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.