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Fractions and decimals: cheat sheet

Number Foundations · The words and methods from the lessons, with an example of each.

mixed numeral

A whole number and a fraction. A negative one is both parts below zero: one and a half below zero.

−112=−1.5-1\tfrac{1}{2} = -1.5

simplest form

Divide the top and the bottom by their highest common factor. In simplest form (lowest terms) they share no factor but 1.

1218=12÷618÷6=23\tfrac{12}{18} = \tfrac{12 \div 6}{18 \div 6} = \tfrac{2}{3}

fraction to percentage

Percent means out of 100: make the fraction out of 100, or work out the decimal and multiply by 100.

34=75100=75%\tfrac{3}{4} = \tfrac{75}{100} = 75\%
3/4 of the bar is 75 of 100 squares
3/4 of the bar is 75 of 100 squares

adding fractions

Cut both into the same parts first (a common denominator; the LCM is the lowest), then add the tops. Never add the bottoms.

13+14=412+312=712\tfrac{1}{3} + \tfrac{1}{4} = \tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12}
Thirds and quarters both cut into twelfths: 4 + 3 = 7 twelfths
Thirds and quarters both cut into twelfths: 4 + 3 = 7 twelfths

dividing by a fraction

How many fit? Dividing by a fraction is multiplying by it flipped (top and bottom swapped): 23÷16=23×6=4\tfrac{2}{3} \div \tfrac{1}{6} = \tfrac{2}{3} \times 6 = 4.

3÷14=3×4=123 \div \tfrac{1}{4} = 3 \times 4 = 12

increase and decrease

15% off leaves 85%: × 0.85. 15% on makes 115%: × 1.15. For what percentage, divide and × 100: 18÷24=75%18 \div 24 = 75\%.

80×0.85=6880×1.15=9280 \times 0.85 = 68 \qquad 80 \times 1.15 = 92
15% off 80 leaves 68
15% off 80 leaves 68

placing a negative fraction

Split each unit into equal parts and count from zero. A negative counts left, and the part counts further from zero, not back towards it.

−114 is between −2 and −1-1\tfrac{1}{4} \text{ is between } {-2} \text{ and } {-1}
Quarters: A at −1 1/4, B at −3/4
Quarters: A at −1 1/4, B at −3/4

equivalent fractions

Multiply (or divide) the top and the bottom by the same number: the same amount, cut into more (or fewer) parts. Adding the same number to both changes the amount.

23=2×43×4=812\tfrac{2}{3} = \tfrac{2 \times 4}{3 \times 4} = \tfrac{8}{12}
2/3 and 8/12 shade the same amount
2/3 and 8/12 shade the same amount

fraction to decimal

Divide the top by the bottom. Some never end: 13=0.333…\tfrac{1}{3} = 0.333\ldots

38=3÷8=0.375\tfrac{3}{8} = 3 \div 8 = 0.375

decimal and percentage

Decimal to percentage: × 100 (0.5 is 50 hundredths, not 5%). Percentage to decimal: ÷ 100.

0.5=50%45%=0.450.5 = 50\% \qquad 45\% = 0.45

percentage to fraction

Put it over 100, then simplify.

16%=16100=42516\% = \tfrac{16}{100} = \tfrac{4}{25}

multiplying fractions

Multiply the tops and the bottoms, then simplify. Multiplying by a fraction less than 1 makes smaller.

23×34=612=12\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}
3/4 of 2/3 of a square: 6 of 12 parts
3/4 of 2/3 of a square: 6 of 12 parts

percentage of an amount

Of means multiply. Mentally: 10% is a tenth (8), 5% is half of that (4).

15% of 80=0.15×80=1215\% \text{ of } 80 = 0.15 \times 80 = 12
15% of a bar of 80 is 12
15% of a bar of 80 is 12

quick strategies

Change the order (4×2.7×25=100×2.74 \times 2.7 \times 25 = 100 \times 2.7), change the form (0.75 = 3/4), or split (12×2.5=24+612 \times 2.5 = 24 + 6). Then estimate to check.

0.75×36=34×36=270.75 \times 36 = \tfrac{3}{4} \times 36 = 27

negative fractions and decimals

The integer sign rules hold for every fraction and decimal. Sign first, then size.

−34×25=−310-\tfrac{3}{4} \times \tfrac{2}{5} = -\tfrac{3}{10}

efficient order

Multiply in any order, pairing friendly numbers; or split a number: 6×(−2.5)=6×(−2)+6×(−0.5)6 \times (-2.5) = 6 \times (-2) + 6 \times (-0.5).

25×(−3.7)×4=100×(−3.7)=−37025 \times (-3.7) \times 4 = 100 \times (-3.7) = -370