How to Find the Whole When You Only Know a Part
Learn to add and subtract mixed numbers, multiply and divide with reciprocals, read the word of correctly, simplify complex fractions, and work backwards from a part to the whole quantity.
If three-fifths of a number is 45, what is the number?
75 — and you get there by multiplying by the reciprocal, not by dividing by 3 and multiplying by 5 in your head:
Working backwards from a part to the whole is the most useful skill in this chapter. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: adding and subtracting fractions and mixed numbers, multiplying and dividing, complex fractions, and word problems.
Working backwards from a part to the whole is the most useful skill in this chapter. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: adding and subtracting fractions and mixed numbers, multiplying and dividing, complex fractions, and word problems.
How do you add and subtract mixed numbers?
Convert every mixed number to an improper fraction first, bring them to the LCM of the denominators, then combine.
Worked example, adding:
Worked example, subtracting from a whole number:
And with unlike denominators:
Measuring cloth makes it real: from a m piece, cutting m leaves m.
Converting first is what avoids the classic mistake. Subtracting the whole numbers and the fractions separately in the last example would give , which is not the answer — because is smaller than , so the fractional part has to borrow from the whole.
Worked example, adding:
Worked example, subtracting from a whole number:
And with unlike denominators:
Measuring cloth makes it real: from a m piece, cutting m leaves m.
Converting first is what avoids the classic mistake. Subtracting the whole numbers and the fractions separately in the last example would give , which is not the answer — because is smaller than , so the fractional part has to borrow from the whole.
How do you multiply and divide fractions, and what does of mean?
Multiply numerators together and denominators together, cancelling common factors first:
The word of means exactly the same as ×. So of is
and of 200 g is g.
To divide, multiply by the reciprocal of the second fraction:
With mixed numbers, convert before anything else:
The placing of of inside a longer expression is what catches students out. In BODMAS, of ranks with brackets, so it is worked before ordinary division.
That matters most when a division comes first. Simplify of :
Working left to right instead would give — a wildly different answer, and the wrong one.
The word of means exactly the same as ×. So of is
and of 200 g is g.
To divide, multiply by the reciprocal of the second fraction:
With mixed numbers, convert before anything else:
The placing of of inside a longer expression is what catches students out. In BODMAS, of ranks with brackets, so it is worked before ordinary division.
That matters most when a division comes first. Simplify of :
Working left to right instead would give — a wildly different answer, and the wrong one.
How do you simplify a complex fraction?
Read the main fraction bar as a division sign, then multiply by the reciprocal.
Worked example.
When one part is a whole number, write it over 1:
For a multi-step expression, follow BODMAS with brackets, of, division, multiplication, addition and subtraction in that order:
The bracket gives . Then
and finally .
The rule that makes complex fractions easy is identifying the main bar — the longest one — since that is the division being asked for. Treating a smaller inner bar as the main one reverses the whole calculation.
Worked example.
When one part is a whole number, write it over 1:
For a multi-step expression, follow BODMAS with brackets, of, division, multiplication, addition and subtraction in that order:
The bracket gives . Then
and finally .
The rule that makes complex fractions easy is identifying the main bar — the longest one — since that is the division being asked for. Treating a smaller inner bar as the main one reverses the whole calculation.
How do you solve fraction word problems?
Three shapes of question appear, and each has its own move.
A fractional part of a quantity — multiply. A student reads of a 350-page book:
The remaining share — subtract from 1. A man spends of his salary on rent and on food. The part spent is
so the part left is . If his salary is ₹24000, that is .
The whole when a part is known — divide. If of a tank holds 60 litres, the full tank holds
The way to tell the first and third apart is to ask what the question gives you. If it gives the whole and asks for a part, multiply; if it gives a part and asks for the whole, divide. A useful check follows automatically: the whole must come out larger than the part you were given, so 160 litres is plausible while 22.5 litres would not be.
A fractional part of a quantity — multiply. A student reads of a 350-page book:
The remaining share — subtract from 1. A man spends of his salary on rent and on food. The part spent is
so the part left is . If his salary is ₹24000, that is .
The whole when a part is known — divide. If of a tank holds 60 litres, the full tank holds
The way to tell the first and third apart is to ask what the question gives you. If it gives the whole and asks for a part, multiply; if it gives a part and asks for the whole, divide. A useful check follows automatically: the whole must come out larger than the part you were given, so 160 litres is plausible while 22.5 litres would not be.
Exam tip
Exam tip: converting mixed numbers before you start
Almost every lost mark in this chapter comes from working with mixed numbers directly.
Convert every mixed number to an improper fraction on its own first line, before adding, subtracting, multiplying or dividing. Handling whole parts and fractions separately fails as soon as borrowing is needed.
For division, write the reciprocal step out: *dividing by means multiplying by * — and flip only the second fraction.
Remember that of means multiply and ranks with brackets in BODMAS, so it is done before ordinary division.
In word problems, decide multiply or divide by asking whether the whole or a part was given, then check the answer's size against that.
And finish properly: reduce to lowest terms, convert an improper result to a mixed number, and attach the unit — ₹10000, 160 litres, 140 pages.
Convert every mixed number to an improper fraction on its own first line, before adding, subtracting, multiplying or dividing. Handling whole parts and fractions separately fails as soon as borrowing is needed.
For division, write the reciprocal step out: *dividing by means multiplying by * — and flip only the second fraction.
Remember that of means multiply and ranks with brackets in BODMAS, so it is done before ordinary division.
In word problems, decide multiply or divide by asking whether the whole or a part was given, then check the answer's size against that.
And finish properly: reduce to lowest terms, convert an improper result to a mixed number, and attach the unit — ₹10000, 160 litres, 140 pages.
Did you know
Why does working backwards from a part need division?
Because multiplication is what made the part in the first place, and division undoes it.
Taking of a tank means multiplying the whole by . So if you know the answer was 60 litres and want the tank, you must reverse that step — divide by , which is the same as multiplying by .
That is why the answer grows: you are multiplying by a number greater than 1. And it is a reliable check, since the whole tank must always hold more than three-eighths of itself.
Taking of a tank means multiplying the whole by . So if you know the answer was 60 litres and want the tank, you must reverse that step — divide by , which is the same as multiplying by .
That is why the answer grows: you are multiplying by a number greater than 1. And it is a reliable check, since the whole tank must always hold more than three-eighths of itself.
Key takeaways
Operations on fractions: quick revision
- Convert mixed numbers to improper fractions first, then use the LCM of the denominators to add or subtract: .
- Multiply numerators and denominators directly, cancelling first; of means the same as × and ranks with brackets in BODMAS.
- Divide by multiplying by the reciprocal of the second fraction only: .
- Simplify a complex fraction by reading the main bar as a division: .
- A fractional part of a quantity means multiply; the remaining share means subtract the used part from 1.
- To find the whole from a part, divide — so if of a tank is 60 litres, the tank holds 160 litres, which must be larger than the part given.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Multiply numerators and denominators directly, cancelling first; of means the same as × and ranks with brackets in BODMAS.
- Divide by multiplying by the reciprocal of the second fraction only: .
- Simplify a complex fraction by reading the main bar as a division: .
- A fractional part of a quantity means multiply; the remaining share means subtract the used part from 1.
- To find the whole from a part, divide — so if of a tank is 60 litres, the tank holds 160 litres, which must be larger than the part given.
You will remember all of this far better after answering five questions on it than after reading it twice.