⅗ Fractions & Percentages Studio

Fractions, decimals and percentages are three ways to represent the same idea: part of a whole or part of a quantity.

• fraction: e.g. \( \tfrac{3}{4} \)
• decimal: e.g. 0.75
• percentage: e.g. 75%

A big goal of the chapter is to move flexibly between these forms and use them to solve problems.

A fraction has a numerator (top) and denominator (bottom). The denominator tells you how many equal parts the whole is divided into. The numerator tells you how many of those parts you have.

• \( \tfrac{3}{8} \): 8 equal parts, 3 of them shaded/used.
• We can talk about fractions of shapes, sets of objects, and measures.
• We can represent fractions with grids, number lines, or bar models.

To simplify a fraction, divide numerator and denominator by the same factor until you cannot any more.

• Find the greatest common divisor (GCD).
• \( \tfrac{24}{30} \): GCD is 6, so \( \tfrac{24}{30} = \tfrac{4}{5} \).
• Fractions that simplify to the same result are equivalent fractions.

An improper fraction has a numerator at least as big as the denominator (e.g. \( \tfrac{11}{4} \)). A mixed number has a whole number plus a proper fraction (e.g. \( 2\tfrac{3}{4} \)).

• To go mixed → improper: \( a\tfrac{b}{c} = \tfrac{ac + b}{c} \).
• To go improper → mixed: divide, use quotient as the whole, remainder over the original denominator.

To add or subtract fractions with the same denominator, add/subtract the numerators and keep the denominator.

• \( \tfrac{3}{10} + \tfrac{4}{10} = \tfrac{7}{10} \).
• For different denominators, use a common denominator (usually the lowest common multiple).
• Always simplify the answer and convert improper answers to mixed numbers when appropriate.

Multiplying is the easy one: multiply numerators, multiply denominators.

• \( \tfrac{2}{3} \times \tfrac{5}{7} = \tfrac{10}{21} \).
• You can often simplify before multiplying by cancelling common factors.
• Mixed numbers should usually be turned into improper fractions first.

Dividing by a fraction is the same as multiplying by its reciprocal.

• Reciprocal of \( \tfrac{a}{b} \) is \( \tfrac{b}{a} \) (flip numerator and denominator).
• \( \tfrac{3}{4} \div \tfrac{2}{5} = \tfrac{3}{4} \times \tfrac{5}{2} = \tfrac{15}{8} = 1\tfrac{7}{8} \).

Mixed numbers appear in real measurements (1½ litres, 2¾ hours). Operations usually go:

• Convert to improper fractions.
• Do the calculation.
• Convert back to a mixed number if needed.

"Percent" means "per hundred". So:

• 25% = \( \tfrac{25}{100} = \tfrac{1}{4} \).
• 40% = \( \tfrac{40}{100} = \tfrac{2}{5} \).
• 60% = \( \tfrac{60}{100} = \tfrac{3}{5} \).

To go from percentage to decimal, divide by 100 (move the decimal point two places left).

To find a percentage of a quantity, turn the percentage into a fraction or decimal and multiply.

• 30% of 80 = \( 0.30 \times 80 = 24 \).
• 15% of 200 = \( \tfrac{15}{100} \times 200 = 30 \).
• For money problems, remember to round sensibly to cents where needed.

To express "A as a percentage of B":

• Compute \( \dfrac{A}{B} \).
• Turn this into a percentage by multiplying by 100.
• Attach % sign and round if necessary.

Example: 18 out of 24: \( \tfrac{18}{24} = 0.75 = 75\% \).

For mental calculations, some percentages are especially friendly:

• 10% = divide by 10.
• 5% = half of 10%.
• 1% = divide by 100.
• 25% = a quarter (÷4).
• 50% = a half (÷2).

You can combine these: 15% = 10% + 5%, 35% = 25% + 10%, and so on.