ACARA v9 CONTENT DESCRIPTION “solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies and using digital tools where appropriate”
Builds on: Multiplying and Dividing by Powers of 10 (AC9M6N06). Place-value skill with decimals supports finding fractions, decimals and percentages of an amount — including the percentage discounts seen when shopping.
A fraction of a quantity
Finding a fraction of a quantity means splitting the quantity into equal parts and taking the number you need. To find one quarter of 20, divide 20 into 4 equal parts of 5, then take 1 part, giving 5. For three quarters of 12, the four parts are 3 each, and taking 3 of them gives 9. The denominator tells you how many parts to make, and the numerator tells you how many to take. This single idea, divide then take, is the foundation for working with percentages too, since a percentage is just a particular kind of fraction.
A fraction of a quantity
To find a fraction of an amount, divide into equal parts, then take the number you need.
Split 30 into 2 equal parts of 15, then take 1 — so 1/2 of 30 is 15.
Percentage as hundredths
A percentage is a number out of 100; the word per cent means per hundred. So 25% means 25 out of 100, which is the same value as the fraction 1/4 and the decimal 0.25. Fifty per cent is 1/2 or 0.5, and ten per cent is 1/10 or 0.1. Percentages, fractions and decimals are three ways of writing the same proportion, and being able to move between them is the key to choosing the easiest method. Recognising that a friendly percentage is really a simple fraction makes most percentage problems much easier.
Percentage as hundredths
A percentage is a number out of 100; the same value can be written as a fraction or decimal.
50% means 50 out of 100 — the same value as the fraction 1/2 and the decimal 0.5.
A percentage of a quantity
To find a percentage of a quantity, the efficient strategy is usually to rewrite the percentage as a simple fraction and find that fraction of the amount. Ten per cent of 50 is one tenth of 50, which is 5; twenty-five per cent of 80 is one quarter of 80, which is 20; fifty per cent of 60 is half of 60, which is 30. Choosing the friendly fraction, a tenth, a quarter, a half, avoids heavy calculation. For familiar percentages this is far quicker than any formula, and it is exactly the kind of efficient strategy this unit encourages.
A percentage of a quantity
Familiar percentages like 10%, 25% and 50% are easiest found as simple fractions.
Press to find the percentage using an easy fraction.
A percentage discount
A discount is a percentage taken off a price, and it is one of the most common everyday uses of percentages. Working it out takes two steps: first find the discount, the percentage of the price; then subtract it from the original price to get the sale price. A $40 item with 20% off has a discount of 20% of $40, which is $8, so the sale price is $40 minus $8, which is $32. Keeping the two steps clear, find the discount, then subtract, prevents the common mistake of stopping at the discount when the question asks for the price paid.
A percentage discount
A discount is a percentage off the price; find the discount, then subtract it.
Press to find the discount, then the sale price.
Finding the amount
Putting it into practice, finding a fraction or percentage of a quantity follows one reliable plan: rewrite the fraction or percentage in its easiest form, then take it of the amount. One quarter of 20 is 5; fifty per cent of 60, seen as a half, is 30; ten per cent of 90, seen as a tenth, is 9. The skill is choosing the most efficient form, a simple fraction is usually easiest, and applying it carefully. With familiar fractions and percentages, this can be done quickly in your head, without reaching for a calculator.
Finding the amount
Rewrite the fraction or percentage in the easiest form, then take it of the quantity.
What is 1/4 of 20? Pick A, B or C.
Solving discount problems
Discount word problems bring these skills together, and the first job is to read what is actually asked. A $40 shirt with 20% off might ask for the discount, which is $8, or for the price you pay, which is $32; these are different answers to different questions. Working through carefully, find the percentage of the price for the discount, then subtract for the sale price, and answer the question that was posed. Reading the question closely and matching the calculation to it is as important as the arithmetic, and it is the heart of solving real percentage problems.
A discount problem
Decide whether the question asks for the discount or the sale price, then calculate.
Solve the discount problem. Pick A, B or C.
Where this leads
Finding fractions, decimals and percentages of quantities is everyday mathematics, used in shopping discounts, tips, taxes, interest and statistics. The skill of moving between fractions, decimals and percentages, and choosing the most efficient form, carries directly into later work on percentage increase and decrease, on financial mathematics, and on rates and proportion. Learning to find a part of a quantity quickly and to read a problem for what it really asks builds confidence with the proportional reasoning that runs through mathematics and through managing money in daily life.
Quick self-check
1. What is 1/4 of 20?
2. What is 50% of 60?
3. 10% of an amount is the same as...
4. A $40 jacket has 20% off. How much is the discount?
5. A $40 jacket has 20% off. What is the sale price?