Yes, 1 &frasl; 2 and 2 &frasl; 4 are equivalent rational numbers. To verify, cross multiply: 1 × 4 = 4 and 2 × 2 = 4. Since both products are equal, the two rational numbers are equivalent. We can also obtain 2 &frasl; 4 by multiplying both the numerator and the denominator of 1 &frasl; 2 by 2.

Equivalent Rational Numbers are a part of rational numbers. A rational number is any number that can be written as $%5Cfrac%7Bp%7D%7Bq%7D$ where $p$ and $q$ are integers and $q$ is not zero. For example, $%5Cfrac%7B3%7D%7B4%7D$ , $%5Cfrac%7B2%7D%7B5%7D$ , and $%5Cfrac%7B7%7D%7B10%7D$ are all rational numbers. Equivalent rational numbers are different fractions that have the same value. For example, $%5Cfrac%7B1%7D%7B2%7D$ , $%5Cfrac%7B2%7D%7B4%7D$ , and $%5Cfrac%7B4%7D%7B8%7D$ are all equal because they represent the same part of a whole. We make equivalent rational numbers by multiplying or dividing both the numerator and the denominator by the same non zero number.

How to Find Equivalent Rational Numbers
Equivalent Rational Numbers by Multiplication
Equivalent Rational Numbers Using Division
Solved Examples

How to Find Equivalent Rational Numbers

To find equivalent rational numbers, multiply (or divide) both the numerator and the denominator of a given fraction by the same non zero integer. For example, from $%5Cfrac%7B2%7D%7B3%7D$ , we can get $%5Cfrac%7B4%7D%7B6%7D$ (multiply by 2), $%20%5Cfrac%7B6%7D%7B9%7D$ (multiply by 3), and so on. All these fractions are equal in value.

Know more about related topics:

Equivalent Rational Numbers by Multiplication

Equivalent rational numbers by multiplication are new fractions that have the same value as the original fraction. We get them by multiplying both the top number (numerator) and the bottom number (denominator) by the same non‑zero number.

For example, take $%5Cfrac%7B2%7D%7B5%7D$ :

Multiply by 2: $%5Cfrac%7B2%20%5Ctimes%202%7D%7B5%20%5Ctimes%202%7D$ = $%5Cfrac%7B4%7D%7B10%7D$
Multiply by 3: $%5Cfrac%7B2%20%5Ctimes%203%7D%7B5%20%5Ctimes%203%7D$ = $%5Cfrac%7B6%7D%7B15%7D$

Here, $%5Cfrac%7B2%7D%7B5%7D$ , $%5Cfrac%7B4%7D%7B10%7D$ , and $%5Cfrac%7B6%7D%7B15%7D$ are all equal in value. They are equivalent rational numbers made by multiplication.

Equivalent Rational Numbers Using Division

Equivalent rational numbers using division are fractions that have the same value as the original, but with smaller numbers. We get them by dividing both the numerator (top number) and the denominator (bottom number) by the same non zero number.

For example, take $%5Cfrac%7B8%7D%7B12%7D$ :

Divide both by 2: $%5Cfrac%7B8%20%5Cdiv%202%7D%7B12%20%5Cdiv%202%7D%20%3D%20%5Cfrac%7B4%7D%7B6%7D$
Divide both by 4: $%5Cfrac%7B8%20%5Cdiv%204%7D%7B12%20%5Cdiv%204%7D%20%3D%20%5Cfrac%7B2%7D%7B3%7D$

Here, $%5Cfrac%7B8%7D%7B12%7D$ , $%5Cfrac%7B4%7D%7B6%7D$ , and $%5Cfrac%7B2%7D%7B3%7D$ are all equal in value. They are equivalent rational numbers made by division.

Solved Examples

Example 1:

Given rational number: $%5Cfrac%7B12%7D%7B16%7D$

Divide numerator and denominator by 4:

$%5Cfrac%7B12%20%5Cdiv%204%7D%7B16%20%5Cdiv%204%7D%20%3D%20%5Cfrac%7B3%7D%7B4%7D$

So, $%5Cfrac%7B12%7D%7B16%7D%20and%20%5Cfrac%7B3%7D%7B4%7D$ are equivalent rational numbers.

Example 2:

Given rational number: $%5Cfrac%7B15%7D%7B25%7D$

Divide numerator and denominator by 5:

$%5Cfrac%7B15%20%5Cdiv%205%7D%7B25%20%5Cdiv%205%7D%20%3D%20%5Cfrac%7B3%7D%7B5%7D$

So, $%5Cfrac%7B15%7D%7B25%7D$ and $%5Cfrac%7B3%7D%7B5%7D$ are equivalent.

Example 3:

Given rational number: $%5Cfrac%7B18%7D%7B24%7D$

Divide numerator and denominator by 6:

$%5Cfrac%7B18%20%5Cdiv%206%7D%7B24%20%5Cdiv%206%7D%20%3D%20%5Cfrac%7B3%7D%7B4%7D$

So, $%5Cfrac%7B18%7D%7B24%7D$ and $%5Cfrac%7B3%7D%7B4%7D$ are equivalent.

Example 4:

Given rational number: $%5Cfrac%7B20%7D%7B30%7D$

Divide numerator and denominator by 10:

$%5Cfrac%7B20%20%5Cdiv%2010%7D%7B30%20%5Cdiv%2010%7D%20%3D%20%5Cfrac%7B2%7D%7B3%7D$

So, $%5Cfrac%7B20%7D%7B30%7D$ and $%5Cfrac%7B2%7D%7B3%7D$ are equivalent rational numbers.

Frequently Asked Questions on Equivalent Rational Numbers

1. What are equivalent rational numbers?

Equivalent rational numbers are rational numbers that have the same value, even though they look different. They are obtained by multiplying or dividing both the numerator and the denominator of a rational number by the same non zero integer. For example, ¹⁄₂, ²⁄₄, ³⁄₆, and ⁴⁄₈ are all equivalent rational numbers because they all represent the same value 0.5.

2. How do you find equivalent rational numbers?

To find equivalent rational numbers, multiply or divide both the numerator and the denominator by the same non zero integer.

By Multiplication: ²⁄₃ × ²⁄₂ = ⁴⁄₆ (equivalent to ²⁄₃)
By Division: ⁸⁄₁₂ ÷ ⁴⁄₄ = ²⁄₃ (equivalent to ⁸⁄₁₂)

Both methods give rational numbers that are equal in value to the original.

3. Are 1⁄2 and 2⁄4 equivalent rational numbers?

Yes, ¹⁄₂ and ²⁄₄ are equivalent rational numbers. To verify, cross multiply: 1 × 4 = 4 and 2 × 2 = 4. Since both products are equal, the two rational numbers are equivalent. We can also obtain ²⁄₄ by multiplying both the numerator and the denominator of ¹⁄₂ by 2.

4. How do you check if two rational numbers are equivalent?

To check if two rational numbers are equivalent, use the cross multiplication method: If ^a⁄_b and ^c⁄_d are two rational numbers, they are equivalent if a × d = b × c.

Example:

Check if ³⁄₄ and ⁹⁄₁₂ are equivalent.

3 × 12 = 36 and 4 × 9 = 36.

Since 36 = 36, ³⁄₄ and ⁹⁄₁₂ are equivalent rational numbers.

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Equivalent Rational Numbers

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