Mixed numbers and fractions of sets

Count the same-sized parts across wholes

One whole contains four quarters. Two wholes and three quarters therefore contain 2 × 4 + 3 = 11 quarters:

A mixed number separates whole units from a fractional part. An improper fraction counts all the parts together, with its numerator at least its denominator.

To reverse 17/5, make groups of five fifths. Three whole groups use fifteen fifths and leave two fifths:

The remainder names the fractional numerator; the denominator still tells the size of each part. If no parts remain, write the whole number alone: 20/5 = 4. Do not add an extra whole just because the original fraction was improper.

Equivalent fractions may use smaller parts or larger groups. For example, 22/4 = 10/4 = 5/2. When a task requests simplest form, reduce the fraction as well as changing its form. Otherwise, an equivalent improper fraction or mixed number still names the same amount.

A set can be the whole

There are 24 counters in a set. To find 3/8 of the set, share it into eight equal groups. Each group has 24÷8 = 3 counters. Select three groups: 3 × 3 = 9 counters.

You can ask the question in the other direction. If 9 of the 24 counters are selected, the selected fraction is 9/24 = 3/8. The denominator is the whole set, including selected and unselected counters. The 15 unselected counters do not become a new denominator.

Check your understanding 1 Name a fraction from the selected count

A tray holds 18 counters. Six are marked. What fraction of the whole tray is marked? What fraction is unmarked? State which count you use as the denominator.

Your marked and unmarked fractions

Show answer

The whole tray contains 18 counters. The marked fraction is 6/18 = 1/3. Twelve are unmarked, so their fraction is 12/18 = 2/3. Both fractions use the same whole. Their sum is one whole tray. Fractions such as 6/12 compare the marked count with the unmarked count instead.

The denominator tells how many equal groups make the set. The numerator tells how many groups to select. Selecting three counters merely because the numerator is three would not find three eighths of the set.

Check your understanding 2 Identify the whole set

A box holds 35 beads. Two fifths are green. How many beads are green?

Number of green beads

Check answer

Show answer

14 beads: 35÷5 × 2 = 14. The 35 beads, not the five groups, form the whole set.

Check your understanding 3 Move between the two forms

Write 41/6 as an improper fraction. Write 23/6 as a mixed number. Use groups of sixths to explain both.

Your answer or drawing on paper

Show answer

Four wholes contain 24 sixths; one more gives 25/6. For 23/6, three wholes use eighteen sixths and leave five, giving 35/6.

Check which whole the fraction uses

Check your understanding 4 The same selected objects can name different fractions

A box holds 35 beads. Fourteen are green. Seven of the green beads have a star. What fraction of the green beads have a star? What fraction of all the beads in the box have a star?

Your two fractions and their wholes

Show answer

Seven out of 14 green beads is 7/14 = 1/2. Seven out of all 35 beads is 7/35 = 1/5. The selected seven beads did not change; the reference whole changed. Neither fraction means that only seven beads are in the box.

A class has 30 pupils. Half are in a choir. One third of the whole class also plays an instrument. These two groups can overlap, so you cannot infer how many do neither unless the question tells you who belongs to both groups.

Do not join fraction parts as disjoint groups unless the context establishes that they are separate.