In this article of ours, you will learn how to find Equivalent Rational Numbers 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.