Table of Contents

Sharing in a ratio Level 7

Introduction

Have you ever shared a pizza with friends? Understanding how to share things fairly is all about ratios! Sharing in a ratio helps us divide quantities in ways that reflect our agreements or needs. In this article, we will explore how to share quantities in a given ratio using real-life problems to understand proportional relationships.

Definition and Concept

A ratio is a way to compare two or more quantities. It shows how much of one thing there is compared to another. When we share in a ratio, we are dividing a total amount into parts that are proportional to the numbers in the ratio.

For example, if we have a ratio of 2:3, it means for every 2 parts of one quantity, there are 3 parts of another quantity.

Relevance:

  • Mathematics: Ratios are foundational in understanding proportions and are used in various mathematical applications.
  • Real-world applications: Ratios are used in cooking, finance, and any situation where quantities need to be divided fairly.

Historical Context or Origin​

Ratios have been used since ancient times, dating back to the Egyptians and Greeks, who utilized them in trade and construction. The concept of ratios has evolved into a fundamental aspect of mathematics, influencing fields from architecture to economics.

Understanding the Problem

To share a quantity in a given ratio, we first need to understand the total parts of the ratio. For example, if we need to share 30 apples in the ratio of 2:3, we must find the total parts, which is 2 + 3 = 5 parts.

Next, we divide the total quantity by the total parts to find the value of one part. In this case, 30 apples ÷ 5 parts = 6 apples per part.

Finally, we multiply the number of parts for each person by the value of one part to find out how many apples each person gets.

Methods to Solve the Problem with different types of problems​

Method 1: Step-by-Step Division

  1. Determine the total parts of the ratio.
  2. Divide the total quantity by the total parts to find one part.
  3. Multiply the number of parts in the ratio for each person by the value of one part.

Example:
Share 30 apples in the ratio of 2:3.
1. Total parts = 2 + 3 = 5.
2. One part = 30 ÷ 5 = 6.
3. For person A (2 parts): 2 × 6 = 12 apples. For person B (3 parts): 3 × 6 = 18 apples.

Exceptions and Special Cases​

  • Zero in Ratio: If one part of the ratio is zero (e.g., 0:3), it means that part does not receive anything.
  • Equal Sharing: If the ratio is the same (1:1), the quantities will be shared equally.

Step-by-Step Practice​

Problem 1: Share 24 candies in the ratio of 1:3.

Solution:

  • Total parts = 1 + 3 = 4.
  • One part = 24 ÷ 4 = 6.
  • For person A (1 part): 1 × 6 = 6 candies. For person B (3 parts): 3 × 6 = 18 candies.

    • Problem 2: Share 50 dollars in the ratio of 2:5.

    Solution:

  • Total parts = 2 + 5 = 7.
  • One part = 50 ÷ 7 ≈ 7.14.
  • For person A (2 parts): 2 × 7.14 ≈ 14.29 dollars. For person B (5 parts): 5 × 7.14 ≈ 35.71 dollars.

    • Examples and Variations

    Example 1: Share 18 pencils in the ratio of 3:2.

    Solution:

  • Total parts = 3 + 2 = 5.
  • One part = 18 ÷ 5 = 3.6 pencils.
  • For person A (3 parts): 3 × 3.6 = 10.8 pencils. For person B (2 parts): 2 × 3.6 = 7.2 pencils.

    • Example 2: Share 100 meters of ribbon in the ratio of 4:1.

    Solution:

  • Total parts = 4 + 1 = 5.
  • One part = 100 ÷ 5 = 20 meters.
  • For person A (4 parts): 4 × 20 = 80 meters. For person B (1 part): 1 × 20 = 20 meters.

    • Interactive Quiz with Feedback System​

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    Common Mistakes and Pitfalls

    • Forgetting to add the parts of the ratio.
    • Not dividing the total quantity correctly.
    • Multiplying by the wrong number of parts.

    Tips and Tricks for Efficiency

    • Always check if the ratio adds up to the total parts before dividing.
    • Practice with real-life scenarios to understand ratios better.
    • Use visual aids like pie charts to represent ratios visually.

    Real life application

    • Cooking: Adjusting recipes based on serving sizes.
    • Finance: Distributing profits or expenses among partners.
    • Construction: Dividing materials for different sections of a project.

    FAQ's

    A ratio is a comparison of two quantities that shows how much of one thing there is compared to another.
    Yes, ratios can be simplified just like fractions to show the simplest form.
    If the total quantity isn’t divisible evenly, you can still distribute the whole number parts and handle the remainder as needed.
    Yes, ratios can be extended to more than two quantities, just add more parts together.
    Ratios help us understand relationships between quantities and are widely used in everyday life, from cooking to budgeting.

    Conclusion

    Sharing in a ratio is a valuable skill that applies to many real-life situations. By practicing how to divide quantities proportionally, you’ll become more adept at handling everyday problems. Remember, understanding ratios not only helps in math but also in making fair decisions in life!

    References and Further Exploration

    • Khan Academy: Lessons on ratios and proportions.
    • Book: Ratio and Proportion by John Smith.

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