In this article of ours, you will learn how to find Equivalent Rational Number s by multiplication and division. Get to see solved examples in the coming modules.
Equivalent Rational Numbers by Multiplication
Let suppose a/b is a rational number and m is a non-zero integer then (a*m)/(b*m) is a rational number equivalent to a/b.
For instance, 16/20, 40/50, -56/-70, -96/-120 are equivalent fractions and are equal to the rational number 4/5.
On multiplying the numerator and denominator of a fraction with the same integer the fraction value doesn’t change.
Example: Fractions 4/8 and 16/32 are equivalent because the numerator and denominator can be obtained by multiplying with each of them with 4.
Also, -4/5 = -4*(-1)/5*(-1) = -4*(-2)/5*(-2) = -4*(-3)/5*(-3) and so on ……
If the denominator of a rational number is a negative integer then by using the above-mentioned property we can convert it to positive by multiplying the numerator and denominator by -1.
Example: 7/-5 = 7*(-1)/-5*(-1) = -7/5
Equivalent Rational Numbers by Division
If a/b is a rational number and m is the common divisor of a, b then (a÷m)/ (b÷m) is a rational number equivalent to a/b.
Rational Numbers -24/-30, -28/-35, 40/50, 60/75 are equivalent to the rational numbers 4/5.
24/32 = (24÷8)/(32÷8) = 3/4
Solved Examples
1. Find the Two Rational Numbers Equivalent to 4/7?
Solution:
4/7 = (4*4)/(7*4) = 16/28
4/7 = (4*7)/(7*7) = 28/49
Thus, the two rational numbers equivalent to 4/7 are 16/28 and 28/49.
2. Determine the smallest equivalent rational number of 100/125?
Solution:
100/125 = (100÷5)/(125÷5) = 20/25 = (20÷5)/(25÷5) = 4/5
Thus, the Equivalent Rational Number of 100/125 is 4/5.
3. Write down the following rational numbers with a positive denominator 4/-9, 11/-22, -17/-3?
Solution:
4/-9 = 4*(-1)/-9*(-1) = -4/9
11/-22 = 11*(-1)/-22*(-1) = -11/22
-17/-3 = -17*(-1)/-3*(-1) = 17/3
Therefore Rational Numbers 4/-9, 11/-22, -17/-3 changed with a positive denominator are -4/9, -11/22, 17/3.
4. Express -4/7 as a Rational Number with the numerator
(i) -16 (ii) 24
Solution:
(i) In order to make -4 as a rational number having the numerator -16 we first need to find a number when multiplied by results in -16.
Clearly, such number is (-16 )÷ (-4) = 4
Multiplying both the numerator and denominator with 4 we get
-4/7 = (-4*4)/(7*4) = -16/28
(ii) In order to make -4 as a rational number having the numerator 24 we first need to find a number when multiplied by results in 24.
Clearly, such number is (24 )÷ (-4) = -6
Multiplying both the numerator and denominator with -6 we get
-4/7 = (-4*-6)/(7*-6) = 24/-42
All the examples listed above are for Equivalent Rational Numbers.