ACARA v9 CONTENT DESCRIPTION “solve problems involving addition and subtraction of fractions using knowledge of equivalent fractions”
Builds on: Comparing and Ordering Fractions (AC9M6N03). Knowing equivalent fractions is exactly what makes adding and subtracting fractions possible — rewriting them so the pieces are the same size.
Same denominator
Adding fractions is simplest when they already share a denominator. Because the pieces are the same size, you just add the numerators and keep the denominator the same. One fifth plus two fifths is three fifths, written 1/5 + 2/5 = 3/5; the fifths do not change, only how many of them you have. Subtraction works the same way, taking one numerator from the other. Seeing that same-denominator fractions combine by counting their equal pieces is the idea that the rest of this unit builds on.
Same denominator
When fractions share a denominator, add the numerators and keep the denominator.
With the same denominator, just add the top numbers — 1/5 + 2/5 = 3/5.
Different denominators
Fractions with different denominators cannot be added directly, because their pieces are different sizes. One half and one quarter are not made of matching parts, so 1/2 + 1/4 is not found by adding tops and bottoms. The solution is to rewrite the fractions so they share a denominator, using equivalent fractions. Since one half equals two quarters, the sum becomes 2/4 + 1/4, which is now straightforward. Recognising when denominators differ, and that they must be made equal first, is the key step in adding and subtracting most fractions.
Different denominators
Fractions with different denominators must be rewritten over a common denominator first.
To add 1/2 and 1/4, rewrite both over a common denominator of 4 — then the pieces are the same size.
Making a common denominator
To add fractions with different denominators, first find a common denominator, a number that both denominators divide into. For halves and quarters, 4 works, since 2 and 4 both divide into 4. Then rewrite each fraction as an equivalent fraction over that common denominator: 1/2 becomes 2/4, while 1/4 already has it. Now both fractions are measured in the same sized pieces and can be added. Finding a common denominator and rewriting with equivalent fractions is the heart of fraction addition and subtraction.
Making a common denominator
Find a denominator both fractions divide into, then rewrite each as an equivalent fraction.
Press to find a common denominator, then rewrite each fraction.
Adding fractions
Once the fractions share a denominator, adding is straightforward: add the numerators and keep the common denominator. For 1/2 + 1/4, rewriting gives 2/4 + 1/4, and adding the tops gives 3/4. If the fractions already match, like 1/4 + 2/4, no rewriting is needed and the answer is 3/4 directly. The full method is: make the denominators the same if they differ, add the numerators, and keep the denominator. Sometimes the answer can then be simplified to lower terms, but the adding step itself is always the same.
Adding fractions
Rewrite over a common denominator if needed, then add the numerators.
What is 1/4 + 2/4? Pick A, B or C.
Subtracting fractions
Subtracting fractions follows the same path as adding. If the denominators match, subtract the numerators and keep the denominator: 3/4 minus 1/4 is 2/4. If they differ, first rewrite over a common denominator, then subtract. For 5/6 minus 1/3, rewrite 1/3 as 2/6, giving 5/6 minus 2/6, which is 3/6, or one half. The method mirrors addition exactly: equal denominators first, then work with the numerators. Keeping the denominators matched ensures you are taking away pieces of the same size.
Subtracting fractions
Over a common denominator, subtract the numerators and keep the denominator.
What is 3/4 − 1/4? Pick A, B or C.
Solving problems
Fraction addition and subtraction appear constantly in real problems: combining parts of a pizza, finding how much liquid is left in a jug, or totalling distances walked. Solving such a problem means first deciding whether to add or subtract, then applying the method, making denominators equal if needed and working with the numerators. Eating 1/4 of a cake then 2/4 more means adding to get 3/4 eaten. The mathematics is the same as the bare calculations; the extra skill is reading the situation to choose the right operation.
A word problem
Decide whether to add or subtract, find a common denominator, then solve.
Solve the problem with fraction addition or subtraction. Pick A, B or C.
Where this leads
Adding and subtracting fractions, built on equivalent fractions, is a cornerstone of all later fraction work. It leads on to multiplying and dividing fractions, to working with mixed numbers, and to the close links between fractions, decimals and percentages. The central habit, rewriting fractions over a common denominator so their pieces match, recurs throughout algebra whenever fractions appear. Mastering these operations gives students the confidence to handle fractions wherever they meet them, in mathematics and in everyday life.
Quick self-check
1. What is 1/5 + 2/5?
2. To add 1/2 + 1/4, you first rewrite 1/2 as...
3. What is 1/2 + 1/4?
4. What is 3/4 − 1/4?
5. You eat 1/4 of a cake, then 2/4 more. How much have you eaten in all?