Section A — Chapters 1–22
Rational Numbers II: Dividing Fractions, Ratio & Proportion
Junior Cycle — 1st Year
- ✓By the end of this lesson students will be able to divide one fraction by another.
- ✓By the end of this lesson students will be able to express ratios in their simplest form.
- ✓By the end of this lesson students will be able to solve problems involving ratio.
- ✓By the end of this lesson students will be able to understand and apply the concept of direct proportion.
Key concepts
To divide one fraction by another, you 'invert' (flip) the second fraction and then multiply the two fractions. Remember, 'invert and multiply'.
A ratio is a way of comparing two or more quantities of the same kind. It shows how much of one quantity there is compared to another. Ratios are often written using a colon (:) or as a fraction. For example, the ratio of 2 apples to 3 oranges can be written as 2:3 or 2/3. Ratios should always be simplified to their lowest terms, just like fractions. All quantities in a ratio must be in the same units.
Proportion describes a relationship between quantities where if one quantity changes, the other quantity changes in a consistent way. Direct proportion means that as one quantity increases, the other quantity increases at the same rate, or as one decreases, the other decreases at the same rate. This means their ratio remains constant. For example, if 2 pens cost €3, then 4 pens (double the quantity) will cost €6 (double the price).
Key facts to remember
- 1To divide fractions, invert the second fraction and multiply.
- 2A ratio compares two or more quantities of the same kind.
- 3Ratios should always be simplified to their lowest terms.
- 4All quantities in a ratio must be expressed in the same units.
- 5Direct proportion means that as one quantity increases, the other increases by the same factor (and vice versa).
- 6Ratios can be written as a:b or a/b.
Worked examples
Example 1
Calculate (3/4) ÷ (1/2).
Answer
3/2
Example 2
Simplify the ratio 15 cm : 3 m. Then, if a recipe requires flour and sugar in the ratio 3:2, and you use 180g of flour, how much sugar do you need?
Answer
The simplified ratio is 1:20. You need 120g of sugar.
Always ensure quantities in a ratio are in the same units before simplifying.
Example 3
If 5 pens cost €7.50, how much would 8 pens cost?
Answer
8 pens would cost €12.00.
This is an example of direct proportion, as the cost increases proportionally with the number of pens.
Common mistakes
- ✗Inverting the first fraction instead of the second when dividing.
- ✗Forgetting to convert units to be the same before simplifying a ratio (e.g., cm and m).
- ✗Not simplifying ratios to their lowest possible terms.
- ✗Confusing direct proportion with inverse proportion (though inverse proportion is not covered in 1st Year).
- ✗Making calculation errors when multiplying or dividing fractions or when solving proportion problems.
Exam tips
- ★Always show all your steps clearly, especially when dividing fractions or solving ratio problems.
- ★Double-check your unit conversions when working with ratios.
- ★Read the question carefully to ensure you understand what quantities are being compared or what is being asked.
- ★Practise regularly to become confident with fraction operations and ratio/proportion problems.
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