If X:y = 4/3 : 5/2 And Y:z = 15/4 : 10/3 , Find X:y:z In The Simplest Form.​

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Understanding the Problem

In this article, we will explore how to simplify ratios given in the form of fractions. We will use the given ratios x:y = 4/3 : 5/2 and y:z = 15/4 : 10/3 to find the simplest form of x:y:z.

What are Ratios?

A ratio is a comparison of two or more numbers. It is a way of expressing the relationship between two or more quantities. Ratios can be expressed as fractions, decimals, or percentages.

Simplifying Ratios

To simplify a ratio, we need to find the greatest common divisor (GCD) of the two numbers in the ratio. The GCD is the largest number that divides both numbers without leaving a remainder.

Finding the GCD

To find the GCD of two numbers, we can use the Euclidean algorithm. The Euclidean algorithm is a method for finding the GCD of two numbers by repeatedly dividing the larger number by the smaller number and taking the remainder.

Example

Let's find the GCD of 4 and 3.

  1. Divide 4 by 3: 4 = 1 × 3 + 1
  2. Divide 3 by 1: 3 = 3 × 1 + 0

The remainder is 0, so the GCD of 4 and 3 is 1.

Simplifying the Ratios

Now that we have found the GCD of 4 and 3, we can simplify the ratios.

x:y = 4/3 : 5/2

To simplify this ratio, we need to find the GCD of 4 and 3, and the GCD of 5 and 2.

The GCD of 4 and 3 is 1. The GCD of 5 and 2 is 1.

Since the GCD of both ratios is 1, we can simplify the ratio by dividing both numbers by 1.

x:y = 4/3 : 5/2 = 4/1 : 5/1 = 4:5

y:z = 15/4 : 10/3

To simplify this ratio, we need to find the GCD of 15 and 4, and the GCD of 10 and 3.

The GCD of 15 and 4 is 1. The GCD of 10 and 3 is 1.

Since the GCD of both ratios is 1, we can simplify the ratio by dividing both numbers by 1.

y:z = 15/4 : 10/3 = 15/1 : 10/1 = 15:10

Combining the Ratios

Now that we have simplified both ratios, we can combine them to find the simplest form of x:y:z.

x:y:z = 4:5 : 15:10

To combine these ratios, we need to find the GCD of 4 and 15, and the GCD of 5 and 10.

The GCD of 4 and 15 is 1. The GCD of 5 and 10 is 5.

Since the GCD of both ratios is 5, we can combine the ratios by dividing both numbers by 5.

x:y:z = 4/5 : 5/5 : 15/5 : 10/5

x:y:z = 4/5 : 1 : 3 : 2

Conclusion

In this article, we have learned how to simplify ratios given in the form of fractions. We have used the given ratios x:y = 4/3 : 5/2 and y:z = 15/4 : 10/3 to find the simplest form of x:y:z. We have found that x:y:z = 4/5 : 1 : 3 : 2.

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 in the ratio.
  • The GCD is the largest number that divides both numbers without leaving a remainder.
  • We can use the Euclidean algorithm to find the GCD of two numbers.
  • We can simplify a ratio by dividing both numbers by the GCD.
  • We can combine two or more ratios by finding the GCD of the two ratios and dividing both numbers by the GCD.

Practice Problems

  1. Simplify the ratio x:y = 6/8 : 9/12.
  2. Simplify the ratio y:z = 12/8 : 15/20.
  3. Combine the ratios x:y = 3/4 : 5/6 and y:z = 10/12 : 15/20.

Answer Key

  1. x:y = 3/4 : 9/12 = 3:9
  2. y:z = 12/8 : 15/20 = 3:5
  3. x:y:z = 3/4 : 5/6 : 10/12 : 15/20 = 3:5 : 5:3
    Frequently Asked Questions: Simplifying Ratios =====================================================

Q: What is a ratio?

A: A ratio is a comparison of two or more numbers. It is a way of expressing the relationship between two or more quantities. Ratios can be expressed as fractions, decimals, or percentages.

Q: Why do we need to simplify ratios?

A: We need to simplify ratios to make them easier to work with. Simplifying ratios helps us to:

  • Compare two or more quantities more easily
  • Perform calculations more efficiently
  • Express the relationship between two or more quantities more clearly

Q: How do we simplify a ratio?

A: To simplify a ratio, we need to find the greatest common divisor (GCD) of the two numbers in the ratio. The GCD is the largest number that divides both numbers without leaving a remainder.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a method for finding the GCD of two numbers by repeatedly dividing the larger number by the smaller number and taking the remainder.

Q: How do we use the Euclidean algorithm to find the GCD?

A: To use the Euclidean algorithm, we follow these steps:

  1. Divide the larger number by the smaller number.
  2. Take the remainder.
  3. Divide the smaller number by the remainder.
  4. Take the remainder.
  5. Repeat steps 3 and 4 until the remainder is 0.
  6. The last non-zero remainder is the GCD.

Q: Can we simplify a ratio by dividing both numbers by a number other than the GCD?

A: No, we cannot simplify a ratio by dividing both numbers by a number other than the GCD. Dividing both numbers by a number other than the GCD will not result in a simplified ratio.

Q: How do we combine two or more ratios?

A: To combine two or more ratios, we need to find the GCD of the two ratios and divide both numbers by the GCD.

Q: What is the simplest form of a ratio?

A: The simplest form of a ratio is a ratio that has been simplified by dividing both numbers by their greatest common divisor (GCD).

Q: Can we simplify a ratio with a variable?

A: Yes, we can simplify a ratio with a variable. To simplify a ratio with a variable, we need to find the GCD of the variable and the other number in the ratio.

Q: How do we simplify a ratio with a variable?

A: To simplify a ratio with a variable, we follow these steps:

  1. Find the GCD of the variable and the other number in the ratio.
  2. Divide both numbers by the GCD.

Q: Can we simplify a ratio with a negative number?

A: Yes, we can simplify a ratio with a negative number. To simplify a ratio with a negative number, we need to find the GCD of the negative number and the other number in the ratio.

Q: How do we simplify a ratio with a negative number?

A: To simplify a ratio with a negative number, we follow these steps:

  1. Find the GCD of the negative number and the other number in the ratio.
  2. Divide both numbers by the GCD.

Q: What are some common mistakes to avoid when simplifying ratios?

A: Some common mistakes to avoid when simplifying ratios include:

  • Dividing both numbers by a number other than the GCD
  • Not finding the GCD of the two numbers in the ratio
  • Not dividing both numbers by the GCD
  • Simplifying a ratio with a variable or a negative number incorrectly

Q: How do we check if a ratio is simplified?

A: To check if a ratio is simplified, we need to make sure that the GCD of the two numbers in the ratio is 1. If the GCD is 1, then the ratio is simplified.