What does a fraction show?
A fraction represents part of a whole, a division, or a ratio. In 3/4, the number on top is the numerator: it tells you how many parts are being considered. The number on the bottom is the denominator: it tells you how many equal parts make up the whole. The parts must be equal-sized for the usual part-of-a-whole interpretation.
Picture the denominator
Imagine a sandwich cut into four equal pieces. One piece is 1/4 of the sandwich; three pieces are 3/4. If the same sandwich were cut into eight equal pieces, one piece would be 1/8. A larger denominator does not automatically mean a larger fraction: when the numerator is fixed and the parts refer to the same whole, more equal parts means each part is smaller.
Equivalent fractions
Fractions can look different and still represent the same amount. For example, 1/2 and 2/4 are equivalent. To make an equivalent fraction, multiply or divide the numerator and denominator by the same non-zero number. Multiplying both parts of 1/2 by 2 gives 2/4; dividing the numerator and denominator of 4/8 by 4 gives 1/2.
Compare fractions thoughtfully
When fractions have the same denominator, the one with the larger numerator is larger: 3/8 is greater than 1/8. When they have the same positive numerator, the fraction with the smaller denominator is larger: 1/3 is greater than 1/5. For different numerators and denominators, you can rewrite them with a common denominator or compare their decimal values when appropriate.
Add and subtract like-sized parts
When denominators match, add or subtract the numerators and keep the denominator. For example, 1/5 + 2/5 = 3/5 because you are combining fifth-sized parts. For different denominators, first rewrite the fractions with a common denominator: 1/2 + 1/4 = 2/4 + 1/4 = 3/4. The denominator describes the size of the parts, so it cannot simply be added.
Fractions in everyday life
Recipes use fractions to adjust quantities; time uses fractions of an hour; and measurements often use halves, quarters, and eighths. If a recipe needs 3/4 cup of milk and you make half the recipe, you need half of 3/4 cup, which is 3/8 cup. Drawing a quick bar or circle can help you check whether an answer is sensible.
A reliable problem-solving routine
First identify what the whole is. Next, check whether the parts are equal. Write the numerator and denominator carefully, then decide whether the task asks you to compare, simplify, add, or subtract. Finally, estimate the answer: adding two positive fractions should not give a negative result, and a part of a whole should usually fall between zero and one.
Check your understanding
- A pizza has 8 equal slices. You eat 3. What fraction remains?
- Simplify 4/8.
- Calculate 2/7 + 3/7.
- Which is larger: 1/3 or 1/5?
Show suggested answers
- 5/8 remains.
- 1/2, dividing top and bottom by 4.
- 5/7.
- 1/3, because thirds are larger pieces than fifths of the same whole.
These are example answers. Other answers may also be correct if they explain the idea clearly.
Key takeaway: Understanding grows when you connect new information to examples and practise explaining it clearly.