QUESTION 3 [Ratio, Rate, Direct And Indirect Proportion]3.1 Write The Following Ratio In Simplified Form: $\[ 3 \frac{2}{3} : 1 \frac{1}{4} \\]3.2 Divide R250 Between Sibongile And Ntoml.3.3 Increase 150 In The Ratio 3:2.

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Introduction

Ratios, rates, direct and indirect proportions are fundamental concepts in mathematics that help us understand the relationships between different quantities. In this article, we will explore these concepts in detail, starting with simplifying ratios, dividing amounts between individuals, and increasing quantities in a given ratio.

3.1 Simplifying Ratios

A ratio is a comparison of two or more numbers. It is usually expressed as a fraction, with the first number as the numerator and the second number as the denominator. To simplify a ratio, we need to find the greatest common divisor (GCD) of the two numbers and divide both numbers by the GCD.

Example 1: Simplifying a Ratio

Suppose we have the ratio 3 2/3 : 1 1/4. To simplify this ratio, we need to find the GCD of the two numbers.

To find the GCD of 3 2/3 and 1 1/4, we can convert the mixed numbers to improper fractions.

3 2/3 = (11/3)
1 1/4 = (5/4)

Now, we can find the GCD of 11 and 5, which is 1.

So, the simplified ratio is:
(11/3) : (5/4) = (11/3) * (4/5) : (5/4) * (4/5)
= 44/15 : 5/5
= 44/15 : 1

Therefore, the simplified ratio is 44/15 : 1.

3.2 Dividing Amounts between Individuals

When dividing an amount between two or more individuals, we need to find the ratio in which the amount is to be divided. For example, suppose we have R250 to divide between Sibongile and Ntoml in the ratio 3:2.

Example 2: Dividing an Amount between Individuals

To divide R250 between Sibongile and Ntoml in the ratio 3:2, we need to find the amount each person will receive.

Let's say Sibongile's share is 3x and Ntoml's share is 2x.

We know that the total amount is R250, so we can set up the equation:

3x + 2x = 250

Combine like terms:

5x = 250

Divide both sides by 5:

x = 50

Now, we can find Sibongile's share:

Sibongile's share = 3x = 3(50) = R150

Ntoml's share = 2x = 2(50) = R100

Therefore, Sibongile will receive R150 and Ntoml will receive R100.

3.3 Increasing Quantities in a Given Ratio

When increasing a quantity in a given ratio, we need to find the factor by which the quantity is to be increased. For example, suppose we have 150 to increase in the ratio 3:2.

Example 3: Increasing a Quantity in a Given Ratio

To increase 150 in the ratio 3:2, we need to find the factor by which the quantity is to be increased.

Let's say the factor is x.

We know that the ratio is 3:2, so we can set up the equation:

3x + 2x = 150 + x

Combine like terms:

5x = 150 + x

Subtract x from both sides:

4x = 150

Divide both sides by 4:

x = 37.5

Now, we can find the increased quantity:

Increased quantity = 150 + x = 150 + 37.5 = 187.5

Therefore, the increased quantity is 187.5.

Conclusion

In conclusion, ratios, rates, direct and indirect proportions are fundamental concepts in mathematics that help us understand the relationships between different quantities. By simplifying ratios, dividing amounts between individuals, and increasing quantities in a given ratio, we can solve a wide range of problems in mathematics and real-life situations.

Key Takeaways

  • A ratio is a comparison of two or more numbers.
  • To simplify a ratio, we need to find the greatest common divisor (GCD) of the two numbers and divide both numbers by the GCD.
  • When dividing an amount between two or more individuals, we need to find the ratio in which the amount is to be divided.
  • When increasing a quantity in a given ratio, we need to find the factor by which the quantity is to be increased.

Practice Problems

  1. Simplify the ratio 2 1/2 : 3 3/4.
  2. Divide R500 between Thembi and Nkosinathi in the ratio 2:3.
  3. Increase 200 in the ratio 4:3.

Answers

  1. 11/4 : 15/4 = 11/4 * (4/15) : (15/4) * (4/15) = 11/15 : 1
  2. Thembi's share = R300, Nkosinathi's share = R400
  3. Increased quantity = 250
    Ratios, Rates, Direct and Indirect Proportions: Frequently Asked Questions ====================================================================

Q: What is a ratio?

A: A ratio is a comparison of two or more numbers. It is usually expressed as a fraction, with the first number as the numerator and the second number as the denominator.

Q: How do I simplify a ratio?

A: To simplify a ratio, you need to find the greatest common divisor (GCD) of the two numbers and divide both numbers by the GCD.

Q: What is the difference between a ratio and a proportion?

A: A ratio is a comparison of two or more numbers, while a proportion is a statement that two ratios are equal.

Q: How do I divide an amount between two or more individuals in a given ratio?

A: To divide an amount between two or more individuals in a given ratio, you need to find the ratio in which the amount is to be divided and then divide the amount accordingly.

Q: How do I increase a quantity in a given ratio?

A: To increase a quantity in a given ratio, you need to find the factor by which the quantity is to be increased and then multiply the quantity by that factor.

Q: What is the difference between a direct proportion and an indirect proportion?

A: A direct proportion is a statement that two ratios are equal, while an indirect proportion is a statement that two ratios are not equal.

Q: How do I solve a proportion problem?

A: To solve a proportion problem, you need to set up an equation using the given ratios and then solve for the unknown variable.

Q: What are some common applications of ratios, rates, direct and indirect proportions?

A: Ratios, rates, direct and indirect proportions have many applications in real-life situations, such as:

  • Cooking: Recipes often require you to mix ingredients in a certain ratio.
  • Building: Architects use ratios to design buildings and ensure that they are proportional.
  • Finance: Investors use ratios to evaluate the performance of stocks and bonds.
  • Science: Scientists use ratios to measure the concentration of substances.

Q: How do I convert a ratio to a percentage?

A: To convert a ratio to a percentage, you need to divide the numerator by the denominator and then multiply by 100.

Q: How do I convert a percentage to a ratio?

A: To convert a percentage to a ratio, you need to divide the percentage by 100 and then express it as a fraction.

Q: What are some common mistakes to avoid when working with ratios, rates, direct and indirect proportions?

A: Some common mistakes to avoid when working with ratios, rates, direct and indirect proportions include:

  • Not simplifying ratios before using them.
  • Not checking for errors in proportions.
  • Not using the correct units when working with rates.
  • Not considering indirect proportions when solving problems.

Q: How do I practice and improve my skills in working with ratios, rates, direct and indirect proportions?

A: To practice and improve your skills in working with ratios, rates, direct and indirect proportions, you can:

  • Practice solving problems using ratios, rates, direct and indirect proportions.
  • Use online resources and tools to help you practice and learn.
  • Join a study group or find a study partner to work with.
  • Review and practice regularly to build your skills and confidence.

Conclusion

In conclusion, ratios, rates, direct and indirect proportions are fundamental concepts in mathematics that have many applications in real-life situations. By understanding and practicing these concepts, you can improve your skills and confidence in solving problems and making informed decisions.