Ratio and Proportion Revision Notes for Leaving Cert Mathematics

Ratio and Proportion

What is a ratio?

A ratio is used to show how things are divided or shared between different parts. It compares one quantity to another quantity and tells us the relationship between them.

For example, if you have 12 apples and 8 oranges, the ratio of apples to oranges is written as 12:8.

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Ratios always show the relationship between quantities in a specific order. The ratio 12:8 is different from 8:12 - the first number corresponds to the first quantity mentioned.

Simplifying ratios

Ratios should normally be expressed in their simplest form using the smallest possible whole numbers. To simplify a ratio, we divide each term by the same number.

The ratio 12:8 can be simplified:

12:8 = 12÷4 : 8÷4 = 3:2

We say 3:2 is the simplest form of the ratio 12:8.

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Always simplify ratios by finding the highest common factor (HCF) of all terms and dividing each term by this number.

Converting fraction ratios to whole numbers

Sometimes ratios contain fractions, but we need to express them as whole numbers. We do this by multiplying each term by the same number to eliminate the fractions.

For the ratio $\frac{1}{3} : \frac{5}{6}$ :

Multiply both terms by 6 (the lowest common multiple of 3 and 6)
$\frac{1}{3} \times 6 : \frac{5}{6} \times 6 = 2:5$

Solving ratio problems

When solving ratio problems, remember that ratios show us the relative sizes of the parts, not the actual amounts.

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The key insight for ratio problems is to find the value of one part first, then use this to calculate the actual quantities.

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Worked Example: Finding the total from one part

Question: A sum of money is divided in the ratio 1:3:5. If the smallest part is €250, find the total sum of money.

Solution:

The ratio 1:3:5 means there are 1 + 3 + 5 = 9 parts in total
The smallest part corresponds to the "1" in the ratio
If 1 part = €250, then 9 parts = €250 × 9 = €2250
Therefore, the total sum of money is €2250

What is proportion?

Proportion is different from ratio. While ratios compare one part to another part, proportion compares a part to the total amount.

For example, if there are 3 goalkeepers in a panel of 24 footballers, the proportion of goalkeepers is: $\frac{3}{24} = \frac{1}{8}$

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Key Difference:

Ratio: part to part comparison (3:21 for goalkeepers to other players)
Proportion: part to whole comparison (3:24 or $\frac{3}{24}$ for goalkeepers to total players)

Direct proportion

When two quantities are in direct proportion, they increase or decrease at the same rate. If one quantity doubles, the other also doubles.

Example: If 1 litre of petrol costs €1.50, then 2 litres cost €3.00, and 5 litres cost €7.50. The cost is directly proportional to the amount of petrol.

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Worked Example: Direct proportion problem

Question: The number of pages in a magazine increased from 64 to 80. The original price of €4.40 increased in the same ratio. What is the new price?

Solution:

Find the cost per page: €4.40 ÷ 64 = €0.06875 per page
Calculate the new cost: €0.06875 × 80 = €5.50
Alternative method: $\frac{€4.40}{64} \times 80 = €5.50$

Therefore, the new price is €5.50.

Proportion and scale

Proportion is often used in scale drawings, maps, and models. The scale tells us the ratio between the drawing/model and the real object.

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Scale problems involve finding actual sizes from model sizes, or vice versa, using the given scale ratio. Always pay attention to the units and ensure they match when making calculations.

Key formulas and methods

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Essential Formulas and Approaches:

For ratio problems:

Simplifying ratios: Divide each term by their highest common factor
Finding totals: Add all parts of the ratio, then use proportion to find the answer
Finding one part: Use the known part to calculate the value of one unit, then multiply

For proportion problems:

Direct proportion: $\frac{x_1}{y_1} = \frac{x_2}{y_2}$
Scale problems: Scale ratio × actual size = model size (or vice versa)

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Worked Example: Mixed ratio and proportion

Question: €80 is divided between two pupils in the ratio 7:3. How much does each pupil get?

Solution:

Total parts = 7 + 3 = 10 parts
Value of 1 part = €80 ÷ 10 = €8
First pupil gets: 7 × €8 = €56
Second pupil gets: 3 × €8 = €24
Check: €56 + €24 = €80 ✓

Exam tips

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Critical Exam Strategies:

Always check that your ratio is in its simplest form
When solving ratio problems, find the value of one part first
Remember the difference: ratios compare parts to parts, proportions compare parts to the total
In direct proportion problems, set up the equation carefully
Always verify your answer makes sense in the context

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Key Points to Remember:

Ratios compare one quantity to another (part to part)
Proportions compare a part to the total amount
Always express ratios in their simplest form using whole numbers
In direct proportion, if one quantity doubles, the other doubles too
To solve ratio problems, find the value of one part first, then multiply to get your answer

Ratio and Proportion (Leaving Cert Mathematics): Revision Notes