COMPARING RATIONAL NUMBERS

A rational number is a number which is in the form of ᵃ⁄b where a and b are integers and b ≠ 0.

Let us learn how to compare two rational numbers.

Comparing Two Positive Rational Numbers with Same Denominator

If two rational numbers are positive with same denominator, compare the numerators and decide the larger and smaller.

For example, consider the two rational numbers ³⁄₇ and ⁵⁄₇.

Both the rational numbers are positive and they have the same denominator 7.

Compare the numerators 3 and 5.

3 < 5

Therefore,

³⁄₇ < ⁵⁄₇

Comparing Two Negative Rational Numbers with Same Denominator

If two rational numbers are negative with same denominator, compare the numerators and decide the larger and smaller.

For example, consider the two rational numbers ⁻³⁄₇ and ⁻⁵⁄₇.

Both the rational numbers are negative and they have the same denominator 7.

Compare the numerators -3 and -5.

-3 > -5

Therefore,

⁻³⁄₇ > ⁻⁵⁄₇

Comparing Two Rational Numbers with Different Signs

In two rational numbers, if one is positive and other one is negative, then it is pretty easy to compare those two rational numbers. The positive rational number is greater than the negative rational number.

For example, consider the two rational numbers ⁻²⁄₅ and ⅛.

Clearly, the positive rational number ⅛ is greater than the negative rational number ⁻²⁄₅.

⁻²⁄₅ < ⅛

Comparing Two Rational with Same Sign and Different Denominators

The following steps will be useful to compare two rational numbers with same sign and different denominators.

Step 1 :

Find the least common multiple of the two different denominators.

Step 2 :

Make the least commom multiple found in step 1 as denominator for each fraction by multiplying the fractions by appropriate numbers.

Step 3 :

Now, compare the numerators and decide the larger and smaller.

For example, consider the two rational numbers ¾ and ⅚.

The least common multiple of the denominators 4 and 6 is 12.

Multiply both numerator and denominator of the fraction ¾ by 3 to get the denominator 12.

⁽³ˣ³⁾⁄₍₄ₓ₃₎ = ⁹⁄₁₂ ----(1)

Multiply both numerator and denominator of the fraction ⅚ by 2 to get the denominator 12.

⁽⁵ˣ²⁾⁄₍₆ₓ₂₎ = ¹⁰⁄₁₂ ----(2)

In (1) and (2), compare the numerators 9 and 10.

9 < 10

Therefore,

⁹⁄₁₂ < ¹⁰⁄₁₂ ----> ¾ < ⅚

Solved Examples

In each of the following examples, compare the two rational numbers :

Example 1 :

⅗ and ⅖

Solution :

Both ⅗ and ⅖ are positive with same denominator.

Compare the numerators 3 and 2.

3 > 2

Therefore,

⅗ > ⅖

Example 2 :

⁻⁵⁄₇ and ⁻⁴⁄₇

Solution :

Both ⁻⁵⁄₇ and ⁻⁴⁄₇ are negative with same denominator.

Compare the numerators -5 and -4.

-5 < -4

Therefore,

⁻⁵⁄₇ < ⁻⁴⁄₇

Example 3 :

⁻²⁄₅ and ⅕

Solution :

Since ⁻²⁄₅ and ⅕ are having different signs, the positive rational number is always greater than the negative rational number.

Therefore,

⁻²⁄₅ < ⅕

Example 4 :

⅙ and ¼

Solution :

Since ⅙ and ¼ are having different denominators, find the least common multiple of the denominators.

Least common multiple of (6, 4) = 12.

Make 12 as denominator for both the fractions by multiplying both the fractions by the approproate numbers. 

Multiply both numerator and denominator of the fraction ⅙ by 2 to get the denominator 12.

⁽¹ˣ²⁾⁄₍₆ₓ₂₎ = ²⁄₁₂ ----(1)

Multiply both numerator and denominator of the fraction ¼ by 3 to get the denominator 12.

⁽¹ˣ³⁾⁄₍₄ₓ₃₎ = ³⁄₁₂ ----(2)

In (1) and (2), compare the numerators.

2 < 3

Therefore,

²⁄₁₂ < ³⁄₁₂ ----> ⅙ < ¼

Example 5 :

⁻¹¹⁄₅ and ⁻²¹⁄₈

Solution :

Since ⁻¹¹⁄₅ and ⁻²¹⁄₈ are having different denominators, find the least common multiple of the denominators.

Least common multiple of (5, 8) = 40.

Make 40 as denominator for both the fractions by multiplying both the fractions by the approproate numbers. 

Multiply both numerator and denominator of the fraction ⁻¹¹⁄₅ by 8 to get the denominator 40.

⁽⁻¹¹ˣ⁸⁾⁄₍₅ₓ₈₎ = ⁻⁸⁸⁄₄₀ ----(1)

Multiply both numerator and denominator of the fraction ⁻²¹⁄₈ by 5 to get the denominator 40.

⁽⁻²¹ˣ⁵⁾⁄₍₈ₓ₅₎ = ⁻¹⁰⁵⁄₄₀ ----(2)

In (1) and (2), compare the numerators.

-88 > -105

Therefore,

⁻⁸⁸⁄₄₀ > ⁻¹⁰⁵⁄₄₀ ----> ⁻¹¹⁄₅ > ⁻²¹⁄₈

Example 6 :

³⁄₋₄ and ⁻¹⁄₂

Solution :

³⁄₋₄ can be written as ⁻³⁄₄.

Since ⁻³⁄₄ and ⁻¹⁄₂ are having different denominators, find the least common multiple of the denominators.

Least common multiple of (4, 2) = 4.

The fraction ⁻³⁄₄ is already having the denominator 4.

⁻³⁄₄ ----(1)

Multiply both numerator and denominator of the fraction ⁻¹⁄₂ by 2 to get the denominator 4.

⁽⁻¹ˣ²⁾⁄₍₂ₓ₂₎ = ⁻²⁄₄ ----(2)

In (1) and (2), compare the numerators.

-3 < -2

Therefore,

⁻³⁄₄ < ⁻²⁄₄ ----> ³⁄₋₄ < ⁻¹⁄₂

Example 7 :

⅘ and ⅔

Solution :

Since ⅘ and ⅔ are having different denominators, find the least common multiple of the denominators.

Least common multiple of (5, 3) = 15.

Make 15 as denominator for both the fractions by multiplying both the fractions by the approproate numbers.

Multiply both numerator and denominator of the fraction ⅘ by 3 to get the denominator 15.

⁽⁴ˣ³⁾⁄₍₅ₓ₃₎ = ¹²⁄₁₅ ----(1)

Multiply both numerator and denominator of the fraction ⅔ by 5 to get the denominator 15.

⁽²ˣ⁵⁾⁄₍₃ₓ₅₎ = ¹⁰⁄₁₅ ----(2)

In (1) and (2), compare the numerators.

12 > 10

Therefore,

¹²⁄₁₅ > ¹⁰⁄₁₅ ----> ⅘ > ⅔

Example 8 :

⁻³⁄₈ and ⁻²⁄₋₅

Solution :

⁻²⁄₋₅ = ²⁄₅

Since ⁻³⁄₈ and ²⁄₅ are having different signs, the positive rational number is always greater than the negative rational number.

Therefore,

⁻³⁄₈ < ²⁄₅ ----> ⁻³⁄₈ < ⁻²⁄₋₅

Arranging the Rational Numbers in Ascending Order and Descending Order

In each of the following examples, compare the given rational numbers and arrange them in ascending and descending orders.

Example 9 :

⁻⁵⁄₁₂, ⁻¹¹⁄₈, ⁻¹⁵⁄₂₄, ⁻⁷⁄₋₉, ¹²⁄₃₆

Solution :

Least common multiple of (12, 8, 24, 9, 36) = 144.

⁽⁻⁵ˣ¹²⁾⁄₍₁₂ₓ₁₂₎ = ⁻⁶⁰⁄₁₄₄

⁽⁻¹¹ˣ¹⁸⁾⁄₍₈ₓ₁₈₎ = ⁻¹⁹⁸⁄₁₄₄

⁽⁻¹⁵ˣ⁶⁾⁄₍₂₄ₓ₆₎ = ⁻⁹⁰⁄₁₄₄

⁽⁻⁷ˣ¹⁶⁾⁄₍₋₉ₓ₁₆₎ = ¹¹²⁄₁₄₄

⁽¹²ˣ⁴⁾⁄₍₃₆ₓ₄₎ = ⁴⁸⁄₁₄₄

⁻¹⁹⁸⁄₁₄₄ > ⁻⁹⁰⁄₁₄₄ > ⁻⁶⁰⁄₁₄₄ > ⁴⁸⁄₁₄₄ > ¹¹²⁄₁₄₄

Ascending Order :

⁻¹¹⁄₈, ⁻¹⁵⁄₂₄, ⁻⁵⁄₁₂, ¹²⁄₃₆, ⁻⁷⁄₋₉

Descending Order :

⁻⁷⁄₋₉, ¹²⁄₃₆, ⁻⁵⁄₁₂, ⁻¹⁵⁄₂₄, ⁻¹¹⁄₈

Example 10 :

⁻¹⁷⁄₁₀, ⁻⁷⁄₅, 0, ⁻²⁄₄, ⁻¹⁹⁄₂₀

Solution :

Least common multiple of (10, 5, 4, 20) = 20.

⁽⁻¹⁷ˣ²⁾⁄₍₁₀ₓ₂₎ = ⁻³⁴⁄₂₀

⁽⁻⁷ˣ⁴⁾⁄₍₅ₓ₄₎ = ⁻²⁸⁄₂₀

⁽⁻²ˣ⁵⁾⁄₍₄ₓ₅₎ = ⁻¹⁰⁄₂₀

⁽⁻¹⁹ˣ¹⁾⁄₍₂₀ₓ₁₎ = ⁻¹⁹⁄₂₀

⁻³⁴⁄₂₀ > ⁻²⁸⁄₂₀ > ⁻¹⁹⁄₂₀ > ⁻¹⁰⁄₂₀ > 0

Ascending Order :

⁻¹⁷⁄₁₀, ⁻⁷⁄₅, ⁻¹⁹⁄₂₀, ⁻²⁄₄, 0

Descending Order :

0, ⁻²⁄₄, ⁻¹⁹⁄₂₀, ⁻⁷⁄₅, ⁻¹⁷⁄₁₀

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