Number and algebra · Lesson 3 of 4

Fractions and mixed numbers

A fraction is a count of equal pieces. Once you see the piece size, adding, subtracting, multiplying and dividing all become one picture each.

About 20 min · Interactive · 8 practice questions

Set two fractions with the + and − buttons, choose an operation, and watch the bars re-cut themselves. The readouts give the exact answer three ways.

Result5/6
Mixed number—
Decimal0.833…
1 first
1 3 second

Pieces and piece sizes

Most people read 3/4 as "three over four", a pair of numbers held together by a line. Instead, read it as a count: three pieces, each of the size called a quarter. The bottom number names the piece size. The top number counts how many pieces you have.

That one reading explains every rule in this lesson. You can add 3 quarters and 2 quarters because they are the same size of piece: 5 quarters. You cannot add 1 half and 1 third by counting, because a half and a third are different sizes. So you cut both into a piece size that fits them both, sixths, and then count: 3 sixths plus 2 sixths is 5 sixths. That is all a common denominator is.

Multiplying is different. 2/3 × 3/4 means "take two thirds of three quarters". It scales an amount rather than combining two counts, so no re-cutting is needed. Dividing asks the opposite question: 3/4 ÷ 1/8 means "how many eighths fit into three quarters". Six do. For the test, the consequence is simple: decide which question is being asked before touching the numbers.

Bars showing one half, one third, the same bar cut into sixths, and the sum five sixths halves thirds sixths sum 1/2 1/3 1/2 = 3/6 1/3 = 2/6 3/6 2/6 = 5/6 Sixths are the first piece size that both the half-cut and the third-cuts land on.
The half-cut and the third-cuts do not line up with each other, but they all line up with sixths. Once both bars are in sixths, adding is just counting: 3 + 2 = 5 sixths.

The six methods

Every fraction question on the test is one of these six jobs, or two of them chained together. Each is a short recipe. The worked line under each card shows the recipe on real numbers.

1. Equivalent fractions and simplifying

  1. Multiplying top and bottom by the same number cuts every piece smaller without changing the amount.
  2. Dividing top and bottom by the same number does the reverse. Keep dividing until no number goes into both.
  3. Fastest: find the largest number that divides both and divide once.

18/24 → both divide by 6 → 3/4. (Or ÷2 then ÷3: 9/12 → 3/4.)

2. Mixed numbers and improper fractions

  1. Mixed → improper: multiply the whole by the bottom, add the top, keep the bottom.
  2. Improper → mixed: divide top by bottom. The quotient is the whole; the remainder stays over the bottom.

2 3/4 → (2 × 4 + 3)/4 = 11/4.   17/5 → 17 ÷ 5 = 3 r 2 → 3 2/5.

3. Add or subtract with the LCD

  1. Turn any mixed numbers into improper fractions first.
  2. Find the lowest common denominator (LCD): the smallest number both bottoms fit into.
  3. Rewrite each fraction with that bottom, scaling the top by the same factor.
  4. Add or subtract the tops only. Simplify, and convert back to a mixed number if asked.

2 1/2 + 1 3/4 → 5/2 + 7/4 → 10/4 + 7/4 = 17/4 = 4 1/4.

4. Multiply

  1. Convert mixed numbers to improper fractions.
  2. Cancel first: divide any top and any bottom by a shared factor.
  3. Multiply the tops together and the bottoms together. No common denominator is needed.

2/3 × 9/10 → cancel 2 with 10, 3 with 9 → 1/1 × 3/5 = 3/5.

5. Divide: keep, flip, multiply

  1. Keep the first fraction as it is.
  2. Flip the second fraction (swap its top and bottom).
  3. Change ÷ to × and multiply as in method 4.

3/4 ÷ 1/8 → 3/4 × 8/1 → cancel 4 with 8 → 3 × 2 = 6.

6. Fraction of an amount, and the whole from a part

  1. "Of" means multiply. Divide the amount by the bottom to find one piece, then multiply by the top.
  2. Reverse it to find the whole: divide the part by the top to get one piece, then multiply by the bottom.

3/8 of 240 → 240 ÷ 8 = 30 → × 3 = 90.   If 3/5 of a class is 18: 18 ÷ 3 = 6 → × 5 = 30 students.

Why cancelling first is allowed

When you multiply fractions, every top ends up multiplied together on the top line and every bottom on the bottom line. A factor that appears in any top and any bottom would cancel at the end anyway. Cancelling it early just keeps the numbers small, which matters most in the non-calculator section.

Calculator that shows its working

Type a fraction calculation

Try 2 1/2 + 1 3/4, 3/4 ÷ 1/8, 2/3 × 9/10 or 3 1/4 − 1 3/4. Use + − × ÷ (or x, *, : and a spaced / for divide). The steps name the LCD and the common factor, not just the answer.

    Why multiplying needs no common denominator

    Adding combines two counts, so the pieces must match. Multiplying takes a part of a part, which is a different action. Draw one fraction across a rectangle and the other down it, and the overlap is the answer. The grid makes its own denominator: 3 rows times 4 columns gives 12 cells, without any searching.

    Area model for two thirds times three quarters: a grid of three rows and four columns with the overlap of two rows and three columns shaded 3 of 4 columns = 3/4 of the width 2 of 3 rows = 2/3 overlap: 2 × 3 = 6 cells whole grid: 3 × 4 = 12 cells 6/12 = 1/2 3/4 (columns) 2/3 (rows) 2/3 of 3/4 = both Tops multiply to count the overlap; bottoms multiply to count the whole grid.
    2/3 × 3/4: the blue columns are 3/4 of the width, the gold rows are 2/3 of the height, and the green overlap is 2/3 of 3/4. It covers 6 of the 12 cells, which is 1/2.

    This is the picture behind "multiply the tops and multiply the bottoms". It also explains the cancelling in method 4: the 3 on top of 3/4 and the 3 underneath 2/3 are the same three rows, counted once in the overlap and once in the whole.

    Denominators and anchors to know

    The non-calculator section rewards knowing these without working. The LCD is the smallest number both denominators fit into, which is often smaller than their product: 4 and 6 need 12, not 24.

    DenominatorsLCDWhy
    2 and 36No shared factor, so multiply: 2 × 3
    3 and 412No shared factor: 3 × 4
    4 and 612Both fit into 12; 24 also works but is not lowest
    2 and 5102 × 5
    3 and 5153 × 5
    4 and 5204 × 5
    6 and 824Share a 2, so half the product: 48 ÷ 2
    5 and 8405 × 8
    8 and 1224Share a 4, so a quarter of the product: 96 ÷ 4

    Mixed number anchors

    Mixed numberImproper fractionHow to see it
    1 1/23/2Two halves plus one half
    1 3/47/4Four quarters plus three
    2 1/49/4Eight quarters plus one
    3 1/310/3Nine thirds plus one
    2 2/512/5Ten fifths plus two

    Hide a column, say each answer aloud, then click a striped cell to check it.

    Drill: which piece size?

    Two short games. The first asks for the lowest common denominator of two bottoms. The second shows a fraction and asks for its simplest form. Both are the two most common places marks are lost in the non-calculator section.

    Common traps

    • Adding tops and bottoms. 1/2 + 1/3 is not 2/5. A half and a third are different piece sizes, so re-cut both into sixths first: 3/6 + 2/6 = 5/6. (2/5 is smaller than 1/2, which should already look wrong.)
    • Scaling the bottom but not the top. If 2 becomes 6 on the bottom, the top must be multiplied by 3 as well: 1/2 = 3/6, not 1/6. Whatever cuts the pieces smaller also makes more of them.
    • Subtracting mixed numbers part by part. 3 1/4 − 1 3/4 is not 2 and "1/4 − 3/4". Convert to improper fractions (13/4 − 7/4 = 6/4 = 1 1/2), or borrow one whole first: 2 5/4 − 1 3/4.
    • Flipping the wrong fraction. Only the fraction after the ÷ is flipped. 3/4 ÷ 1/8 = 3/4 × 8/1 = 6, but 4/3 × 1/8 gives 1/6, which is the answer to a different question.
    • Missing that "of" means multiply. "3/8 of 240" is 3/8 × 240 = 90. Students who add or divide here get numbers that are not close to the sensible estimate (a bit less than half of 240).
    • Stopping at an improper or unsimplified fraction. 17/4 is correct, but if the question asks for a mixed number the answer is 4 1/4, and if it asks for simplest form 6/4 must become 3/2 or 1 1/2. Read the required form before you write the answer.
    • Taking a fraction of the wrong whole. "She used 2/3, then 1/4 of the remainder" means 1/4 of what was left, not 1/4 of the original. Work out the remainder first, then take the fraction of that.

    ACER-style practice

    Eight questions in the formats the real test uses: multiple choice, type your answer, and true-or-false tables. Questions marked "no calculator" belong to the test's final section, where you cannot use one. Check each answer for a full explanation.

    0 of 8 checked

    What to remember

    • The bottom number names the piece size; the top counts the pieces.
    • Add or subtract only matching piece sizes: rewrite with the LCD, then add or subtract the tops.
    • Mixed to improper: (whole × bottom + top) / bottom. Improper to mixed: divide, remainder stays over the bottom.
    • Multiply: cancel first, then top × top / bottom × bottom. No common denominator.
    • Divide: keep, flip, multiply. Only the second fraction flips.
    • "Of" means multiply: amount ÷ bottom × top. Whole from a part: part ÷ top × bottom.
    • Finish in the form the question asks for: simplest form, or a mixed number.

    All questions on this page are original and are not reproduced from ACER material. LANTITE is administered by the Australian Council for Educational Research.