Fractions

Mixed and Improper Fractions

Convert the mixed number 2 3/5 to the improper fraction 13/5, then convert back. Multiply the whole part by the denominator, add the numerator, and place the result over the denominator.

Example

Convert 2 3/5 to an improper fraction and back.

highlighted = computed this step

Step 1 — Set up

Set up the fraction expression.

2 352\ \frac{3}{5}
Start with 2 wholes and 3/5whole 1whole 2extra 3/52 wholes and 3 extra pieces of size 1/5Each whole is split into 5 equal pieces.

Step 2 — Multiply whole by denominator

Multiply the whole number by the denominator: 2 x 5 = 10.

2×5=102 \times 5 = \hl{10}
2 wholes x 5 pieces each = 10 pieceswhole 1whole 2extra 3/510 whole pieces + 3 extra pieces = 13 piecesThe whole-number part contributes 10 denominator-sized pieces.

Step 3 — Add numerator

Add the numerator: 10 + 3 = 13.

10+3=1310 + 3 = \hl{13}
Add the extra 3 pieces: 10 + 3 = 13whole 1whole 2extra 3/510 whole pieces + 3 extra pieces = 13 piecesNow there are 13 total shaded pieces.

Step 4 — Write improper fraction

Write the improper fraction with numerator 13 and denominator 5.

2 35=1352\ \frac{3}{5} = \hlmath{\frac{13}{5}}
All pieces use denominator 5: 13/5whole 1whole 2extra 3/510 whole pieces + 3 extra pieces = 13 piecesThe improper numerator counts every shaded 1/5-sized piece.

Step 5 — Divide back

Divide back: 13 ÷ 5 = 2 remainder 3.

13÷5=2 r 313 \div 5 = \hl{2} \ \mathrm{r}\ \hl{3}
13 divided by 5 gives 2 remainder 3whole 1whole 2extra 3/510 whole pieces + 3 extra pieces = 13 piecesGrouping 13 pieces by 5 recovers the mixed number.

Step 6 — Result

Read the final result.

135=2 35\frac{13}{5} = 2\ \frac{3}{5}
13/5 is 2 3/5whole 1whole 2extra 3/510 whole pieces + 3 extra pieces = 13 piecesThe value did not change: 2 3/5 = 13/5.13/513/5 = 2 3/5
mixed-improper-conversion A mixed number like 2 3/5 means 2 whole pieces plus 3/5. To write it as a single fraction, count all fifths: 2 × 5 = 10, plus 3 = 13 fifths. To go back, divide: 13 ÷ 5 = 2 remainder 3, so 13/5 = 2 3/5.