A fraction of a set compares how many objects meet a condition with how many objects there are in total. Picture a board with 20 cards, of which 5 are marked. Those five make up 5/20 of the board, which simplifies to 1/4. The numerator counts the marked cards and the denominator counts all the cards.
Whole: 20 cards
Selected part: 5 marked cards
Here every object counts as one unit. A large card and a small card count the same, so 1/4 describes the number of cards, not the surface they cover. If the question were about area or weight, you would have to measure those quantities instead of counting. Before writing anything, decide what the whole is and which condition picks out the part.
In the fruit box example below, the pears did not disappear; the whole changed. A fraction of a set describes a relationship with one specific whole, so it helps to always say “of the box now” or “of the original box.”
Finding how many objects match a fraction
When you know the fraction and the total, split the total into equal groups and take as many groups as the numerator says. If 2/5 of 30 beads are green, each fifth holds 30 ÷ 5 = 6 beads and two fifths hold 2 × 6 = 12. The fraction of a number calculator does the same operation as a product, 2/5 × 30.
When you count whole pieces, the answer must be a whole number of objects. If you get 2.5 marbles, recheck the problem: the fraction may be approximate, the total may be different, or the problem may be about a measurement rather than a count. A fraction greater than 1 can multiply a quantity, but it cannot be the part of a single set, because there cannot be more selected objects than objects in the set.
Worked example: pears in a box of fruit
A box holds 18 pieces of fruit and 6 are pears. What fraction of the box is pears? How does it change if you add 6 apples?
- Count the total. The box has 18 pieces of fruit, which will be the denominator.
- Count the part. There are 6 pears, which will be the numerator.
- Write and simplify. Divide 6/18 by 6 on top and bottom to get 1/3.
- Change the whole. With 6 more apples there are 24 pieces of fruit, and the same pears are 6/24 = 1/4.
Result. The pears are 1/3 of the box, and 1/4 once the apples are added.
Check. 1/3 of 18 is 6 and 1/4 of 24 is also 6, so both fractions describe the same six pears.
Part-whole and part-part
A basket holds 3 red balls and 7 blue ones. The red balls are 3/10 of the total, and the ratio of red to blue is 3 : 7. The first uses the total as the denominator. The second compares two groups with each other and is not a fraction of the basket. To go from a ratio to a fraction of the total, add the groups first: 3 + 7 = 10. The what fraction is one amount of another tool calculates the quotient of part and total, and ratio to fraction converts a relationship between two groups.
If objects belong to two categories at once, count each one only once before forming the fraction. To see the same number as a part, a quotient, a measure, an operator, and a ratio, read meanings of a fraction.
Practice
To practice, count the objects in a collection of your own, change the total without changing the part, and watch how the fraction moves. The visual fractions tool lets you explore sets alongside other models.