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📐 Mathematics - Class 7
Lesson 16 · Fractions and Decimals

Word Problems with Fractions

MathematicsSubject
15 minEstimated read
One asymmetry explains everything

You can only add pieces that are the same size. But multiplying is taking a part of a part, and that works whatever the sizes are.

Adding and subtracting

Rewrite both with a common bottom, then add the tops and leave the bottom alone. Three fifths plus one fifth is four fifths, just as three apples plus one apple is four apples.

Never add the bottoms

One half plus one half is not two quarters. That would say half a chapati plus half a chapati leaves you with half a chapati.

Multiplying

Multiply the tops and the bottoms; no common bottom is needed. Multiplying by a fraction smaller than 1 makes the answer smaller.

Why dividing means turning upside down

Six divided by a half asks how many halves fit into six, and the answer is twelve. Small pieces means many of them fit.

Operation What to do Example
Add or subtractCommon bottom first, then the tops only2/3 + 1/4 = 8/12 + 3/12 = 11/12
MultiplyTops together, bottoms together2/3 × 3/4 = 6/12 = 1/2
DivideTurn the second one upside down, then multiply2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9

Mixed numbers

Change a mixed number into an improper fraction first: two and a half becomes five halves. Do all the work in improper form and change back at the end.

Two and a half is not two times a half

Two and a half means two PLUS a half, which is five halves. Two TIMES a half is one. Everywhere else in mathematics, symbols side by side mean multiply.

Cancel before you multiply, not after

For 8/15 × 5/12, cancel first: it becomes 2/3 × 1/3 = 2/9, all with small numbers.

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Worked Example 1

A jug holds 2/3 litre and another holds 1/4 litre.

Find a common bottom2/3 = 8/12 and 1/4 = 3/12
Add the tops only11/12 litre altogether
Subtract the same way5/12 litre more in the first
Sense check11/12 is just under a litre, which is right. If the answer had come out over 1, something went wrong.
Worked Example 2

A field is 3/4 hectare, and two thirds of it is planted with maize.

The word OF means multiply2/3 × 3/4
Cancel before multiplyingThe 3s cancel, leaving 2/4 = 1/2 hectare
Notice what happenedMultiplying made the answer SMALLER, which is normal when multiplying by a fraction less than 1.
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Activity 16.1

The paper strip kitchen

What you needSeveral strips of paper all cut to exactly the same length, scissors, pencil, ruler.
Make your piecesFold strips into halves, thirds, quarters, sixths and twelfths, cut along the folds and label every piece.
Now try to addLay a third and a quarter end to end and try to say what the total is, using only thirds or only quarters.
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1

You can only ADD pieces that are the same size. That is why a common bottom is needed.

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1Why does adding fractions need a common bottom number when multiplying does not?
2Why does dividing by a fraction smaller than 1 give a bigger answer?
3Explain why 1/2 + 1/2 is not 2/4, using an everyday example.
4How do you turn a mixed number into an improper fraction, and why is it worth doing first?
5Why is it better to cancel before multiplying rather than after?

Question 1 of 10

1What is 1/2 + 1/2?
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Slide 1 of 4
Unit 16 · Fractions

Fractions

Mathematics · Grade 7

What You Will Be Able To Do

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Add and subtract fractions, and say why a common bottom is needed

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Multiply them, and see why the word OF means multiply

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Divide them, and explain why the answer often gets bigger

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Handle mixed numbers safely by converting them first

“

You can only add pieces that are the same size. But multiplying is taking a part of a part, and that works whatever the sizes are.

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Presenter notes: Open with the question the class has probably been carrying for a year without asking: why does adding fractions need a common bottom when multiplying does not? Write both rules on the board and ask whether they seem consistent. They do not, which is honest, and by the end of the lesson the class will see that both come out of a single difference. Promise that, because it converts a set of rules into one idea.