GED Mathematical Reasoning Quantitative Reasoning Ratios and Proportions Setting Up and Solving

What Is This?

Ratios and proportions are a fundamental concept in Quantitative Reasoning, involving the comparison of two or more quantities. A ratio is a statement of the relationship between two or more numbers, usually expressed as a fraction or a ratio of the form a:b or a:b:c. A proportion is a statement that two ratios are equal.

This topic appears in exams to test your ability to set up and solve problems involving ratios and proportions, often in real-world contexts such as finance, engineering, or science. You can expect questions that involve finding equivalent ratios, solving proportion problems, and applying ratios to solve problems.

Why It Matters

This topic is commonly tested in exams such as the GMAT, GRE, and SAT, and typically carries 10-20% of the total marks. The skill being tested is your ability to understand and apply the underlying math concepts, as well as your ability to reason and solve problems under time pressure.

Core Concepts

To master this topic, you need to own the following foundational ideas:

• Equivalent ratios: Two ratios are equivalent if they have the same value, even if they are expressed differently. For example, 2:3 and 4:6 are equivalent ratios.
• Proportion problems: A proportion problem is a problem that involves setting up a proportion to solve for an unknown value. For example, "If 2 pounds of flour makes 4 cookies, how many pounds of flour will make 12 cookies?"
• Cross-multiplication: Cross-multiplication is a technique used to solve proportion problems by multiplying the means and the extremes of the proportion.

Prerequisites

Before tackling this topic, you need to understand the following key concepts:

• Fractions and decimals
• Basic algebra (e.g. solving linear equations)
• Percents and percentages

If you are missing these prerequisites, you may struggle to understand the underlying math concepts and may make errors in your calculations.

The Rule-Book (How It Works)

The primary rule for setting up and solving problems involving ratios and proportions is:

If a:b = c:d, then ad = bc

This rule can be applied to solve proportion problems by cross-multiplying the means and the extremes of the proportion.

Sub-rules and exceptions:

• If the ratios are not equivalent, the proportion is false.
• If the ratios are equivalent, the proportion is true.
• If the proportion is true, the means and extremes are equal.

Visual pattern:

Imagine a balance scale with two pans, one with a ratio of a:b and the other with a ratio of c:d. If the two pans are balanced, then the two ratios are equivalent.

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 questions.

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

The three most important rules for this topic are:

1. If a:b = c:d, then ad = bc
2. Equivalent ratios: Two ratios are equivalent if they have the same value, even if they are expressed differently.
3. Cross-multiplication: Cross-multiplication is a technique used to solve proportion problems by multiplying the means and the extremes of the proportion.

Worked Examples (Step-by-Step)

Example 1: Easy

Question: If 2 pounds of flour makes 4 cookies, how many pounds of flour will make 12 cookies?

Answer: 6 pounds

Key rule applied: Proportion problems

Step-by-step:

• Set up the proportion: 2 pounds / 4 cookies = x pounds / 12 cookies
• Cross-multiply: 2x = 48
• Solve for x: x = 24

Example 2: Medium

Question: A car travels 250 miles in 5 hours. How many miles will it travel in 10 hours?

Answer: 500 miles

Key rule applied: Equivalent ratios

Step-by-step:

• Set up the ratio: 250 miles / 5 hours = x miles / 10 hours
• Find the equivalent ratio: 250 miles / 5 hours = 500 miles / 10 hours
• Solve for x: x = 500

Example 3: Hard

Question: A bakery sells 240 loaves of bread per day at $2.50 per loaf. How much will they sell for per loaf if they sell 300 loaves per day?

Answer: $2.00 per loaf

Key rule applied: Cross-multiplication

Step-by-step:

• Set up the proportion: 240 loaves / $2.50 per loaf = 300 loaves / x per loaf
• Cross-multiply: 240x = 750
• Solve for x: x = 3.125

Common Exam Traps & Mistakes

Error 1: Forgetting to cross-multiply

• Wrong answer: 6 pounds (from Example 1)
• Correct approach: Cross-multiply to solve for x

Error 2: Not checking for equivalent ratios

• Wrong answer: 200 miles (from Example 2)
• Correct approach: Find the equivalent ratio

Error 3: Not considering the proportion as a whole

• Wrong answer: 400 miles (from Example 3)
• Correct approach: Consider the proportion as a whole and cross-multiply

Error 4: Not checking for proportionality

• Wrong answer: $3.00 per loaf (from Example 3)
• Correct approach: Check for proportionality before solving

Error 5: Not using the correct units

• Wrong answer: 500 pounds (from Example 3)
• Correct approach: Use the correct units (dollars per loaf)

Shortcut Strategies & Exam Hacks

Memory aid: Use the acronym PROPORTION to remember the steps:

P - Problem R - Ratio O - Opposite sides P - Product O - Opposite sides R - Ratio T - Transpose I - Inverse O - Opposite sides N - Number

Elimination strategy: Eliminate options that are clearly incorrect or outside the range of possible answers.

Pattern recognition tip: Look for patterns in the numbers or ratios, such as equivalent ratios or proportionality.

Question-Type Taxonomy

The three distinct question formats for this topic are:

Practice Set (MCQs)

Question 1: Easy

Question: If 3 pounds of flour makes 6 cookies, how many pounds of flour will make 12 cookies?

A) 4 pounds B) 5 pounds C) 6 pounds D) 8 pounds

Correct answer: C) 6 pounds

Explanation: The ratio of flour to cookies is 3:6, which is equivalent to 1:2. To make 12 cookies, we need 6 pounds of flour.

Why the distractors are tempting:

• A) 4 pounds: This option is tempting because it is close to the correct answer, but it is not the correct answer.
• B) 5 pounds: This option is tempting because it is a plausible answer, but it is not the correct answer.
• D) 8 pounds: This option is tempting because it is a larger amount of flour, but it is not the correct answer.

Question 2: Medium

Question: A bakery sells 240 loaves of bread per day at $2.50 per loaf. How much will they sell for per loaf if they sell 300 loaves per day?

A) $2.00 per loaf B) $2.50 per loaf C) $3.00 per loaf D) $3.50 per loaf

Correct answer: A) $2.00 per loaf

Explanation: The ratio of loaves to price is 240:2.50, which is equivalent to 300:x. To find the new price per loaf, we need to cross-multiply and solve for x.

Why the distractors are tempting:

• B) $2.50 per loaf: This option is tempting because it is the original price per loaf, but it is not the correct answer.
• C) $3.00 per loaf: This option is tempting because it is a plausible answer, but it is not the correct answer.
• D) $3.50 per loaf: This option is tempting because it is a larger price per loaf, but it is not the correct answer.

Question 3: Hard

Question: A car travels 250 miles in 5 hours. How many miles will it travel in 10 hours?

A) 375 miles B) 400 miles C) 500 miles D) 600 miles

Correct answer: C) 500 miles

Explanation: The ratio of miles to hours is 250:5, which is equivalent to x:10. To find the new distance, we need to cross-multiply and solve for x.

Why the distractors are tempting:

• A) 375 miles: This option is tempting because it is a plausible answer, but it is not the correct answer.
• B) 400 miles: This option is tempting because it is a larger distance, but it is not the correct answer.
• D) 600 miles: This option is tempting because it is a much larger distance, but it is not the correct answer.

Question 4: Easy

Question: If 2 pounds of flour makes 4 cookies, how many pounds of flour will make 8 cookies?

A) 3 pounds B) 4 pounds C) 5 pounds D) 6 pounds

Correct answer: B) 4 pounds

Explanation: The ratio of flour to cookies is 2:4, which is equivalent to 1:2. To make 8 cookies, we need 4 pounds of flour.

Why the distractors are tempting:

• A) 3 pounds: This option is tempting because it is close to the correct answer, but it is not the correct answer.
• C) 5 pounds: This option is tempting because it is a larger amount of flour, but it is not the correct answer.
• D) 6 pounds: This option is tempting because it is an even larger amount of flour, but it is not the correct answer.

Question 5: Medium

Question: A bakery sells 240 loaves of bread per day at $2.50 per loaf. How much will they sell for per loaf if they sell 180 loaves per day?

A) $2.00 per loaf B) $2.50 per loaf C) $3.00 per loaf D) $3.50 per loaf

Correct answer: B) $2.50 per loaf

Explanation: The ratio of loaves to price is 240:2.50, which is equivalent to 180:x. To find the new price per loaf, we need to cross-multiply and solve for x.

Why the distractors are tempting:

• A) $2.00 per loaf: This option is tempting because it is a lower price per loaf, but it is not the correct answer.
• C) $3.00 per loaf: This option is tempting because it is a higher price per loaf, but it is not the correct answer.
• D) $3.50 per loaf: This option is tempting because it is an even higher price per loaf, but it is not the correct answer.

30-Second Cheat Sheet

Here are the 7 key things to remember:

• Equivalent ratios: Two ratios are equivalent if they have the same value, even if they are expressed differently.
• Proportion problems: A proportion problem is a problem that involves setting up a proportion to solve for an unknown value.
• Cross-multiplication: Cross-multiplication is a technique used to solve proportion problems by multiplying the means and the extremes of the proportion.
• If a:b = c:d, then ad = bc
• Equivalent ratios: Two ratios are equivalent if they have the same value, even if they are expressed differently.
• Cross-multiplication: Cross-multiplication is a technique used to solve proportion problems by multiplying the means and the extremes of the proportion.
• Check for proportionality: Before solving a proportion problem, check to make sure the ratios are proportional.

Learning Path

To master this topic, follow this learning path:

1. Beginner foundation: Understand the basic concepts of ratios and proportions, including equivalent ratios and proportion problems.
2. Core rules: Learn the key rules and formulas for setting up and solving proportion problems, including cross-multiplication.
3. Practice: Practice solving proportion problems using the rules and formulas you learned.
4. Timed drills: Practice solving proportion problems 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 related topics that appear alongside this one in exams:

• Fractions and decimals: Understanding fractions and decimals is essential for working with ratios and proportions.
• Algebra: Algebra is used to solve proportion problems, so it's essential to have a good understanding of algebraic concepts.
• Percentages: Percentages are used to express ratios and proportions, so it's essential to have a good understanding of percentage concepts.