Equality of Rational Numbers using Cross Multiplication

We will learn about the equality of rational numbers using cross multiplication.

How to determine whether the two given rational numbers are equal or not using cross multiplication?

We know there are many methods to determine the equality of two rational numbers but here we will learn the method of equality of two rational numbers using cross multiplication.

In this method, to determine the equality of two rational numbers a/b and c/d, we use the following result:

\(\frac{a}{b}\) = \(\frac{c}{d}\)

⇔ a × d = b × c
⇔ Numerator of first × Denominator of second = Denominator of first × Numerator of second

Solved examples on equality of rational numbers using cross multiplication:

1. Which of the following pairs of rational numbers are equal?

(i) \(\frac{-8}{32}\) and \(\frac{6}{-24}\) (ii) \(\frac{-4}{-18}\) and \(\frac{8}{24}\)

Solution:

(i) The given rational numbers are \(\frac{-8}{32}\) and \(\frac{6}{-24}\)

Numerator of first × Denominator of second = (-8) × (-24) = 192 and, Denominator of first × Numerator of second = 32 × 6 = 192.

Clearly,

Numerator of first × Denominator of second = Denominator of first × Numerator of second

Hence, \(\frac{-8}{32}\) = \(\frac{6}{-24}\)

Therefore, the given rational numbers \(\frac{-8}{32}\) and \(\frac{6}{-24}\) are equal.

(ii) The given rational numbers are \(\frac{-4}{-18}\) and \(\frac{8}{24}\)

Numerator of first × Denominator of second = -4 × 24 = -96 and, Denominator of first × Numerator of second = (-18) × 8 = -144

Clearly,

Numerator of first × Denominator of second ≠ Denominator of first × Numerator of second

Hence, \(\frac{-4}{-18}\) ≠ \(\frac{8}{24}\).

Therefore, the given rational numbers \(\frac{-4}{-18}\) and \(\frac{8}{24}\) are not equal.

2. If \(\frac{-6}{8}\) = \(\frac{k}{64}\), find the value of k.

Solution :

We know that \(\frac{a}{b}\) = \(\frac{c}{d}\) if ad = bc

Therefore, \(\frac{-6}{8}\) = \(\frac{k}{64}\)

⇒ -6 × 64 = 8 × k, [Numerator of first × Denominator of second = Denominator of first × Numerator of second]

⇒ -384 = 8k

⇒ 8k = -384

⇒ \(\frac{8k}{8}\) = \(\frac{-384}{8}\), [Dividing both sides by 8]

⇒ k = -48

Therefore, the value of k = -48

3. If \(\frac{7}{m}\) = \(\frac{49}{63}\), find the value of m.

Solution:

In order to write \(\frac{49}{63}\) as a rational number with numerator 7, we first find a number which when divided 49 gives 7.

Clearly, such a number is 49 ÷ 7 = 7.

Dividing the numerator and denominator of 49/63 by 7, we have

\(\frac{49}{63}\) = \(\frac{49 ÷ 7}{63 ÷ 7}\) = \(\frac{7}{9}\)

Therefore, \(\frac{7}{m}\) = \(\frac{49}{63}\)

⇒ \(\frac{7}{m}\) = \(\frac{7}{9}\)

⇒ m = 9

4. Fill in the blank: \(\frac{-7}{15}\) = \(\frac{.....}{135}\)

Solution:

In order to fill the required blank, we have to express -7 as a rational number with denominator 135. For this, we first find an integer which when multiplied with 15 gives us 135.

Clearly, such an integer is 135 ÷ 15 = 9

Multiplying the numerator and denominator of \(\frac{-7}{15}\) by 9, we get

\(\frac{-7}{15}\) = \(\frac{(-7) × 9}{15 × 9}\) = \(\frac{-63}{135}\)

Therefore, the required number is -63.

● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number