Maharashtra Board Class 7 Maths part 1 Chapter 2 Multiplication and Division of Integers PDF Download

Multiplication and Division of Integers

Let's Recall

In the previous class, we have learnt to add and subtract integers. Using those methods, fill in the blanks below.

(1) 5 + 7 = ____

(2) 10 + (−5) = ____

(3) −4 + 3 = ____

(4) (−7) + (−2) = ____

(5) (+ 8) − (+ 3) = ____

(6) (+ 8) − (−3) = ____

Write a number in each bracket to obtain the answer '3' in each operation.

(−6) + ( ) = 3

4 − ( ) = 3

7 + ( ) = 3

(−5) − ( ) = 3

−8 + ( ) = 3

9 − ( ) = 3

Teacher's Note

In our daily life, we use negative numbers for debt. If your friend lends you 50 rupees, you have −50 rupees. This is like Mayuri's bicycle repair story.

Exam Trick

Remember: Negative number means you owe money. Positive number means you have money. When you add and subtract, think about money you have and money you owe.

Points to Remember

Integers include positive numbers, negative numbers, and zero.
We use minus sign (−) to show negative numbers or debt.
When we add the same negative number many times, we can use multiplication.

Multiplication of Integers

Mayuri's bicycle got punctured on the way back from school and she did not have enough money to get it repaired. Sushant, Snehal and Kalpana lent her five rupees each. Thus she borrowed 15 rupees altogether and got the bicycle repaired. We show borrowed money, or a debt, using the '−' (minus) sign. That is, Mayuri had a debt of 15 rupees or Mayuri had −15 rupees.

We see here that (−5) + (−5) + (−5) = −15

Hence note that (−5) × 3 = 3 × (−5) = −15

Of course, Mayuri paid back her debt the next day.

We have learnt the multiplication and division of whole numbers. We have even made tables to carry out the multiplication. Now let us learn to multiply integers i.e. multiplication of numbers in the set that includes negative numbers, positive numbers and zero.

(−3) + (−3) + (−3) + (−3) This addition is the addition of (−3) taken 4 times. It equals −12. It can be written as (−3) × 4 = −12.

Similarly, (−5) × 6 = −30, (−7) × 2 = −14, 8 × (−7) = −56

Teacher's Note

Multiplication of negative numbers follows a pattern, just like the multiplication table you learned in primary school. If you learn the pattern, it becomes very easy.

Exam Trick

Remember: Negative × Positive = Negative. Like: debt × time = more debt. If you owe 5 rupees and this happens 3 times, you owe 15 rupees total.

Points to Remember

Negative number added many times equals negative result.
Multiplication is a faster way to add the same number many times.
(−5) × 3 means add (−5) three times: (−5) + (−5) + (−5) = −15

Let's Learn: Making Tables

Now let us make the table of (−4).

(−4) × 0 = 0

(−4) × 1 = −4

(−4) × 2 = −8

(−4) × 3 = −12

Observe the pattern here. As the multiplier of (−4) increases by 1, the product is reduced by 4.

Keeping the same pattern, if we extend the table upwards, decreasing the multiplier, this is what we will get.

(−4) × (−2) = 8

(−4) × (−1) = 4

(−4) × 0 = 0

As the multiplier of (−4) decreases by one unit, the product increases by 4.

The table for (−5) is given below. Complete the tables of (−6) and (−7).

Now I Know!

The product of two positive (+ve) integers is a positive (+ve) integer.

The product of one positive (+ve) and one negative (−ve) integer is a negative integer.

The product of two negative (−ve) integers is a positive (+ve) integer.

Even number × (−ve) number = (−ve) number

Even number × (−ve) number = (−ve) number

(−ve) number × (+ve) number = (−ve) number

(−ve) number × (−ve) number = (+ve) number

Teacher's Note

The pattern of multiplication with negative numbers is like a rule that never changes. Two negatives together make a positive, just like in English grammar: "not bad" means "good".

Exam Trick

Count the negative signs: If you have an even number of negatives (2, 4, 6...), the answer is positive. If you have an odd number of negatives (1, 3, 5...), the answer is negative.

Points to Remember

(−) × (+) = (−)
(+) × (−) = (−)
(−) × (−) = (+)
(+) × (+) = (+)

Division of Integers

We have learnt how to divide one positive integer by another. We also know that the quotient of such a division may be an integer or a fraction.

Example: \(6 \div 2 = \frac{6}{2} = 3\), \(5 \div 3 = \frac{5}{3} = 1 + \frac{2}{3}\)

On the number line, we can show negative integers on the left of the zero. We can show parts of integers also in the same way.

Here, the numbers \(-\frac{5}{2}\), \(-\frac{3}{2}\), \(\frac{3}{2}\), \(\frac{5}{2}\) are shown on the number line.

Note that \(\left(-\frac{1}{2}, \frac{1}{2}\right)\), \(\left(-\frac{3}{2}, \frac{3}{2}\right)\), \(\left(-\frac{5}{2}, \frac{5}{2}\right)\) are mutually opposite numbers.

That is, \(\frac{1}{2} + \left(-\frac{1}{2}\right) = 0\), \(\frac{3}{2} + \left(-\frac{3}{2}\right) = 0\), \(-\frac{5}{2} + \frac{5}{2} = 0\)

Pairs of opposite numbers are also called pairs of additive inverse numbers. We have seen that (−1) × (−1) = 1. If the two sides of this equation are divided by (−1) we get the equation \((−1) = \frac{1}{(−1)}\). Therefore, the quotient of the division \(\frac{1}{(−1)}\) is (−1).

Hence, we see that \(6 \times (−1) = 6 \times \frac{1}{(−1)} = \frac{6}{(−1)}\)

To Divide Any Positive Integer By a Negative Integer

\(\frac{7}{-2} = \frac{7 \times 1}{(-1) \times 2} = 7 \times \frac{1}{(-1)} \times \frac{1}{2} = \frac{7}{1} \times (-1) \times \frac{1}{2} = \frac{(7) \times (-1)}{2} = \frac{-7}{2}\)

To Divide Any Negative Integer By a Negative Integer

\(\frac{-13}{-2} = \frac{(-1) \times 13}{(-1) \times 2} = \frac{(-1)}{(-1)} \times 13 \times \frac{1}{2} = (-1) \times \frac{(-1)}{1} \times \frac{13}{2} = \frac{1 \times 13}{2} = \frac{13}{2}\)

Similarly, verify that \(\frac{-25}{-4} = \frac{25}{4}\), \(\frac{-18}{-2} = \frac{18}{2} = 9\) etc.

This explains the division of negative integers.

When one integer is divided by another non-zero integer, it is customary to write the denominator of the quotient as a positive integer.

Hence we write \(\frac{7}{-2} = \frac{-7}{2}\), \(\frac{-11}{-3} = \frac{11}{3}\)

Teacher's Note

Division of negative numbers follows the same rules as multiplication of negative numbers. Two negatives make a positive. If you have a debt (negative) and you share it equally (also like a negative operation), it becomes positive in a way.

Exam Trick

Division rules are just like multiplication rules: (+) ÷ (−) = (−), (−) ÷ (+) = (−), (−) ÷ (−) = (+), (+) ÷ (+) = (+). Also remember: Always write the bottom number as positive.

Points to Remember

Division of integers follows the same sign rules as multiplication.
When dividing, the bottom number (denominator) should always be positive.
Two negative numbers divided give a positive result.
You cannot divide any number by zero.
A positive and negative number divided give a negative result.

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