One Number Has No Reciprocal at All
Learn to add and subtract rational numbers with unlike denominators, multiply and divide using the reciprocal, combine all four operations in the right order, and solve word problems.
Which number has no reciprocal?
Zero. Flipping would give , which is undefined — so zero is the one rational number with no reciprocal at all.
That single exception matters, because division depends on reciprocals. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: adding and subtracting rational numbers, multiplying and dividing them, combined expressions, and word problems.
That single exception matters, because division depends on reciprocals. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: adding and subtracting rational numbers, multiplying and dividing them, combined expressions, and word problems.
How do you add and subtract rational numbers?
With like denominators, add or subtract the numerators and keep the denominator:
With unlike denominators, first rewrite both with the LCM of the denominators.
Worked example. Add and . The LCM of 4 and 6 is 12:
For subtraction, add the additive inverse — that is, change the sign of the second number and add:
A shopping context makes it concrete: buying kg of tomatoes and kg of onions gives kg in all.
The error to guard against is the one carried over from multiplication. Denominators are never added: is , not . A common denominator is required for addition and subtraction only.
With unlike denominators, first rewrite both with the LCM of the denominators.
Worked example. Add and . The LCM of 4 and 6 is 12:
For subtraction, add the additive inverse — that is, change the sign of the second number and add:
A shopping context makes it concrete: buying kg of tomatoes and kg of onions gives kg in all.
The error to guard against is the one carried over from multiplication. Denominators are never added: is , not . A common denominator is required for addition and subtraction only.
How do you multiply and divide rational numbers?
Multiply numerators together and denominators together, apply the sign rule, and reduce:
Unlike signs gave a negative answer, and the HCF 15 reduced it. No common denominator is needed here.
The reciprocal — also called the multiplicative inverse — of is , and a number times its reciprocal is always 1:
Notice the reciprocal keeps the same sign: the reciprocal of is , not .
To divide, multiply by the reciprocal of the divisor:
Another: .
Only the second number is flipped. Flipping the first, or both, changes the question — and since zero has no reciprocal, division by zero remains impossible here too.
Unlike signs gave a negative answer, and the HCF 15 reduced it. No common denominator is needed here.
The reciprocal — also called the multiplicative inverse — of is , and a number times its reciprocal is always 1:
Notice the reciprocal keeps the same sign: the reciprocal of is , not .
To divide, multiply by the reciprocal of the divisor:
Another: .
Only the second number is flipped. Flipping the first, or both, changes the question — and since zero has no reciprocal, division by zero remains impossible here too.
How do you simplify an expression with all four operations?
Follow BODMAS: brackets first, then division and multiplication left to right, then addition and subtraction left to right.
Worked example.
Multiplication first:
Then the addition:
A second example, with a bracket and a division:
The bracket gives , so
The order is what decides the answer, and reversing it gives something else entirely. Working the first example left to right would give — a completely different result, and a wrong one.
Worked example.
Multiplication first:
Then the addition:
A second example, with a bracket and a division:
The bracket gives , so
The order is what decides the answer, and reversing it gives something else entirely. Working the first example left to right would give — a completely different result, and a wrong one.
How do you solve word problems on rational numbers?
Decide which operation the situation calls for, then work with the fractions exactly as above.
Distance. A cyclist covers km in the morning and km in the evening. The total is
Cost. If kg of dal costs ₹60, then 1 kg costs
Share of a quantity. A rope m long is cut into pieces each m long. The number of pieces is
Remaining part. A tank is full and more is filled. The part filled is , so the empty part is .
The choice of operation is where marks are won or lost. "How many pieces?" means divide the total by one piece, while "what is the total length?" means multiply — and the 9 pieces above being a whole number is a useful sign the right operation was chosen.
Distance. A cyclist covers km in the morning and km in the evening. The total is
Cost. If kg of dal costs ₹60, then 1 kg costs
Share of a quantity. A rope m long is cut into pieces each m long. The number of pieces is
Remaining part. A tank is full and more is filled. The part filled is , so the empty part is .
The choice of operation is where marks are won or lost. "How many pieces?" means divide the total by one piece, while "what is the total length?" means multiply — and the 9 pieces above being a whole number is a useful sign the right operation was chosen.
Exam tip
Exam tip: showing the LCM and the reciprocal step
Every operation in this chapter has one line that carries the method mark, so write it down.
For addition and subtraction, show the LCM conversion — — before combining.
For division, show the reciprocal step on its own line: *dividing by means multiplying by .* And flip only the second number.
Turn every subtraction into an addition of the additive inverse first, so two minus signs never collide.
Always give the final answer in standard form — a positive denominator, reduced by the HCF — and convert an improper result to a mixed number when the question is about a quantity.
And remember the two zero facts: zero has no reciprocal, and division by zero is undefined.
For addition and subtraction, show the LCM conversion — — before combining.
For division, show the reciprocal step on its own line: *dividing by means multiplying by .* And flip only the second number.
Turn every subtraction into an addition of the additive inverse first, so two minus signs never collide.
Always give the final answer in standard form — a positive denominator, reduced by the HCF — and convert an improper result to a mixed number when the question is about a quantity.
And remember the two zero facts: zero has no reciprocal, and division by zero is undefined.
Did you know
Why does dividing by a number smaller than one give a bigger answer?
Because dividing asks how many of the smaller pieces fit inside the larger amount, and small pieces fit many times.
In the rope example, m divided by m gave 9 — far more than itself. Each piece is less than a metre, so a m rope yields plenty of them.
That is also a free check on every division you do. If the divisor is less than 1 the answer must be larger than the number you started with, and if it is greater than 1 the answer must be smaller.
In the rope example, m divided by m gave 9 — far more than itself. Each piece is less than a metre, so a m rope yields plenty of them.
That is also a free check on every division you do. If the divisor is less than 1 the answer must be larger than the number you started with, and if it is greater than 1 the answer must be smaller.
Key takeaways
Operations on rational numbers: quick revision
- Add and subtract only after converting to the LCM of the denominators; denominators are never added.
- Subtraction is addition of the additive inverse, so .
- Multiply numerators and denominators directly with the sign rule: .
- The reciprocal of is and keeps the same sign; a number times its reciprocal is 1, and zero has no reciprocal.
- Divide by multiplying by the reciprocal of the second number only: .
- Follow BODMAS in combined expressions, and give every answer in standard form.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Subtraction is addition of the additive inverse, so .
- Multiply numerators and denominators directly with the sign rule: .
- The reciprocal of is and keeps the same sign; a number times its reciprocal is 1, and zero has no reciprocal.
- Divide by multiplying by the reciprocal of the second number only: .
- Follow BODMAS in combined expressions, and give every answer in standard form.
You will remember all of this far better after answering five questions on it than after reading it twice.