Rates & Ratios - Varsity Tutors

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Rates & Ratios

Study Guide

Key Definition

A ratio is a comparison of two numbers, such as $3 : 5$ or $\frac{3}{5}$. A rate is a comparison of two quantities with different units, like $5 \, \text{miles per} \, 3 \, \text{hours}$.

Important Notes

  • A ratio can be simplified by dividing both terms by their greatest common factor, e.g., $6 : 21$ simplifies to $2 : 7$.
  • The total of all parts in a ratio is the sum of the individual parts.
  • Rates often describe speed, cost, or density.
  • Understanding ratios helps in comparing different quantities.
  • Simplifying rates and ratios makes them easier to interpret.

Mathematical Notation

$\frac{a}{b}$ represents a fraction$:$ is used to denote a ratio$+$ represents addition$-$ represents subtractionRemember to use proper notation when solving problems

Why It Works

Ratios and rates provide a way to compare quantities, allowing us to understand relationships and solve problems involving proportions and conversions.

Remember

Ratios can be simplified just like fractions, and rates can help compare quantities with different units.

Quick Reference

Simplified Ratio:$\frac{6}{21} = \frac{2}{7}$Rate Example:$34 \, \text{dollars per square foot}$

Understanding Rates & Ratios

Choose your learning level

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

Ratios compare two numbers, like $3 : 5$, and can be written as fractions $\frac{3}{5}$.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

What is the simplified form of the ratio $6 : 21$?

A$1 : 3$B$2 : 7$C$3 : 10$D$3 : 5$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You have a part-time job that pays $10$ dollars per hour. How much do you earn if you work $15$ hours?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

If the ratio of boys to girls in a class is $3 : 4$ and there are $28$ students, how many are boys?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

If the rate of speed is $60 \, \text{miles per hour}$, how far will you travel in $2.5$ hours?

A$120 \, \text{miles}$B$150 \, \text{miles}$C$180 \, \text{miles}$D$210 \, \text{miles}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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Review key concepts and takeaways

recap. Use space or enter to play video.recap thumbnail

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