Ratios: Fundamentals and Simplification

Ratios are one of the most frequently tested topics in UCAT Quantitative Reasoning. They appear in questions about mixing, sharing, scaling, and comparing quantities. This lesson covers what ratios mean, how to express and simplify them, and the critical difference between part-to-part and part-to-whole ratios.

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What Is a Ratio?

A ratio compares two or more quantities of the same type. It tells you how much of one thing there is relative to another.

Example: A class has 18 boys and 12 girls. The ratio of boys to girls is 18:12, which simplifies to 3:2.

This means for every 3 boys, there are 2 girls.

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Writing Ratios

Ratios can be written in three ways:

Format	Example
Using a colon	3:2
As a fraction	3/2
In words	"3 to 2"

Important: The order matters. "Boys to girls = 3:2" is different from "Girls to boys = 2:3". Always check which quantity comes first.

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Simplifying Ratios

Simplify a ratio by dividing all parts by their highest common factor (HCF).

Method

1. Find the HCF of all parts
2. Divide each part by the HCF

Worked Examples

Original Ratio	HCF	Simplified
18:12	6	3:2
45:30:15	15	3:2:1
120:80	40	3:2
250:150:100	50	5:3:2

Simplifying in Steps

If the HCF is not obvious, simplify in stages:

Example: 84:56

• Both are even → 42:28
• Both are even → 21:14
• Both divisible by 7 → 3:2

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Part-to-Part vs Part-to-Whole

This distinction is crucial and frequently tested.

Part-to-Part Ratio

Compares one part to another part.

Example: Boys to girls = 3:2

This means there are 3 parts boys and 2 parts girls.

Part-to-Whole Ratio

Compares one part to the total.

Example: Boys to total = 3:5 (because total parts = 3 + 2 = 5)

Part-to-Part	Part-to-Whole
Boys : Girls = 3 : 2	Boys : Total = 3 : 5
	Girls : Total = 2 : 5

Converting Between Them

Given: The ratio of cats to dogs is 4:3.

• Total parts = 4 + 3 = 7
• Cats are 4/7 of the total
• Dogs are 3/7 of the total

Given: 2/5 of the class are left-handed.

• Left-handed : Right-handed = 2 : 3 (since 2/5 left-handed means 3/5 right-handed)

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Equivalent Ratios

Just like equivalent fractions, ratios can be scaled up or down.

3:2 = 6:4 = 9:6 = 12:8 = 15:10 = 30:20

All of these represent the same relationship.

Finding an Equivalent Ratio with a Specific Total

Example: The ratio of red to blue beads is 3:5. If there are 120 beads in total, how many are red?

• Total parts = 3 + 5 = 8
• One part = 120 ÷ 8 = 15
• Red = 3 × 15 = 45 beads

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Three-Part Ratios

Some questions involve three or more quantities.

Example: A drink is mixed in the ratio orange juice : lemonade : water = 2 : 3 : 5.

• Total parts = 2 + 3 + 5 = 10
• To make 2 litres (2,000 ml) of the drink:
  • Orange juice = (2/10) × 2,000 = 400 ml
  • Lemonade = (3/10) × 2,000 = 600 ml
  • Water = (5/10) × 2,000 = 1,000 ml

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Combining Ratios

Sometimes you need to combine two ratios that share a common term.

Example: A : B = 2 : 3 and B : C = 3 : 4. Find A : B : C.

Since B is already 3 in both ratios:

A : B : C = 2 : 3 : 4

Example: A : B = 3 : 4 and B : C = 2 : 5. Find A : B : C.

B is 4 in the first ratio and 2 in the second. Find the LCM of 4 and 2, which is 4.

• Scale the second ratio: B : C = 4 : 10

Now A : B : C = 3 : 4 : 10

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UCAT-Style Worked Examples

Worked Example 1

Data: A hospital employs doctors, nurses, and support staff in the ratio 2 : 5 : 3. There are 450 staff in total.

Question: How many nurses are there?

• Total parts = 2 + 5 + 3 = 10
• One part = 450 ÷ 10 = 45
• Nurses = 5 × 45 = 225

Worked Example 2

Data: In a survey, the ratio of respondents who preferred Option A to Option B was 7:4. 132 people preferred Option B.

Question: How many preferred Option A?

• B = 4 parts = 132 people
• One part = 132 ÷ 4 = 33
• A = 7 × 33 = 231 people

Worked Example 3

Data: Concrete is mixed in the ratio cement : sand : gravel = 1 : 2 : 4. A builder needs 2,100 kg of concrete.

Question: How many kilograms of sand are needed?

• Total parts = 1 + 2 + 4 = 7
• One part = 2,100 ÷ 7 = 300
• Sand = 2 × 300 = 600 kg

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Practice Exercises

Simplify:

• 24:36 = ? (2:3)
• 150:100:50 = ? (3:2:1)
• 48:64 = ? (3:4)

Part-to-whole:

• Ratio A:B = 5:3. What fraction of the total is A? (5/8)
• Ratio X:Y:Z = 2:3:5. What percentage is Y? (30%)

Finding quantities:

• Ratio 4:7, total = 220. Find the smaller part. (80)
• Ratio 3:5:2, total = 600. Find each part. (180, 300, 120)

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Summary

• A ratio compares quantities of the same type — order matters
• Simplify by dividing all parts by the highest common factor
• Part-to-part compares one part to another; part-to-whole compares one part to the total
• To find quantities from a ratio: Find total parts, find one part (÷ total parts), multiply by the number of parts
• Three-part ratios: Same method — add all parts for the total
• Combining ratios: Make the shared term equal using the LCM