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Number · Fractions & decimals

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The four operations with fractions

Why adding fractions needs a common denominator but multiplying doesn't, why dividing flips the second fraction, and why mixed numbers must become improper before multiplying or dividing.

Number · Fractions & decimals

The four operations with fractions

Why adding fractions needs a common denominator but multiplying doesn't, why dividing flips the second fraction, and why mixed numbers must become improper before multiplying or dividing.

Why it works

A fraction is a count of pieces of a particular size: 34\frac{3}{4} is three quarter-sized pieces. Every rule follows from that reading.

Adding needs same-sized pieces. You can't add 1 third and 1 quarter as a count — the pieces are different sizes (27\frac{2}{7} is the classic wrong answer, and it's smaller than 13\frac{1}{3}, which no sum of positives can be). Recut both into twelfths:

23+34=812+912=1712=1512.\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\tfrac{5}{12}.

The denominator is the piece size — it does NOT add; only the counts add.

Multiplying is "a fraction OF a fraction" — no common denominator needed. 23×34\frac{2}{3} \times \frac{3}{4} means two-thirds of three-quarters: cutting quarters into thirds multiplies the piece count top and bottom, so tops multiply and bottoms multiply:

23×34=612=12.\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}.

Cancel before multiplying when you can — same answer, smaller numbers.

Dividing flips the SECOND fraction. "÷25\div \frac{2}{5}" asks *how many two-fifths fit?* Fitting pieces of size 25\frac{2}{5} is the same as multiplying by its reciprocal 52\frac{5}{2} — because dividing by 2 halves and dividing by 15\frac{1}{5} quintuples. The first fraction is the amount being divided; it never flips:

34÷25=34×52=158.\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}.

Mixed numbers go improper first (for × and ÷). 212×1122\tfrac{1}{2} \times 1\tfrac{1}{2} is NOT 2×1+12×122 \times 1 + \tfrac{1}{2} \times \tfrac{1}{2} — that misses the cross terms (2×122 \times \tfrac12 and 1×121 \times \tfrac12), exactly like expanding brackets term-by-term. Convert:

223×112=83×32=246=4.2\tfrac{2}{3} \times 1\tfrac{1}{2} = \frac{8}{3} \times \frac{3}{2} = \frac{24}{6} = 4.

(For + and −, you may handle wholes and parts separately — addition has no cross terms.)

Never cancel across a + or −. In 3+44\frac{3 + 4}{4} the 4s don't cancel: cancelling is dividing top AND bottom, and the top's 4 is glued into a sum. Only factors of the whole top and whole bottom cancel.