Fractions and decimals: cheat sheet

mixed numeral

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

-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.

\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.

\tfrac{3}{4} = \tfrac{75}{100} = 75\%

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.

\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

dividing by a fraction

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

3 \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: $18 \div 24 = 75\%$ .

80 \times 0.85 = 68 \qquad 80 \times 1.15 = 92

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.

-1\tfrac{1}{4} \text{ is between } {-2} \text{ and } {-1}

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.

\tfrac{2}{3} = \tfrac{2 \times 4}{3 \times 4} = \tfrac{8}{12}

fraction to decimal

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

\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\% \qquad 45\% = 0.45

percentage to fraction

Put it over 100, then simplify.

16\% = \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.

\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}

percentage of an amount

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

15\% \text{ of } 80 = 0.15 \times 80 = 12

quick strategies

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

0.75 \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.

-\tfrac{3}{4} \times \tfrac{2}{5} = -\tfrac{3}{10}

efficient order

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

25 \times (-3.7) \times 4 = 100 \times (-3.7) = -370