Number

Fractions and percentages

Explore equivalence, percentages and all four operations.

Predict, test and explain

Step 1 of 3

Predict

Split each shaded quarter into two. What represents 3/4?

14=28\frac{1}{4}=\frac{2}{8}
14\frac{1}{4}
28\frac{2}{8}

Scroll the diagram sideways to see every label.

Decimal14=0.25\frac{1}{4}=0.25Percentage14=25%\frac{1}{4}=25\%

InvestigateWhy must added fractions use the same sized parts?

Explore and compare
Can you break the rule?

Multiplying a positive number by a non-negative number makes it bigger.

Always, sometimes or never? Test it in the simulation.

Hint

Choose Four operations and Multiply. Try a multiplier less than one, equal to one, and greater than one.

Check the reasoning

Sometimes. A multiplier greater than one increases a positive number. One leaves it unchanged; a multiplier between zero and one reduces it.

2×12=1,2×1=2,2×3=62\times\tfrac12=1,\quad2\times1=2,\quad2\times3=6

One counterexample disproves “always”. Examples alone do not prove it.

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Subtract and give the answer in its simplest form.

34−16\frac34-\frac16

Hint

Use a common denominator of 12.

Reveal answer

Seven twelfths. Subtract the numerators after making equal-sized parts.

912−212=712\frac9{12}-\frac2{12}=\frac7{12}

Try a short practice

Use these fixed values. The diagram controls are separate.

Check your understanding

Which sum is correct?

12+13\frac12+\frac13

Complete the missing step

Simplify the result to complete the final step.

34÷12\frac34\div\frac12

  1. 34×21\frac34\times\frac21
  2. 64\frac64

Spot the first mistake

This is an attempted solution containing a mistake.

Which step contains the first mistake?

12+13\frac12+\frac13

  1. 1+12+3\frac{1+1}{2+3}
  2. 25\frac25
  3. 0.40.4

Build an explanation

Hint 1

Equivalent fractions name the same amount: multiply or divide the numerator and denominator by the same non-zero number.

Hint 2

For addition or subtraction, first use a common denominator so the parts have the same size. Add or subtract the numerators and keep that denominator.

Hint 3

For multiplication, multiply numerators and denominators. To divide by a non-zero fraction, multiply by its reciprocal. Simplify common factors.

Worked example

This example uses fixed values, separate from the diagram controls.

Calculate (1/2 + 1/3 − 1/6) × 3/4 ÷ 1/2.

  1. Use sixths inside the brackets: 1/2 = 3/6 and 1/3 = 2/6.
  2. 3/6 + 2/6 − 1/6 = 4/6 = 2/3.
  3. Multiply: 2/3 × 3/4 = 6/12 = 1/2.
  4. Divide by 1/2 by multiplying by 2/1: 1/2 × 2/1 = 1. This asks how many halves fit into one half.

Answer: 1.

Connect this idea

Watch out: Do not add denominators when adding fractions. Division by a fraction uses its reciprocal, and division by zero is undefined.