EQUIVALENT RATIONAL NUMBERS

Examples 1-7 : Write four rational numbers equivalent to the given rational number.

Example 1 :

⅔ 

Solution :

To find four equivalent rational numbers, multiply both numerator and denominator of the fraction ⅔ by 2, 3, 4 and 5.

⅔ x ²⁄₂ = ⁴⁄₁₂

⅔ x ³⁄₃ = ⁶⁄₉

⅔ x ⁴⁄₄ = ⁸⁄₁₂

⅔ x ⁵⁄₅ = ¹⁰⁄₁₅

Example 2 :

⁴⁄₇

Solution :

⁴⁄₇ x ²⁄₂ = ⁸⁄₁₄

⁴⁄₇ x ³⁄₃ = ¹²⁄₂₁

⁴⁄₇ x ⁴⁄₄ = ¹⁶⁄₂₈

⁴⁄₇ x ⁵⁄₅ = ²⁰⁄₃₅

Example 3 :

⁻⁶⁄₇

Solution :

⁻⁶⁄₇ x ²⁄₂ = ⁻¹²⁄₁₄

⁻⁶⁄₇ x ³⁄₃ = ⁻¹⁸⁄₂₁

⁻⁶⁄₇ x ⁴⁄₄ = ⁻²⁴⁄₂₈

⁻⁶⁄₇ x ⁵⁄₅ = ⁻³⁰⁄₃₅

Example 4 :

⁷⁄₉

Solution :

⁷⁄₉ x ²⁄₂ = ¹⁴⁄₁₈

⁷⁄₉ x ³⁄₃ = ²¹⁄₂₇

⁷⁄₉ x ⁴⁄₄ = ²⁸⁄₃₆

⁷⁄₉ x ⁵⁄₅ = ³⁵⁄₄₅

Example 5 :

2⅓ 

Solution :

2⅓ is a mixed number. To get four equivalent rational numbers to 2⅓, convert 2⅓ to  an improper fraction.

2⅓ = ⁷⁄₃

To find four equivalent rational numbers, multiply both numerator and denominator of the fraction ⁷⁄₃ by 2, 3, 4 and 5.

⁷⁄₃ x ²⁄₂ = ¹⁴⁄6

⁷⁄₃ x ³⁄₃ = ²¹⁄9

⁷⁄₃ x ⁴⁄₄ = ²⁸⁄12

⁷⁄₃ x ⁵⁄₅ = ³⁵⁄15

The four rational numbers equivalent to 2⅓ are

¹⁴⁄6, ²¹⁄9, ²⁸⁄12, ³⁵⁄15

Example 6 :

1.2 

Solution :

Convert the given decimal number to a fraction.

1.2 = ¹²⁄₁₀

= ⁶⁄₅

To find four equivalent rational numbers, multiply both numerator and denominator of the fraction ⁷⁄₃ by 2, 3, 4 and 5.

⁶⁄₅ x ²⁄₂ = ¹²⁄₁₀

⁶⁄₅ x ³⁄₃ = ¹⁸⁄₁₅

⁶⁄₅ x ⁴⁄₄ = ²⁴⁄₂₀

⁶⁄₅ x ⁵⁄₅ = ³⁰⁄₂₅

The four rational numbers equivalent to 1.2 are

¹²⁄₁₀, ¹⁸⁄₁₅, ²⁴⁄₂₀, ³⁰⁄₂₅

Example 7 :

Write a rational number equivalent to ¾ with denominator 20. 

Solution :

In the given fraction ¾, the denominator is 4. To get denominator 20, we have to multiply 4 by 5.

To get a rational number equivalent to ¾ with denominator 20, multiply both numerator and denominator of the fraction ¾ by 5.

¾ = ¾ x ⁵⁄₅

= ¹⁵⁄₂₀

Example 8 :

Write a rational number equivalent to ⁻⁷⁄₈ with denominator 56. 

Solution :

In the given fraction ⁻⁷⁄₈, the denominator is 8. To get denominator 56, we have to multiply 8 by 7.

To get a rational number equivalent to ⁻⁷⁄₈ with denominator 56, multiply both numerator and denominator of the fraction ⁻⁷⁄₈ by 7.

⁻⁷⁄₈ = ⁻⁷⁄₈ x ⁷⁄₇

= ⁻⁴⁹⁄₅₆

Example 9 :

Write a rational number equivalent to ¹⁰⁄₂₈ with denominator 14.

Solution :

In the given fraction ¹⁰⁄₂₈, the denominator is 28. To get denominator 14, we have to divide 28 by 2.

To get a rational number equivalent to ¹⁰⁄₂₈ with denominator 14, divide both numerator and denominator of the fraction ¹⁰⁄₂₈ by 2.

¹⁰⁄₂₈ = ⁽¹⁰^÷²⁾⁄₍₂₈÷₂₎

= ⁵⁄₁₄

Example 10 :

Write a rational number equivalent to ¹⁵⁄₂₁ with denominator 7.

Solution :

In the given fraction ¹⁵⁄₂₁, the denominator is 7. To get denominator 7, we have to divide 21 by 3.

To get a rational number equivalent to ¹⁵⁄₂₁ with denominator 7, divide both numerator and denominator of the fraction ¹⁵⁄₂₁ by 3.

¹⁵⁄₂₁ = ⁽¹⁵^÷³⁾⁄₍₂₁÷₃₎

= ⁵⁄₇

Example 11 :

If the two rational numbers ½ and ˣ⁄₈ are equivalent, find the value of x.

Solution :

Since ½ and ˣ⁄₈ are equivalent rational numbers,

½ = ˣ⁄₈

The denominator on the right side is 8. In the fraction ½, the denominator is 2. To get denominator 8, we have to multiply 2 by 4.

To get a rational number equivalent to ½ with denominator 8, multiply both numerator and denominator of the fraction ½ by 4.

⁽¹ˣ⁴⁾⁄₍₂ₓ₄₎ = ˣ⁄₈

⁴⁄₈ = ˣ⁄₈

The above two rational numbers are equivalent with the same denominator. Then, the numerators must be equal.

Therefore,

x = 4

Example 12 : 

If the two rational numbers ⁻³⁄₄ and ʸ⁄₂₀ are equivalent, find the value of y.

Solution :

Since ⁻³⁄₄ and ʸ⁄₂₀ are equivalent rational numbers,

⁻³⁄₄ = ʸ⁄₂₀

The denominator on the right side is 20. In the fraction ⁻³⁄₄, the denominator is 4. To get denominator 20, we have to multiply 4 by 5.

To get a rational number equivalent to ⁻³⁄₄ with denominator 20, multiply both numerator and denominator of the fraction ⁻³⁄₄ by 5.

⁻⁽³ˣ⁵⁾⁄₍₄ₓ₅₎ = ʸ⁄₂₀

⁻¹⁵⁄₂₀ = ʸ⁄₂₀

Therefore,

y = -15

Example 13 : 

If the two rational numbers ˣ⁄₅ and ⁹⁄₁₅ are equivalent, find the value of x.

Solution :

Since ˣ⁄₅ and ⁹⁄₁₅ are equivalent rational numbers,

ˣ⁄₅ = ⁹⁄₁₅

The denominator on the right side is 15. In the fraction ⁹⁄₁₅, the denominator is 15. To get denominator 5, we have to divide 15 by 3.

To get a rational number equivalent to ⁹⁄₁₅ with denominator 5, divide both numerator and denominator of the fraction ⁹⁄₁₅ by 3.

ˣ⁄₅ = ⁽⁹^÷³⁾⁄₍₁₅÷₃₎

ˣ⁄₅ = ⅗

Therefore,

x = 3

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