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Adding and subtracting fractions - Mathematics Year 5

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Introduction

Equal-length wholes divided into equal parts. A half and two quarters represent the same length; the colour alone is not the evidence.

One team uses three eighths of a metre of ribbon and another uses four eighths. How much have they used together? The answer counts eighths: seven of them. Changing the denominator to sixteen would change what each counted piece means.

Learn to add and subtract fractions by keeping track of a common whole and equal-sized units. Move from fraction strips to equations, cross one whole, and solve mixed-number problems with a check. Your final work includes eight independent calculations, a labelled diagram and a practical-context explanation with units.

This course follows England's Year 5 fraction work using equal denominators and cases where one denominator is a multiple of the other. General unlike-denominator methods are developed further in Year 6. Before starting, recall equivalent fractions, multiplication facts and the meaning of numerator and denominator. Revisit Equivalent fractions and comparing fractions - Mathematics Year 5 if changing a fraction's name changes its value in your working.

Learning goals

Add and subtract with equal or compatible denominators and explain the unit being counted.
Convert an answer greater than one between improper fraction and mixed-number forms.
Choose an efficient visual or symbolic method for an unfamiliar problem.
Communicate a complete solution with a diagram, meaningful unit and independent check.

You can draw strips, use existing classroom fraction pieces, or give an accessible spoken explanation with a partner recording the diagram. The calculation does not require cutting food or buying equipment.

Count equal parts of a common whole

A fraction refers to a whole or unit. If one whole strip represents one metre, one eighth is one of eight equal lengths in that metre. The denominator names how many equal parts make one whole; the numerator tells how many of those parts are being counted. A fraction can also be understood as a number on a number line.

The opening Commons strip chart uses the same total length in every row. Compare one half with two quarters and four eighths. The end points line up because the values are equal, even though the part sizes differ. If you make your own strips, start with equal-length wholes. Counting half of a large sheet plus half of a small sheet as one small sheet would mix units.

Worked addition: three eighths plus four eighths

Draw a strip divided into eight equal cells. Mark three cells with one pattern and four different cells with another pattern, so that the two amounts do not overlap. Seven cells are now marked. Describe the result as seven eighths of the original whole. Both shading pattern and labels should show the groups, so that colour is not the only clue.

$\frac{3}{8} + \frac{4}{8} = \frac{7}{8}$

The denominator remains eight because the parts have not become smaller. You added counts of the same unit, just as three pencils plus four pencils is seven pencils. The analogy explains the count, but the eighth is a fractional unit rather than a whole pencil.

Worked subtraction: remove or compare

If seven tenths of a length are present and three tenths are removed, four tenths remain. The same subtraction can describe the difference between seven tenths and three tenths of a common whole. State which interpretation matches the problem.

$\frac{7}{10} - \frac{3}{10} = \frac{4}{10} = \frac{2}{5}$

Four tenths and two fifths are equal values. Simplifying both numerator and denominator by two changes the name, not the remaining length. A strip divided into ten parts groups each two adjacent tenths into one fifth. You do not remove a further two tenths when simplifying.

Crossing one whole

Five sixths plus four sixths is nine sixths. Since six sixths make one whole, nine sixths is one whole and three sixths, or one and a half. A picture needs more than one equal-sized strip; squeezing nine shaded sixths into a single six-cell strip hides the amount.

$\frac{5}{6} + \frac{4}{6} = \frac{9}{6} = 1 \frac{3}{6} = 1 \frac{1}{2}$

An improper fraction has a numerator greater than or equal to its denominator. Its value is at least one; it is not an invalid fraction. A mixed number combines a whole-number part and a proper fraction. In the mixed notation one and a half, the whole and fractional parts are added, not multiplied.

Error clinic

A learner writes three eighths plus four eighths equals seven sixteenths. The first wrong step is adding the denominators. It replaces eighth-sized parts with sixteenth-sized parts while keeping the count seven. Seven sixteenths is less than one half, yet the intended sum contains four eighths already, plus three more eighths. This size check exposes the error as well as the strip model.

Try it: Find two sevenths plus three sevenths, then six ninths minus two ninths. State the part name before the answer. Checks: five sevenths and four ninths. Sketch the same whole for each question; the two questions need not use the same subdivision.

Rename compatible fractions before combining

You can add half a metre and three eighths of a metre, but the written fractions initially count different-sized parts. Rewrite them as equivalent fractions with a common denominator before combining their numerators. The underlying unit, a metre in this example, must also match.

Worked compatible-denominator addition

One half is four eighths: each half is split into four equal eighths. The three eighths already use the intended part size.

$\frac{1}{2} + \frac{3}{8} = \frac{4}{8} + \frac{3}{8} = \frac{7}{8}$

Multiply both the numerator and denominator of one half by four. Multiplying only its denominator would produce one eighth and change the quantity. A number line offers another view: start at one half and move forward three steps of one eighth to finish at seven eighths. Use equal eighth-sized steps throughout.

A sum greater than one

To combine two thirds and three sixths, rewrite the thirds as sixths. Each third contains two sixths, so two thirds is four sixths.

$\frac{2}{3} + \frac{3}{6} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6} = 1 \frac{1}{6}$

Six sixths make the first whole, leaving one sixth. On a number line, the addition passes through one rather than stopping there. An answer above one is expected: two thirds is greater than one half, and the added three sixths is another half.

Worked subtraction

Five sixths minus one third uses the same exchange of part sizes. Convert one third to two sixths, then subtract equal units.

$\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$

Check by the inverse operation: one half plus one third equals three sixths plus two sixths, or five sixths. Restoring the original amount is a different operation from simply repeating the subtraction. A rough size check also helps: removing about one third from a value close to one should leave a positive amount smaller than the starting value.

Choose an efficient denominator

For thirds and sixths, six is already a common denominator. Eighteen would also work: twelve eighteenths plus nine eighteenths equals twenty-one eighteenths, again one and one sixth. The longer calculation is valid, but the larger numbers and extra simplification usually make six more efficient. The smallest common denominator is useful, not a requirement for a correct value.

When one denominator is a multiple of the other, try the larger denominator first. For fourths and eighths, use eighths; for tenths and hundredths, use hundredths. Do not apply this shortcut to any two denominators without checking. Four is not a multiple of three; using fourths for thirds would not create an equivalent fraction with a whole-number numerator. That general case belongs to later work.

Watch the original Khan Academy explanation of adding unlike denominators. Follow how each fraction is renamed without changing its value. Pause before the numerators are combined, predict the common part size and compare the method with the half-and-eighths example. Use the written model if the film is unavailable.

Try it: Three tenths plus seven hundredths uses thirty hundredths plus seven hundredths, giving thirty-seven hundredths. Seven eighths minus one quarter uses seven eighths minus two eighths, giving five eighths. Verify each with a labelled number line or inverse calculation.

Add and subtract mixed numbers with an exchange

A mixed number names whole units plus a fraction of another unit. One and three quarters is one whole plus three quarters. To express it as an improper fraction, rename the whole as four quarters and combine the seven quarters. Do not multiply one by three quarters and call the result the same mixed number.

A cake diagram labelled in quarters. For a mixed-number model, imagine further cakes of the same size. Perspective does not change the intended equal-quarter model.

The cake illustration models equal quarters of one cake; it does not prove that real uneven slices have equal mass. One and three quarters of these model cakes means a whole equal-sized cake plus three quarters of another. Half of a much larger cake would not necessarily be the same amount as two quarters of this cake. All calculations below keep the whole unit consistent.

Addition by separating whole and fractional parts

Add one and three quarters to two and one eighth. The whole-number parts total three. Rename three quarters as six eighths, then add the extra eighth. The fractional total is seven eighths, so the answer is three and seven eighths.

$1 \frac{3}{4} + 2 \frac{1}{8} = 3 + \frac{6}{8} + \frac{1}{8} = 3 \frac{7}{8}$

An alternative is to convert both mixed numbers first: one and three quarters is fourteen eighths; two and one eighth is seventeen eighths. Fourteen plus seventeen is thirty-one eighths, or three and seven eighths. Both methods preserve value. Separating the whole parts uses smaller numbers here; conversion can be easier to organise in some subtraction problems.

Subtraction that needs an exchange

Find two and one quarter minus five eighths. Rename the quarter as two eighths. You cannot remove five eighths from the fractional part alone while leaving the two whole units unchanged. Exchange one whole for eight eighths: two and two eighths becomes one and ten eighths.

$2 \frac{1}{4} - \frac{5}{8} = 2 \frac{2}{8} - \frac{5}{8} = 1 + \frac{10}{8} - \frac{5}{8} = 1 \frac{5}{8}$

The exchange changes the representation, not the starting quantity. One whole plus ten eighths still equals two and one quarter. After removing five eighths, one whole and five eighths remain. A two-whole strip drawing with a further quarter can show the same exchange.

Second method and inverse check

Convert the starting amount to an improper fraction: two and one quarter is nine quarters, or eighteen eighths. Subtracting five eighths leaves thirteen eighths, again one and five eighths. The inverse check adds the removed amount to the result.

$\frac{18}{8} - \frac{5}{8} = \frac{13}{8} = 1 \frac{5}{8}$ $1 \frac{5}{8} + \frac{5}{8} = 1 + \frac{10}{8} = 2 \frac{1}{4}$

A common incorrect shortcut takes five minus two to get three eighths and leaves two wholes, giving two and three eighths. This reverses only the fractional subtraction and produces an answer larger than the starting amount, despite removing a positive quantity. Correct the first invalid step by exchanging a whole or by converting the entire mixed number.

Subtracting from a whole number

Three minus seven tenths can be represented as two wholes and ten tenths, minus seven tenths. The result is two and three tenths. The missing fraction to the next whole gives a quick check: three tenths plus seven tenths restores one, and with the two whole units restores three.

Try it: One and one quarter plus two and one half equals three and three quarters. One and one half minus three quarters equals three quarters. Choose either mixed-number method, write the exchange or conversion explicitly, and explain why the result's size makes sense.

Choose a method and solve a complete problem

An equation is part of a solution. First identify the quantity, unit and question; then choose a representation, calculate, check and interpret. Adding two positive lengths finds a total length. Subtracting used length from starting length finds what remains. A word such as 'more' does not always tell you to add: asking how much more one length is than another asks for a difference.

Ribbon-planning example: invented quantities

A display team starts with two and a half metres of ribbon. One border needs three quarters of a metre and another needs seven eighths of a metre. Assume the two lengths can be cut without extra cutting loss in this simplified planning problem. How much is used, and how much remains?

Use eighths because eight is a multiple of four and two. Three quarters is six eighths; two and a half is twenty eighths.
Total used is six eighths plus seven eighths, or thirteen eighths of a metre: one and five eighths of a metre.
Remaining length is twenty eighths minus thirteen eighths, or seven eighths of a metre.
Check that thirteen eighths used plus seven eighths remaining is twenty eighths, the original two and a half metres.

$\frac{3}{4} m + \frac{7}{8} m = \frac{13}{8} m = 1 \frac{5}{8} m$ $2 \frac{1}{2} m - 1 \frac{5}{8} m = \frac{20}{8} m - \frac{13}{8} m = \frac{7}{8} m$

An alternative subtracts the two lengths one after the other: two and a half minus three quarters is one and three quarters, then subtract seven eighths to leave seven eighths. This works because both requested pieces are removed. Subtracting the difference between the two requested lengths would answer a different question. In real material planning, include any stated trimming allowance; none is supplied here.

A missing-amount question

A bottle contains one and one quarter litres in a fictional capacity problem. After some is poured out, seven eighths of a litre remains. The removed amount is ten eighths minus seven eighths, or three eighths of a litre. Check by adding three eighths to seven eighths: ten eighths, or one and one quarter. These are supplied quantities, not measurements you have made. The answer needs litres, not metres or an unexplained three eighths.

Plan before calculating

For each problem, say whether you are finding a total, a remainder, a difference or a missing amount. A labelled strip or number line can expose a mistaken operation before arithmetic starts. A written method is efficient when it uses a sensible denominator and records enough intermediate work to be checked. A correct answer without reasoning does not show why a method fits an unfamiliar context.

Independent eight-question check

Work these before opening the answers. Give simplest proper fractions or mixed numbers where appropriate; keep an improper-fraction intermediate when useful. Explain question 5 with two equal-sized whole strips and question 8 with an inverse check.

Question	Calculation
1	Three eighths plus four eighths
2	Seven tenths minus three tenths
3	One half plus three eighths
4	Five sixths minus one third
5	Two thirds plus three sixths
6	Seven eighths plus three quarters
7	One and one quarter plus two and one half
8	Two and one quarter minus five eighths

Worked answers, after attempting all eight

Seven eighths: count seven eighth-sized parts.
Four tenths, simplified to two fifths: the part size remains a tenth until simplification.
Seven eighths: rename the half as four eighths.
Three sixths, simplified to one half: one third is two sixths.
Seven sixths, or one and one sixth: two thirds is four sixths; one whole absorbs six sixths.
Thirteen eighths, or one and five eighths: three quarters is six eighths.
Three and three quarters: two and one half contributes two wholes and two quarters.
Thirteen eighths, or one and five eighths: eighteen eighths minus five eighths. Adding the removed five eighths restores eighteen eighths, or two and one quarter.

Return to any question where the answer is implausible. Adding positive fractions must give a result greater than either addend. Subtracting a positive amount gives less than the starting quantity; the present examples stay non-negative. These bounds can reject some mistakes, but they do not prove every plausible answer correct. Combine a size check with equivalence, inverse calculation or a diagram.

Final explanation: Choose an unfamiliar length or capacity problem, declare one common unit and show two valid approaches. Explain which is clearer for you and why. If an answer only changes because a diagram's whole changed size, repair the representation rather than forcing the arithmetic to match it.

Interactive Tasks

Quiz: Test Your Knowledge

What is 3/8 plus 4/8? (7/8) (!7/16) (!12/8) (!1/8)

Which denominator can directly name both thirds and sixths using whole-number numerators? (6) (!3) (!5) (!9)

Which fraction is equivalent to one third? (2/6) (!1/6) (!3/6) (!2/3)

What is 5/6 minus 1/3 in simplest form? (1/2) (!4/3) (!4/6) (!2/6)

What is 2/3 plus 3/6 as a mixed number? (1 1/6) (!1 5/6) (!1 1/3) (!1 2/3)

Why does adding three eighths and four eighths keep the denominator eight? (The counted parts are still eighths of the same whole) (!Adding always doubles the part size) (!The denominator counts the seven shaded parts) (!Every sum must have a denominator larger than both starting denominators)

What is 2 1/4 minus 5/8? (1 5/8) (!2 3/8) (!1 3/8) (!1 7/8)

Which feature makes two fraction-strip rows suitable for comparing parts of one common whole? (The total whole-strip lengths are equal) (!The rows have different colours) (!The rows have the same number of labels) (!The rows are drawn on different pages at unknown scales)

From 2 1/2 metres, a team uses 3/4 metre and 7/8 metre. How much remains under the stated no-loss assumption? (7/8 metre) (!13/8 metres) (!3/8 metre) (!1/8 metre)

Which inverse check tests the result of 5/6 minus 1/3 equalling 1/2? (Add 1/2 and 1/3 and obtain 5/6) (!Add 5/6 and 1/3 and obtain 1/2) (!Subtract 1/3 from 1/2 and obtain 5/6) (!Add the denominators and keep the numerator unchanged)

Memory Game

Numerator	Counts the stated fractional parts
Denominator	Names the equal subdivision of one whole
Equivalent fractions	Different names with equal value
Mixed number	Whole-number part plus a proper fraction
Inverse check	Restore a subtraction's starting amount by addition
Common whole	Consistent unit behind the compared quantities

Drag and Drop

One half as eighths	4/8
One third as sixths	2/6
Three quarters as eighths	6/8
Two and one quarter as eighths	18/8
One and one sixth as sixths	7/6

Crossword Puzzle

NUMERATOR	Fraction component that counts the parts
DENOMINATOR	Fraction component naming the equal subdivision
EQUIVALENT	Describes fractions with the same value
ADDITION	Operation used to find a combined total
INVERSE	Describes an operation that undoes another
WHOLE	Complete unit against which the parts are defined

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Count equal fractional units: Draw an eight-cell whole, combine three eighths and four eighths without overlap, and explain why the denominator remains eight.
Remove equal fractional units: Show seven tenths minus three tenths with a strip. Explain the equivalent simplified result without removing more length.
Rename a half without changing it: Use equal-length strips to show one half as four eighths. Label both names and the whole.
Recognise a mixed amount: Represent nine sixths using two equal-sized whole strips. Explain the mixed number and its simplified form.

Standard

Calculate with compatible parts: Solve one half plus three eighths and five sixths minus one third. Show the renaming step and check one answer by the inverse operation.
Cross a whole with thirds and sixths: Explain two thirds plus three sixths using a diagram, an equation and an estimate that predicts a result above one.
Exchange a whole in subtraction: Solve two and one quarter minus five eighths by exchanging one whole. Explain why the starting value is preserved.
Complete the independent fraction set: Attempt all eight questions in the final teaching chapter before looking at the checks. Show a diagram for question five and an inverse check for question eight.

Advanced

Compare two valid fraction methods: For one and three quarters plus two and one eighth, compare separating whole parts with converting to improper fractions. Discuss efficiency and a possible error in each method.
Solve and audit the ribbon plan: Calculate used and remaining ribbon from the stated two-and-a-half-metre supply. Show two solution paths, units, an inverse check and the no-cutting-loss assumption.
Repair the first invalid fraction step: Critique a learner who adds three eighths and four eighths to make seven sixteenths, and another who gets two and three eighths from two and one quarter minus five eighths. Find each first invalid step.
Design an unfamiliar missing-amount problem: Create a length or capacity question with compatible denominators and a mixed-number start. Supply a complete solution, an independently different check and a clear common-whole explanation.

Learning Check

Complete eight fraction calculations independently: Solve the eight listed calculations in the final teaching chapter. Show a common-unit diagram for question five and an inverse check for question eight.
Explain a common denominator: Explain why one half plus three eighths uses eighths, and why changing only a denominator changes the value.
Compare efficient and longer valid methods: For two thirds plus three sixths, compare sixths and eighteenths and express the answer as a mixed number.
Justify a mixed-number exchange: Explain two and one quarter minus five eighths with two methods and reject the result two and three eighths.
Interpret a ribbon solution with units: From two and a half metres, remove three quarters of a metre and seven eighths of a metre. Give total used, remainder, check and one modelling assumption.
Find a missing poured amount: A fictional bottle starts with one and one quarter litres and retains seven eighths of a litre. Find the amount removed, with an inverse check.
Evaluate a visual representation: Explain why two different-sized cakes cannot automatically supply a common-whole diagram for half plus half, and why colour alone is insufficient evidence.

Worked checks and assessment guidance

The results in order are 7/8, 2/5, 7/8, 1/2, 1 1/6, 1 5/8, 3 3/4 and 1 5/8. Equivalent unsimplified intermediate fractions are valid when identified as equivalent. For question five, use four sixths plus three sixths across two equal wholes. For question eight, add five eighths to one and five eighths to restore two and one quarter.
One half is four eighths because both its numerator and denominator are multiplied by four. The added units are therefore four eighths and three eighths, making seven eighths. Replacing one half by one eighth changes the amount rather than renaming it. Both quantities must also use the same underlying whole or measurement unit.
Sixths give four sixths plus three sixths, or seven sixths, equal to one and one sixth. Eighteenths give twelve eighteenths plus nine eighteenths, or twenty-one eighteenths, the same value. The larger denominator is valid but needs larger numbers and further simplification. Using six is usually clearer for this case.
Two and one quarter is two and two eighths. Exchange one whole to get one plus ten eighths; subtract five eighths to obtain one and five eighths. Alternatively, eighteen eighths minus five eighths is thirteen eighths. The incorrect result reverses only the fractional subtraction and exceeds the starting value despite removing a positive amount.
The used total is six eighths plus seven eighths, or thirteen eighths of a metre, equal to one and five eighths. The starting length is twenty eighths, so seven eighths of a metre remains. Used plus remaining restores twenty eighths. The simplified model assumes no additional cutting loss; a real allowance would need to be supplied and included.
The starting amount is ten eighths of a litre. Ten eighths minus seven eighths leaves three eighths of a litre removed. Adding three eighths removed and seven eighths remaining gives ten eighths, or one and one quarter litres. These are supplied mathematical quantities, not actual pupil measurements.
A half is relative to a specified whole. Half of a large cake and half of a small cake need not have equal size or mass, so they cannot simply be treated as two halves of the same cake unit. Equal-sized reference wholes and equal subdivisions support the intended model. Labels, boundaries or patterns make the relationship available without relying only on colour.

Learning evidence

Knowledge: Explain numerator, denominator, equal parts, common whole, equivalence, common denominator, improper fraction and mixed number.

Skills: Add and subtract equal or compatible denominators, exchange a whole without changing value, choose a sensible representation, and compare two valid solution methods.

Products: Keep the eight independent calculations, a two-whole strip diagram, a mixed-number exchange, the ribbon solution with metres, a missing-capacity solution with litres, and one corrected error.

Assessment: Correct values alone are insufficient for the reasoning objectives. Look for an explicit common unit, value-preserving renaming, a justified operation, a diagram and an independent inverse or size check. Accept a spoken, tactile or digital explanation demonstrating the same relationships. A screen score does not establish practical cutting or measuring competence.

Transfer: Explain how the method changes when the written part sizes differ, why a larger common denominator can be valid but less efficient, and why changing the underlying whole can invalidate an otherwise neat picture.

OER and Wikipedia

This is an original teaching sequence and set of exercises. The uploaded editorial objectives were checked against Department for Education: mathematics programmes of study, Year 5, fractions. The course concentrates on equal denominators and compatible multiples, with mixed and improper representations. It does not claim to complete the whole national programme.

Images and credits

FractionStrips.PNG, Zapotz, CC0. Original equal-length strip chart, unchanged.
Cake-quarters.jpg, Ian Tresman, own illustration based on Hemera Photo Objects as described on the Commons page; CC BY 2.5. Original file unchanged; the course treats the labels as a model, not a weighing record.

Video

Adding fractions with unlike denominators, original Khan Academy publication. Embedded from the original provider; no video or transcript copied. Pause and follow the value-preserving renaming, then apply it to the written Year-5 examples.

All ribbon, bottle and independent-practice quantities are invented mathematical examples. No actual learner performance or physical experiment is claimed. The following encyclopedia page is optional reference, not the course's answer key.