A fraction is a relationship
A fraction describes equal parts of a whole. In 3/5, the denominator 5 tells you how many equal pieces make the whole; the numerator 3 tells you how many pieces you have. The equal-size condition matters: three unequal pizza slices do not necessarily make 3/5 of a pizza. You can also read a fraction as division: 3/5 means 3 ÷ 5, or 0.6. A percentage expresses that same relationship out of 100, so 0.6 is 60%. Fractions, decimals, and percentages are different ways of writing the same value.
Equivalent fractions and a common denominator
Multiplying the numerator and denominator by the same nonzero number preserves a fraction’s value. For example, 2/3 = 8/12 because both numbers were multiplied by 4. To add 1/3 and 1/4, first describe the pieces using the same size: 4/12 + 3/12 = 7/12. Adding denominators would mix up the size of the pieces. For subtraction, use the same process. To simplify 18/24, divide both numbers by their greatest common factor, 6, giving 3/4. Simplification changes the notation, not the amount.
Multiply, divide, and check the size
For multiplication, multiply the numerators and multiply the denominators: 2/3 × 3/5 = 6/15 = 2/5. You can interpret this as two thirds of three fifths. For division by a nonzero fraction, multiply by its reciprocal: 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2. The result makes sense because one and a half half-sized portions fit into three quarters. Dividing does not always make a number smaller. Before calculating, estimate the result so that a misplaced reciprocal is easier to notice.
Percentages in everyday decisions
To find 15% of 80, calculate 0.15 × 80 = 12. A 15% discount on an $80 item gives a price of $68 before any tax. Percentage change compares the difference with the original amount: (new − original) ÷ original × 100%. A change from 40 to 50 is a 25% increase, since 10/40 = 0.25. Returning from 50 to 40 is a 20% decrease, since the starting value is now 50. Always identify the reference whole before you divide.
Further reading
These resources offer a fuller treatment of the underlying concepts. The lesson, quiz wording, and flashcards here were created for this project.
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