Subtraction of Rational Numbers | Properties, FAQs For SSC CGL CHSL CPO

Anand Classes provides complete study material on the subtraction of rational numbers for SSC Exams including CGL, CHSL, CPO, and GD Constable. In this lesson, we explain subtraction of rational numbers with solved examples, step-by-step solutions, properties (closure, commutative, associative), FAQs, and important tricks. Students preparing for SSC exams will strengthen their arithmetic and quantitative aptitude skills with these notes. Click the print button to download study material and notes.

Subtraction of Rational Numbers – SSC CGL CHSL Exam Notes

Subtracting one rational number from another rational number is the same as adding the additive inverse (negative) of the rational number that is being subtracted to the other rational number.

If $\frac{p}{q}$ and $\frac{r}{s}$ are two rational numbers, then

$$
\frac{p}{q} – \frac{r}{s} = \frac{p}{q} + \left(-\frac{r}{s}\right)
$$

That is, subtraction means adding the additive inverse.

Ex.1 : Subtract: (i) $\frac{-5}{12}$ from $\frac{-3}{8}$ (ii) $\frac{-2}{47}$ from $\frac{8}{15}$

Solution

(i)
$$
\frac{-5}{12} – \frac{-3}{8} = \frac{-5}{12} + \left(-\frac{-3}{8}\right) = \frac{-5}{12} + \frac{3}{8}
$$

LCM of $12$ and $8$ is $24$.

$$
\frac{-5}{12} = \frac{-10}{24}, \quad \frac{3}{8} = \frac{9}{24}
$$

So,
$$
\frac{-10}{24} + \frac{9}{24} = \frac{-1}{24}
$$

(ii)
$$
\frac{8}{15} – \frac{-2}{47} = \frac{8}{15} + \frac{2}{47}
$$

LCM of $15$ and $47$ is $705$.

$$
\frac{8}{15} = \frac{376}{705}, \quad \frac{2}{47} = \frac{30}{705}
$$

So,
$$
\frac{376}{705} + \frac{30}{705} = \frac{406}{705}
$$

Ex.2 : What number should be added to $\frac{-7}{12}$ to get $\frac{5}{6}$?

Solution

Let the required number be $x$.

$$
\frac{-7}{12} + x = \frac{5}{6}
$$

So,
$$
x = \frac{5}{6} – \frac{-7}{12}
$$

Taking LCM $= 12$:
$$
\frac{5}{6} = \frac{10}{12}
$$

Thus,
$$
x = \frac{10}{12} + \frac{7}{12} = \frac{17}{12}
$$

Hence, the required number is $\frac{17}{12}$.

Ex.3 : What number should be subtracted from $-4$ to get $\frac{-11}{5}$?

Solution

Let the required number be $x$.

$$
-4 – x = \frac{-11}{5}
$$

So,
$$
-4 = \frac{-11}{5} + x
$$

Thus,
$$
x = -4 – \frac{-11}{5} = -4 + \frac{11}{5}
$$

Taking LCM $= 5$:
$$
-4 = \frac{-20}{5}
$$

So,
$$
x = \frac{-20}{5} + \frac{11}{5} = \frac{-9}{5}
$$

Hence, the required number to be subtracted is $\frac{-9}{5}$.

What is Closure Property of Rational Numbers ?

Let us subtract any two rational numbers and check whether the difference is a rational number.

Thus, we find that the difference of two rational numbers is also a rational number.

In other words:
If $\frac{p}{q}$ and $\frac{r}{s}$ $(q, s \neq 0)$ are two rational numbers, then

$$
\frac{p}{q} – \frac{r}{s}
$$

is also a rational number.

This is called the closure property of subtraction of rational numbers.

What are Properties of Subtraction of Rational Numbers ?

Why Subtraction is Not Commutative ?

Consider the rational numbers $\frac{-2}{3}$ and $\frac{4}{5}$.

$$
\frac{-2}{3} – \frac{4}{5} = \frac{-10}{15} – \frac{12}{15} = \frac{-22}{15}
$$

But,
$$
\frac{4}{5} – \frac{-2}{3} = \frac{12}{15} – \frac{-10}{15} = \frac{22}{15}
$$

Since,
$$
\frac{-2}{3} – \frac{4}{5} \neq \frac{4}{5} – \frac{-2}{3}
$$

Thus, subtraction is not commutative for rational numbers.

Why Subtraction is Not Associative ?

Consider the rational numbers $\frac{7}{6}, \frac{2}{3}$ and $\frac{-5}{4}$.

First,
$$
\frac{7}{6} – \left( \frac{2}{3} – \frac{-5}{4} \right)
= \frac{7}{6} – \left( \frac{8}{12} – \frac{-15}{12} \right)
= \frac{7}{6} – \frac{23}{12}
$$

Taking LCM $= 12$:
$$
\frac{14}{12} – \frac{23}{12} = \frac{-9}{12} = \frac{-3}{4}
$$

Now,
$$
\left( \frac{7}{6} – \frac{2}{3} \right) – \frac{-5}{4}
= \left( \frac{14}{12} – \frac{8}{12} \right) – \frac{-15}{12}
= \frac{6}{12} – \frac{-15}{12}
$$

So,
$$
\frac{6}{12} + \frac{15}{12} = \frac{21}{12} = \frac{7}{4}
$$

Clearly,
$$
\frac{7}{6} – \left( \frac{2}{3} – \frac{-5}{4} \right) \neq \left( \frac{7}{6} – \frac{2}{3} \right) – \frac{-5}{4}
$$

Thus, subtraction is not associative for rational numbers.

Another Example

Consider the rational numbers $\frac{-3}{2}, \frac{-5}{4}$ and $\frac{1}{2}$.

First,
$$
\frac{-3}{2} – \left( \frac{-5}{4} – \frac{1}{2} \right)
= \frac{-3}{2} – \left( \frac{-5}{4} – \frac{2}{4} \right)
= \frac{-3}{2} – \frac{-3}{4}
$$

Taking LCM $= 4$:
$$
\frac{-6}{4} – \frac{-3}{4} = \frac{-6}{4} + \frac{3}{4} = \frac{-3}{4}
$$

Now,
$$
\left( \frac{-3}{2} – \frac{-5}{4} \right) – \frac{1}{2}
= \left( \frac{-6}{4} – \frac{-5}{4} \right) – \frac{1}{2}
= \left( \frac{-1}{4} \right) – \frac{1}{2}
$$

Taking LCM $= 4$:
$$
\frac{-1}{4} – \frac{2}{4} = \frac{-3}{4}
$$

Since both give the same result here, but in general subtraction does not satisfy the associative law.

Thus, subtraction is not associative for rational numbers.

FAQs on Subtraction of Rational Numbers for SSC CGL CHSL Exams

Q1. Is subtraction of two rational numbers always a rational number?

Answer: Yes ✅. The difference of any two rational numbers is always a rational number. This is called the closure property.

Q2. Is subtraction of rational numbers commutative?

Answer: No ❌. For rational numbers $a$ and $b$, in general
$$
a – b \neq b – a
$$

Example:
$\frac{-2}{3} – \frac{4}{5} = \frac{-22}{15}$
but
$\frac{4}{5} – \frac{-2}{3} = \frac{22}{15}$

Q3. Is subtraction of rational numbers associative?

Answer: No ❌. For rational numbers $a, b, c$, in general
$$
(a – b) – c \neq a – (b – c)
$$

Example:
$(\frac{7}{6} – \frac{2}{3}) – \frac{-5}{4} = \frac{7}{4}$
but
$\frac{7}{6} – (\frac{2}{3} – \frac{-5}{4}) = \frac{-3}{4}$

Q4. What is the identity element for subtraction of rational numbers?

Answer: There is no identity element for subtraction.
(Though $a – 0 = a$ works one way, $0 – a \neq a$.)

Q5. What is the difference between subtraction and addition properties for rational numbers?

Answer:

• Addition is closed, commutative, associative, and has an identity element ($0$).
• Subtraction is closed, but not commutative, not associative, and has no identity element.

Q6. Is subtraction the same as adding the negative?

Answer: Yes ✅.
For any rational numbers $a$ and $b$,
$$
a – b = a + (-b)
$$

Example:
$5 – 7 = 5 + (-7) = -2$

Q7. Can subtraction of two positive rational numbers be negative?

Answer: Yes ✅.
If the number being subtracted is larger, the result is negative.

Example:
$\frac{3}{5} – \frac{7}{5} = \frac{-4}{5}$

Q8. Is $a – b = b – a$ ever true?

Answer: Only when $a = b$.

Example:
If $a = b = \frac{2}{7}$, then
$$
a – b = \frac{2}{7} – \frac{2}{7} = 0
$$
and
$$
b – a = \frac{2}{7} – \frac{2}{7} = 0
$$

So, subtraction is equal only when the two rational numbers are the same.

To learn NDA Polity Notes Click : Citizenship Amendment Act, 2015

Q9. Can subtraction give zero as the answer?

Answer: Yes ✅.
If we subtract a number from itself,
$$
a – a = 0
$$

Example:
$\frac{9}{11} – \frac{9}{11} = 0$

To learn NDA Polity Notes (Citizenship Amendment Act (CAA)), click here

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