Fractions: keep the whole in view
Add, multiply and divide fractions while keeping track of what each quantity represents.
In this guide
A fraction names both an amount and a unit
In 3/8, the denominator says that one whole has been divided into eight equal parts. The numerator counts three of those parts. Before calculating, identify the whole: a litre, a budget, a class or an entire task. Two fractions can only be combined as portions of a whole when their reference whole is the same. Half of a small container need not equal half of a larger container. Equivalent fractions preserve the amount by multiplying or dividing both numerator and denominator by the same nonzero number.
Worked example: combine equal-sized pieces
A reading group finishes 5/12 of one booklet on Monday and another 1/8 of that same booklet on Tuesday. What fraction is finished? Twelfths and eighths name different-sized pieces, so first express both in twenty-fourths. Multiply 5/12 by 2/2 to obtain 10/24. Multiply 1/8 by 3/3 to obtain 3/24. Adding gives 13/24 of the booklet. The unfinished fraction is 24/24 - 13/24 = 11/24. A rough check helps: 5/12 is slightly less than one-half and 1/8 is small, so a total slightly greater than one-half is reasonable.
Multiplication answers a different question
Suppose 3/5 of a storage box is occupied, and 2/3 of that occupied space holds books. The books occupy 2/3 × 3/5 = 6/15 = 2/5 of the entire box. Here 'of' describes a portion of a portion. It does not ask you to combine 2/3 and 3/5. You can simplify before multiplying: the factor 3 in the first denominator cancels the factor 3 in the second numerator. Cancellation applies to factors in a product; it does not allow deleting numbers across an addition sign.
Division counts how many groups fit
There are 7/8 litre of juice, and each cup takes 1/4 litre. The number of cup-sized portions is (7/8) ÷ (1/4) = (7/8) × 4 = 7/2 = 3.5. Interpret the result: this fills three complete cups and half of a fourth. If the question asks for full cups only, the answer is three, not four. Dividing by a proper fraction can increase the numerical value because the unit being counted is smaller. The physical amount of juice has not increased.
Common traps
Keep exact fractions until rounding is actually requested. A decimal approximation can make later subtraction less accurate. When subtracting from a whole, write that whole with a suitable denominator first. For mixed numbers, converting to improper fractions often makes multiplication and division easier. A denominator of zero is never allowed; a fraction such as 0/7 is valid and equals zero.
- Do not add denominators: 1/3 + 1/3 counts two thirds, not two sixths.
- Use a shared denominator for addition and subtraction, not as a compulsory first step for multiplication.
- After calculating, ask whether the answer counts whole objects, partial objects or a fraction of a stated total.
Try it yourself
A student uses 2/5 of a roll of paper for labels and 1/6 of the same roll for covers. What fraction remains? Then work out how many portions of 1/8 kilogram fit into 3/4 kilogram. Show the operation and unit for each answer.
Reveal the worked answer
A student uses 2/5 of a roll of paper for labels and 1/6 of the same roll for covers. What fraction remains? Pause before reading the solution. Convert to thirtieths: 2/5 = 12/30 and 1/6 = 5/30. The used fraction is 17/30, leaving 30/30 - 17/30 = 13/30. For a second check, divide 3/4 kilogram into portions of 1/8 kilogram: (3/4) × 8 = 6 portions. Both calculations should keep the named unit attached to the final answer.
A student uses 2/5 of a roll of paper for labels and 1/6 of the same roll for covers. What fraction remains? Pause before reading the solution. Convert to thirtieths: 2/5 = 12/30 and 1/6 = 5/30. The used fraction is 17/30, leaving 30/30 - 17/30 = 13/30. For a second check, divide 3/4 kilogram into portions of 1/8 kilogram: (3/4) × 8 = 6 portions. Both calculations should keep the named unit attached to the final answer.
Put one idea into practice.
Try a short mathematics session, then use the explanations to choose what to review next.
Start a short sessionSources & further reading
OpenStax Elementary Algebra 2e: Add and subtract fractionsOpenStax Prealgebra 2e: Fractions and related topicsSomething unclear or incorrect? Suggest a correction to this guide.
Original FreeReviewer.ph guidance, informed by the linked sources. Exam requirements and procedures should always be checked with the examining body.