Compare Two Sums Without Working Either of Them Out
Learn to build an expression from a situation, rearrange terms safely to make arithmetic easy, compare expressions by reasoning, and open brackets without losing a sign.
What is an arithmetic expression?
An arithmetic expression is a combination of numbers joined by the operations , , and , with brackets used to show which operation is carried out first. Writing a situation as an expression turns a word problem into something you can calculate — and often simplify before you calculate.
This page covers everything in the CBSE Class 7 Mathematics chapter on arithmetic expressions: building them from situations, identifying terms, rearranging them safely, comparing them by reasoning, and removing brackets.
This page covers everything in the CBSE Class 7 Mathematics chapter on arithmetic expressions: building them from situations, identifying terms, rearranging them safely, comparing them by reasoning, and removing brackets.
How do you write an expression for a situation?
Translate the situation phrase by phrase, then use brackets to protect whatever must happen first.
You buy 3 notebooks at Rs 40 each and 2 pens at Rs 15 each. The expression is
Now a situation where brackets matter. You buy 4 boxes, each containing 1 pen at Rs 15 and 1 pencil at Rs 5. The cost of one box is , so
Without the brackets, gives 65 — a different and wrong answer, because it buys four pens and a single pencil.
So brackets are not decoration. They record which quantities belong together, and choosing where they go is part of understanding the problem rather than a step that follows it.
You buy 3 notebooks at Rs 40 each and 2 pens at Rs 15 each. The expression is
Now a situation where brackets matter. You buy 4 boxes, each containing 1 pen at Rs 15 and 1 pencil at Rs 5. The cost of one box is , so
Without the brackets, gives 65 — a different and wrong answer, because it buys four pens and a single pencil.
So brackets are not decoration. They record which quantities belong together, and choosing where they go is part of understanding the problem rather than a step that follows it.
What are the terms of an expression, and how can you rearrange them?
The terms of an expression are the parts joined by and signs, and each sign belongs to the term that follows it.
In , the terms are , and .
Two properties let you rearrange them safely:
Swapping (commutativity): the order of terms may be changed. becomes .
Grouping (associativity): terms may be bracketed together in any way. .
That second form is much easier: , with no borrowing needed.
Another example: reorders to , which needs no written working at all.
The limit to remember: when you move a term you must carry its sign with it. Rewriting as changes the value entirely, because the signs were left behind.
In , the terms are , and .
Two properties let you rearrange them safely:
Swapping (commutativity): the order of terms may be changed. becomes .
Grouping (associativity): terms may be bracketed together in any way. .
That second form is much easier: , with no borrowing needed.
Another example: reorders to , which needs no written working at all.
The limit to remember: when you move a term you must carry its sign with it. Rewriting as changes the value entirely, because the signs were left behind.
How do you compare two expressions without evaluating them?
By comparing their terms and reasoning about the difference — which is faster and shows more understanding than grinding out both answers.
Compare and . Both contain 63, and , so
Compare and . Both start from 145, but the second subtracts more, leaving less:
Compare and . The same 23 taken more times is larger, so .
A subtler one: compare and . The first multiplies both parts by 4, the second only the 50 — so the first is larger, without computing either.
The reasoning rule is: find what the two expressions share, then judge only how they differ.
Compare and . Both contain 63, and , so
Compare and . Both start from 145, but the second subtracts more, leaving less:
Compare and . The same 23 taken more times is larger, so .
A subtler one: compare and . The first multiplies both parts by 4, the second only the 50 — so the first is larger, without computing either.
The reasoning rule is: find what the two expressions share, then judge only how they differ.
Formula
How do you use the distributive property and remove brackets?
The distributive property spreads multiplication across addition or subtraction:
It works in both directions, and each direction is useful.
Expanding: — a quick way to do mentally.
Factorising: .
For removing brackets, the sign in front decides what happens:
- a plus before a bracket leaves every sign unchanged:
- a minus before a bracket reverses every sign inside:
Check that second one: , and as well.
The most common error in the chapter is reversing only the first sign: writing , which gives 18 and is wrong.
It works in both directions, and each direction is useful.
Expanding: — a quick way to do mentally.
Factorising: .
For removing brackets, the sign in front decides what happens:
- a plus before a bracket leaves every sign unchanged:
- a minus before a bracket reverses every sign inside:
Check that second one: , and as well.
The most common error in the chapter is reversing only the first sign: writing , which gives 18 and is wrong.
Exam tip
Exam tip: carrying the sign when you move a term
Rearranging is meant to make a calculation easier, but it goes wrong the moment a sign is left behind.
Before swapping anything, rewrite the expression with every sign attached to its term: becomes the terms , , . Now move them freely — they carry their signs with them.
The same habit fixes bracket removal. Write out the bracket fully expanded with the signs already flipped, and only then collect the numbers. Doing the sign change and the arithmetic in one line is where marks vanish.
And when a question says "compare", it usually wants your reasoning, not two computed answers — so state what the expressions share and how they differ.
Before swapping anything, rewrite the expression with every sign attached to its term: becomes the terms , , . Now move them freely — they carry their signs with them.
The same habit fixes bracket removal. Write out the bracket fully expanded with the signs already flipped, and only then collect the numbers. Doing the sign change and the arithmetic in one line is where marks vanish.
And when a question says "compare", it usually wants your reasoning, not two computed answers — so state what the expressions share and how they differ.
Did you know
Why does putting a minus before a bracket flip every sign?
Because subtracting a whole group means removing everything in it. If you take away , you are taking away 12 and then giving back 5, since the 5 was never part of what the group was worth.
That is why becomes . The flipped sign on the 5 is not a rule to memorise — it is what "remove the whole group" actually means.
That is why becomes . The flipped sign on the 5 is not a rule to memorise — it is what "remove the whole group" actually means.
Key takeaways
Arithmetic expressions: quick revision
- An arithmetic expression joins numbers with , , and , with brackets recording which operation happens first — and brackets change the answer, as against shows.
- Terms are the parts joined by and , and each sign belongs to the term after it.
- Swapping and grouping let you reorder terms to make arithmetic easier, provided every sign travels with its term.
- Compare expressions by finding what they share and reasoning about how they differ, rather than evaluating both.
- The distributive property gives ; a plus before a bracket keeps the signs, a minus reverses every one of them.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Terms are the parts joined by and , and each sign belongs to the term after it.
- Swapping and grouping let you reorder terms to make arithmetic easier, provided every sign travels with its term.
- Compare expressions by finding what they share and reasoning about how they differ, rather than evaluating both.
- The distributive property gives ; a plus before a bracket keeps the signs, a minus reverses every one of them.
You will remember all of this far better after answering five questions on it than after reading it twice.