AC9M7N06 · Year 7 · Number

Operations with rational numbers

ACARA v9 CONTENT DESCRIPTION “use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies”

Once you know that a fraction, a decimal and a percentage are three names for the same value, the next move is doing arithmetic with them. Adding, subtracting, multiplying and dividing rational numbers is the everyday maths behind discounts, recipes, splitting bills and measuring ingredients. The skill that matters most here is choosing the easiest form to calculate with, then converting back at the end.

The single biggest stumbling block is adding fractions with different denominators. Students often add straight across, turning 1/3 plus 1/4 into 2/7, which is wrong. The reason it fails is worth seeing rather than memorising.

Why unlike fractions need a common denominator

A third of a bar and a quarter of a bar are pieces of different sizes. You cannot count them together any more than you can add 3 apples and 4 oranges and call the answer 7 apples. The fix is to cut both bars into the same size pieces. Twelfths work for thirds and quarters, because 12 divides evenly by both. One third becomes 4 twelfths, one quarter becomes 3 twelfths, and now the pieces match and can be added to give 7 twelfths.

A common denominator
Unlike pieces must be made the same size before they can be added.
Thirds and quarters are different-sized pieces, so they cannot be combined as they are. Rewriting both over twelfths gives matching pieces, and then the shaded parts simply add: 4 twelfths plus 3 twelfths is 7 twelfths.

This is the whole logic of a common denominator: rewrite each fraction so the pieces are identical, then the numerators add directly. The same idea governs subtraction. Multiplication is gentler, since you simply multiply the tops together and the bottoms together, and division flips the second fraction and multiplies. Each operation has its own rule, but they all rest on keeping track of what one piece actually represents.

Sliding between forms to make work easier

Percentages, decimals and fractions are interchangeable, and a skilled problem-solver picks whichever form is quickest. To find 25% of 80, it is faster to read 25% as one quarter and take a quarter of 80, giving 20. To find 0.6 of a quantity, the decimal is easiest. The art is in the choosing, not in being forced down one path.

Sliding between forms
Divide to reach the decimal, multiply by 100 to reach the percentage.
The same value travels between three forms by simple operations: divide to get the decimal, multiply by 100 to get the percentage. Solving a percentage problem is just choosing which form is easiest to calculate with.

A discount problem shows all of this at once. A jacket marked 80 dollars with 25% off means finding 25% of 80, which is 20, then subtracting to reach a sale price of 60 dollars. Notice the two operations working together: a percentage calculation followed by a subtraction. Real problems almost always chain operations like this, which is why fluency with each one, and confidence moving between forms, pays off across the rest of the year.

Teaching tip: the add-across error (1/3 + 1/4 = 2/7) is so common it is worth catching head on. Ask the student whether 1/2 plus 1/2 should equal 2/4. They will say it must equal 1 whole. Add-across gives 2/4, which is only a half, so the rule visibly breaks. That contradiction does more than any correction.

For percentage work, encourage friendly benchmarks: 25% is a quarter, 10% is found by moving the decimal one place, 50% is half. Most real-world percentages can be built from these without a calculator, which keeps the focus on reasoning.

Builds on: Equivalent representations of rational numbers (AC9M7N04). That unit showed one rational number wearing three names; this unit puts those forms to work in calculation.
Quick self-check
1. To add 1/3 and 1/4, what is the necessary first step?
2. A jacket costs 80 dollars and is reduced by 25%. What is the sale price?
3. Which calculation correctly converts the fraction 3/5 to a decimal?
4. What is 0.6 written as a percentage?
5. A recipe needs 3/4 of a cup of flour but you are tripling it. How much flour in total?