Data Analytics: Analytics Fundamentals Ratios

What Is This?

Ratios are mathematical expressions showing the relationship between two or more quantities. They are often used to compare the proportions of different things, such as the ratio of boys to girls in a class or the ratio of ingredients in a recipe.

This topic appears in exams because ratios are a fundamental concept in mathematics, and understanding them is essential for solving problems in various fields, including science, engineering, finance, and more. Exams will typically test your ability to identify and work with ratios, as well as to apply them to solve problems in different contexts.

Why It Matters

Ratios are tested in various exams, including mathematics, science, and engineering exams. They appear frequently, often carrying 10-20% of the total marks. The skill being tested is your ability to understand and apply the concept of ratios, as well as to identify and work with different types of ratios.

Core Concepts

To master ratios, you need to understand the following core concepts:

• Equivalent ratios: Ratios that have the same value, but are expressed in different terms. For example, 2:3 and 4:6 are equivalent ratios.
• Simplified ratios: Ratios that have been reduced to their simplest form, often by dividing both terms by their greatest common divisor. For example, 6:8 can be simplified to 3:4.
• Ratio of ratios: The ratio of two or more ratios, often used to compare the proportions of different things. For example, the ratio of the ratio of boys to girls in a class to the ratio of boys to girls in another class.
• Percentage change: The change in a quantity as a percentage of its original value. For example, if a price increases from $10 to $12, the percentage change is 20%.

Prerequisites

Before tackling ratios, you need to understand the following prerequisites:

• Fractions: You need to understand how to work with fractions, including adding, subtracting, multiplying, and dividing them.
• Decimals: You need to understand how to work with decimals, including converting between fractions and decimals.
• Percentages: You need to understand how to work with percentages, including calculating percentage changes.

The Rule-Book (How It Works)

The primary rule for working with ratios is:

• The ratio of two or more quantities is equal to the ratio of their corresponding parts.

For example, if you have a ratio of 2:3, and you want to find the ratio of the corresponding parts, you can multiply both terms by 2 to get 4:6.

Sub-rules and exceptions include:

• The ratio of a quantity to itself is always 1:1. For example, the ratio of 5 to 5 is 1:1.
• The ratio of a quantity to zero is undefined. For example, the ratio of 5 to 0 is undefined.

A simple visual pattern to help you remember the rule is to think of a ratio as a balance scale, where the ratio of the two quantities is equal to the ratio of their corresponding parts.

Exam / Job / Audit Weighting

Frequency: 20-30% Difficulty Rating: Intermediate Question Type or Real-World Task Type: Multiple-choice questions, short-answer questions, and problem-solving exercises.

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

The following are the three most important rules, formulas, and standards for working with ratios:

• The ratio of two or more quantities is equal to the ratio of their corresponding parts.
• The ratio of a quantity to itself is always 1:1.
• The ratio of a quantity to zero is undefined.

Worked Examples (Step-by-Step)

Here are three worked examples that escalate in difficulty:

Example 1 (Easy)

What is the ratio of 4:6?

• Step 1: Identify the two quantities: 4 and 6.
• Step 2: Divide both quantities by their greatest common divisor, which is 2.
• Step 3: Simplify the ratio to get 2:3.

Answer: 2:3

Key rule applied: Simplified ratios.

Example 2 (Medium)

What is the ratio of the ratio of boys to girls in a class to the ratio of boys to girls in another class?

• Step 1: Identify the two ratios: 3:5 and 2:3.
• Step 2: Divide both ratios by their greatest common divisor, which is 1.
• Step 3: Simplify the ratio of ratios to get 6:10.

Answer: 6:10

Key rule applied: Ratio of ratios.

Example 3 (Hard)

A company has a ratio of 3:5 of male to female employees. If the company hires 20 new employees, and 12 of them are male, what is the new ratio of male to female employees?

• Step 1: Identify the original ratio: 3:5.
• Step 2: Calculate the new number of male employees: 3/8 x 32 = 12.
• Step 3: Calculate the new number of female employees: 5/8 x 32 = 20.
• Step 4: Simplify the new ratio to get 6:10.

Answer: 6:10

Key rule applied: Percentage change.

Common Exam Traps & Mistakes

Here are four common exam traps and mistakes:

• Mistake 1: Failing to simplify the ratio.
• Mistake 2: Failing to identify the corresponding parts.
• Mistake 3: Failing to calculate the percentage change.
• Mistake 4: Failing to simplify the ratio of ratios.

Shortcut Strategies & Exam Hacks

Here are three shortcut strategies and exam hacks:

• Hack 1: Use a ratio calculator to simplify the ratio.
• Hack 2: Use a percentage change calculator to calculate the percentage change.
• Hack 3: Use a ratio chart to identify the corresponding parts.

Question-Type Taxonomy

Here are three distinct question formats that ratios appear in:

Practice Set (MCQs)

Here are five multiple-choice questions at mixed difficulty levels:

Question 1 (Easy)

What is the ratio of 2:4?

A) 1:2 B) 2:4 C) 3:6 D) 4:8

Correct answer: B) 2:4

Explanation: The correct answer is B) 2:4 because the ratio of 2:4 is equal to the ratio of their corresponding parts.

Why the distractors are tempting: The distractors A) 1:2, C) 3:6, and D) 4:8 are all plausible answers because they are equivalent ratios or simplified ratios.

Question 2 (Medium)

What is the ratio of the ratio of boys to girls in a class to the ratio of boys to girls in another class?

A) 3:5 B) 6:10 C) 9:15 D) 12:20

Correct answer: B) 6:10

Explanation: The correct answer is B) 6:10 because the ratio of the ratio of boys to girls in a class to the ratio of boys to girls in another class is equal to the ratio of their corresponding parts.

Why the distractors are tempting: The distractors A) 3:5, C) 9:15, and D) 12:20 are all plausible answers because they are equivalent ratios or simplified ratios.

Question 3 (Hard)

A company has a ratio of 3:5 of male to female employees. If the company hires 20 new employees, and 12 of them are male, what is the new ratio of male to female employees?

A) 3:5 B) 6:10 C) 9:15 D) 12:20

Correct answer: B) 6:10

Explanation: The correct answer is B) 6:10 because the new ratio of male to female employees is equal to the ratio of their corresponding parts.

Why the distractors are tempting: The distractors A) 3:5, C) 9:15, and D) 12:20 are all plausible answers because they are equivalent ratios or simplified ratios.

Question 4 (Easy)

What is the ratio of 6:8?

A) 1:2 B) 2:4 C) 3:4 D) 4:6

Correct answer: C) 3:4

Explanation: The correct answer is C) 3:4 because the ratio of 6:8 is equal to the ratio of their corresponding parts.

Why the distractors are tempting: The distractors A) 1:2, B) 2:4, and D) 4:6 are all plausible answers because they are equivalent ratios or simplified ratios.

Question 5 (Medium)

What is the ratio of the ratio of boys to girls in a class to the ratio of boys to girls in another class?

A) 2:3 B) 3:5 C) 4:6 D) 6:10

Correct answer: B) 3:5

Explanation: The correct answer is B) 3:5 because the ratio of the ratio of boys to girls in a class to the ratio of boys to girls in another class is equal to the ratio of their corresponding parts.

Why the distractors are tempting: The distractors A) 2:3, C) 4:6, and D) 6:10 are all plausible answers because they are equivalent ratios or simplified ratios.

30-Second Cheat Sheet

Here are the five things you need to remember walking into the exam hall:

• The ratio of two or more quantities is equal to the ratio of their corresponding parts.
• The ratio of a quantity to itself is always 1:1.
• The ratio of a quantity to zero is undefined.
• The ratio of ratios is equal to the ratio of their corresponding parts.
• The ratio of a quantity to a quantity is equal to the ratio of their corresponding parts.

Learning Path

Here is a suggested study sequence to master ratios from scratch to exam-ready:

1. Beginner foundation: Understand the basics of fractions, decimals, and percentages.
2. Core rules: Learn the core rules of ratios, including the ratio of two or more quantities, the ratio of a quantity to itself, and the ratio of a quantity to zero.
3. Practice: Practice solving problems involving ratios, including multiple-choice questions, short-answer questions, and problem-solving exercises.
4. Timed drills: Practice solving problems involving ratios under timed conditions to improve your speed and accuracy.
5. Mock tests: Take mock tests to assess your knowledge and identify areas for improvement.

Related Topics

Here are three closely connected topics that appear alongside ratios in exams:

• Fractions: Fractions are a fundamental concept in mathematics, and understanding them is essential for working with ratios.
• Decimals: Decimals are a fundamental concept in mathematics, and understanding them is essential for working with ratios.
• Percentages: Percentages are a fundamental concept in mathematics, and understanding them is essential for working with ratios.