Page 20 - Start Up Mathematics_7

Property of zero

Zero divided by any non-zero integer a is equal to zero, i.e., 0 ÷ a = 0 provided a ≠ 0.
For example, 0 ÷ 5 = 0 and 0 ÷ (–7) = 0.

a ÷ 0 is meaningless, i.e., division by 0 is not defined.
Division by 1 keeps the integer unchanged

If a is any integer, then a ÷ 1 = a, i.e., any integer divided by 1 remains as it is.
For integer 5, we have 5 ÷ 1 = 5 and for integer –3, we have –3 ÷ 1 = –3.
Division of integer by itself
a
Any non-zero integer a, divided by itself gives 1 as the answer, i.e., a ÷ a = = 1.
a
For integer –2, we have –2 ÷ (–2) = 1 and for integer 3, we have 3 ÷ 3 = 1.

Example 22: Evaluate each of the following:
(a) (–36) ÷ (–9) (b) (–39) ÷ (39) (c) 13 ÷ {(–3) + 2}
(d) 0 ÷ (–8) (e) {(–36) ÷ 12} ÷ 3 (f) {(–7) + 5} ÷ {(–3) + 1}

Solution: (a) (–36) ÷ (–9) = 4
(b) (–39) ÷ (39) = –1
(c) 13 ÷ {(–3) + 2} = 13 ÷ (–1) = –13

(d) 0 ÷ (–8) = 0
(e) {(–36) ÷ 12} ÷ 3 = (–3) ÷ 3 = –1
(f) {(–7) + 5} ÷ {(–3) + 1} = (–2) ÷ (–2) = 1

Example 23: Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for a = 12, b = 1 and c = –2.
Solution: To verify a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
LHS = a ÷ (b + c) = 12 ÷ {1 + (–2)} = 12 ÷ (–1) = –12
12 12
RHS = (a ÷ b) + (a ÷ c) = 1 + –2 = 12 – 6 = 6

 LHS ≠ RHS, hence verified.
Example 24: Fill in the blanks:
(a) 469 ÷ _______ = 469 (b) (–205) ÷ _______ = 1 (c) –89 ÷ _______ = 89

(d) _______ ÷ 1 = –83 (e) _______ ÷ 48 = –1 (f) 30 ÷ _______ = –2
Solution: (a) 469 ÷ 1 = 469 (b) (–205) ÷ (–205) = 1 (c) –89 ÷ (–1) = 89
(d) (–83) ÷ 1 = –83 (e) (–48) ÷ 48 = –1 (f) 30 ÷ (–15) = –2

Example 25: Write five pairs of integers (a, b) such that b ÷ a = –2. One such pair is (3, –6)
because –6 ÷ 3 = –2.
Solution: The required pairs of integers are (1, –2), (–2, 4), (4, –8), (–5, 10) and (7, –14).

There are infinite pairs of integers (a, b) satisfying b = –2a.

Example 26: The temperature at noon was 10°C. If it decreases at the rate of 2°C per hour until
midnight, at what time would the temperature be 6°C below zero? What would be
the temperature at midnight?

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