Comparison of Rational Numbers by Cross Multiplication


 
 
Concept Explanation
 

Comparison of Rational Numbers by Cross Multiplication

Comparison:

Comparison means to analyse the relation between two numbers. There can be three situations

1. Both the numbers are equal

2. First number is less than the second number

3. First number is more than the second number.

Cross Multiplication:

The method of cross multiplication means multiplying the numerator of one number with the denominator of the other number. For two rational numbers

frac{p}{q}; and;frac{r}{s}

Cross multiplication means multiplying the numerator of first number with the denominator of second number and the denominator of first number with the numerator of second number and we get

ptimes s;and;qtimes r.

Comparison of Rational Numbers by Cross Multiplication:

 When ever we are given two rational numbers and we have to check whether the given rational numbers are equal, less than or greater than, the comparison is done by the method of cross multiplication.

For two rational numbers frac{p}{q};and;frac{r}{s}  there can be three situations

Case 1. They are equal if: ptimes s=qtimes r.

i.e. ,       frac{p}{q}=frac{r}{s}

Leftrightarrow ;;ptimes s=qtimes r

Numerator of first times Denominator of second = Numerator of second times Denominator of first.

Case 2. The first rational number is greater than the second rational number if: ptimes s> qtimes r.

i.e. ,       frac{p}{q}>frac{r}{s}

Leftrightarrow ;;ptimes s>qtimes r

Numerator of first times Denominator of second  >  Numerator of second times Denominator of first.

Case 3. The first rational number is less than the second rational number if: ptimes s<qtimes r.

i.e. ,       frac{p}{q}<frac{r}{s}

Leftrightarrow ;;ptimes s=qtimes r

Numerator of first times Denominator of second < Numerator of second times Denominator of first.

Illustration: Compare the rational numbers frac{-7}{21};and;frac{3}{-9} for equality.

Solution:To compare the rational numbers for equality we will perform cross multiplication

(-7)times (-9)=21times 3  

Rightarrow     63 = 63

Hence, these rational numbers are equal.

Illustration: Compare the rational numbers , frac{5}{7};and;frac{20}{28} for equality.

Solution: To compare the rational numbers for equality we will perform cross multiplication

5times 28=7times 20.  

Rightarrow   140 = 140

Hence, these rational numbers are equal.

Illustration: Do frac{4}{-9};and; frac{-16}{36}represent the same rational number? 

Solution: To compare the rational numbers for equality we will perform cross multiplication

frac{4}{-9};and; frac{-16}{36}

4 x 36 = 144

(-9) x (-16) = 144

These rational numbers are equal.

Illustration: Compare the rational numbers frac{2}{3};and;frac{5}{8}

Solution: To compare the rational numbers for equality we will perform cross multiplication

frac{2}{3};and;frac{5}{8}

: 2 x 8 = 16 and 3 x 5 = 15

Hence, frac{2}{3} >;frac{5}{8}

Sample Questions
(More Questions for each concept available in Login)
Question : 1

Which of the following relation holds for  frac{3}{-11}, frac{4}{-9} ?

Right Option : A
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Explanation
Question : 2

Which of the following relation holds for  frac{5}{-11} and frac{-4}{13}?

Right Option : B
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Explanation
Question : 3

For two rational numbers that are in standard form frac{A}{B}, , and, , frac{C}{D}; which of the following condition (s) is/are true ?

Right Option : D
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Explanation
 
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