You Can Multiply 103 by 97 in Your Head
Learn to expand and factorise with the distributive property, multiply two binomials without missing a term, use the same idea to do awkward products mentally, and verify the commutative and associative properties.
How do you multiply 103 by 97 without writing anything down?
Rewrite the numbers so the awkwardness cancels. Since and :
The distributive property is what lets you split and recombine like that. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: expanding and factorising, multiplying binomials, mental products, and the commutative and associative properties.
The distributive property is what lets you split and recombine like that. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: expanding and factorising, multiplying binomials, mental products, and the commutative and associative properties.
How do you expand and factorise with the distributive property?
The distributive property says a multiplier spreads across every term inside a bracket:
Expanding means removing the bracket:
Factorising is the same property read backwards — take the common factor out:
To factorise, find the HCF of all the terms and place it outside. For the HCF of 6 and 15 is 3, so 3 comes out.
Checking is easy in both directions: expand your factorised answer and you should get back the original.
The two errors to avoid are mirror images. When expanding, the multiplier must reach every term — writing misses the second. When factorising, take out the highest common factor — writing is not wrong but is unfinished, since could come out.
Expanding means removing the bracket:
Factorising is the same property read backwards — take the common factor out:
To factorise, find the HCF of all the terms and place it outside. For the HCF of 6 and 15 is 3, so 3 comes out.
Checking is easy in both directions: expand your factorised answer and you should get back the original.
The two errors to avoid are mirror images. When expanding, the multiplier must reach every term — writing misses the second. When factorising, take out the highest common factor — writing is not wrong but is unfinished, since could come out.
Formula
How do you multiply two binomials?
Apply the distributive property twice, so every term of the first bracket multiplies every term of the second:
Two terms times two terms gives four products, and then like terms are collected.
Worked examples.
The sign travels with each term, so in the last example .
The same works on numbers, which is the link to the next section:
and indeed .
Counting the products is the check that prevents most errors. Two terms by two terms must give four; two by three gives six. Writing only three products — the commonest slip — means one pairing was skipped, and the missing term is usually one of the two middle ones.
Two terms times two terms gives four products, and then like terms are collected.
Worked examples.
The sign travels with each term, so in the last example .
The same works on numbers, which is the link to the next section:
and indeed .
Counting the products is the check that prevents most errors. Two terms by two terms must give four; two by three gives six. Writing only three products — the commonest slip — means one pairing was skipped, and the missing term is usually one of the two middle ones.
How do you use the same idea for mental arithmetic?
Split an awkward number into a round number plus or minus a small one, then distribute.
Near a hundred.
The two middle terms cancel, leaving .
Or done the other way:
Multiplying by a number just above a round one.
Multiplying by a number just below.
Splitting the other factor.
A shopkeeper pricing 98 items at ₹45 each does exactly this — a hundred lots less two lots.
Justifying each step is part of the question, and the justification is always the same: the distributive property allows the split, because multiplying by is the same as multiplying by 100 and then subtracting two lots. Choosing the split so one part is a round number is what makes the arithmetic doable in the head.
Near a hundred.
The two middle terms cancel, leaving .
Or done the other way:
Multiplying by a number just above a round one.
Multiplying by a number just below.
Splitting the other factor.
A shopkeeper pricing 98 items at ₹45 each does exactly this — a hundred lots less two lots.
Justifying each step is part of the question, and the justification is always the same: the distributive property allows the split, because multiplying by is the same as multiplying by 100 and then subtracting two lots. Choosing the split so one part is a round number is what makes the arithmetic doable in the head.
How do you verify the commutative and associative properties?
Compute both sides and show they agree.
Commutative property — the order of multiplication does not matter:
Verifying with algebraic expressions:
Both sides give , so the property holds.
Another: , and . Equal.
Associative property — the grouping does not matter:
Verifying:
Both give , since .
These properties are what let you regroup a product into whichever order is easiest:
The restriction must be stated, and it is what a question tests. These properties hold for multiplication and addition only — not for subtraction or division. So , and while . Regrouping a division is not allowed.
Commutative property — the order of multiplication does not matter:
Verifying with algebraic expressions:
Both sides give , so the property holds.
Another: , and . Equal.
Associative property — the grouping does not matter:
Verifying:
Both give , since .
These properties are what let you regroup a product into whichever order is easiest:
The restriction must be stated, and it is what a question tests. These properties hold for multiplication and addition only — not for subtraction or division. So , and while . Regrouping a division is not allowed.
Exam tip
Exam tip: counting the products before collecting terms
Expansion questions are marked line by line, so keep the lines.
Before collecting like terms, count the products you should have. Two brackets of two terms give four; if you have three, one pairing was missed.
Write the four products out before simplifying, then collect. The unsimplified line carries a method mark.
When factorising, take out the highest common factor and check by expanding your answer back.
For a mental product, show the split on its own line — ** — and name the distributive property as the justification. The reason is usually the mark.
And when verifying a property, compute both sides separately and state that they are equal. Remember that commutative and associative apply to multiplication and addition only, never to subtraction or division.
Before collecting like terms, count the products you should have. Two brackets of two terms give four; if you have three, one pairing was missed.
Write the four products out before simplifying, then collect. The unsimplified line carries a method mark.
When factorising, take out the highest common factor and check by expanding your answer back.
For a mental product, show the split on its own line — ** — and name the distributive property as the justification. The reason is usually the mark.
And when verifying a property, compute both sides separately and state that they are equal. Remember that commutative and associative apply to multiplication and addition only, never to subtraction or division.
Did you know
Why do the middle terms vanish in 103 times 97?
Because the two numbers sit the same distance either side of 100.
Expanding gives four products: , then , then , then . The two middle terms are equal in size and opposite in sign, so they cancel exactly — leaving .
That cancelling only happens when the offsets match, which is why is easy while is not. It is also the beginning of the identity , which turns this convenience into a general rule.
Expanding gives four products: , then , then , then . The two middle terms are equal in size and opposite in sign, so they cancel exactly — leaving .
That cancelling only happens when the offsets match, which is why is easy while is not. It is also the beginning of the identity , which turns this convenience into a general rule.
Key takeaways
The distributive property: quick revision
- — the multiplier reaches every term, so .
- Factorising is the reverse: take out the highest common factor, so .
- gives four products — after collecting.
- Count the products before simplifying; three products means one pairing was skipped.
- For mental arithmetic, split to a round number: , and because the middle terms cancel.
- Commutative and associative properties hold for multiplication and addition only — verify by computing both sides, and never regroup a subtraction or division.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Factorising is the reverse: take out the highest common factor, so .
- gives four products — after collecting.
- Count the products before simplifying; three products means one pairing was skipped.
- For mental arithmetic, split to a round number: , and because the middle terms cancel.
- Commutative and associative properties hold for multiplication and addition only — verify by computing both sides, and never regroup a subtraction or division.
You will remember all of this far better after answering five questions on it than after reading it twice.