/2 by 1/4 becomes 1/2 × 4/1."
Kabir worked it through: (1×4)/(2×1) = 4/2 = 2. "Dividing by a fraction less than 1 makes the answer BIGGER? That's the opposite of multiplying!"
"Right again — division by a small fraction asks 'how many of these small pieces fit into this amount,' and the answer is naturally a bigger count," said Ms. Rao. "Half a pizza divided into quarter-slices gives you 2 slices — more pieces than you started with as a count, even though the total pizza amount hasn't changed."
To multiply two fractions: multiply the numerators together, multiply the denominators together, then simplify. a/b × c/d = (a×c)/(b×d).
Multiplying by a fraction less than 1 makes the result SMALLER than the original — because you are taking a part of a part, not combining full amounts.
To divide by a fraction: flip (find the reciprocal of) the second fraction, then multiply instead. a/b ÷ c/d = a/b × d/c.
Dividing by a fraction less than 1 makes the result BIGGER — the opposite effect of multiplying, because you're counting how many small pieces fit into the whole.
Multiply 9/7 × 6 and 11 × 4/5.
Convert 4 2/5 to improper, then multiply by 15/11.
A student studies 1¾ hours daily for 7 days — use 7/4 × 7 = 49/4.
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