Fractions: worked examples and method

Foundation and Higher · Grade 3 · Non-calculator

This page sets out the method for fractions in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Fractions topic page.

Key facts

Equivalent: $\frac{5}{12} = \frac{15}{36}$ because both 5 and 12 were multiplied by 3.
Add/subtract: $\frac{a}{b} \pm \frac{c}{d} = \frac{a d \pm b c}{b d}$ , then simplify.
Multiply: $\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$ . Cancel diagonally before multiplying.
Divide: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ .
Mixed $\to$ improper: $3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$ .
Improper $\to$ mixed: $\frac{33}{20} = 1 \frac{13}{20}$ because $33 \div 20 = 1$ remainder 13.

The method, step by step

Method: add or subtract with a lowest common denominator

When you see $+$ or $-$ between two fractions (including mixed numbers).

Convert any mixed number to an improper fraction.
List multiples of each denominator until you meet the smallest number that appears in both lists — that is the LCD.
Rewrite each fraction with that denominator: whatever you multiply the bottom by, multiply the top by too.
Add or subtract the numerators. The denominator stays the same.
Simplify by dividing top and bottom by their highest common factor. If the answer is top-heavy, write it as a mixed number.

Method: multiply fractions

When you see $\times$ , or the word 'of' between two fractions.

Convert mixed numbers to improper fractions. Do not multiply the whole parts and the fraction parts separately.
Cancel any numerator with any denominator that share a factor (they do not need to be in the same fraction).
Multiply the remaining numerators. Multiply the remaining denominators.
Simplify if anything is left to cancel, and convert to a mixed number if needed.

Method: divide fractions

When you see $\div$ , or a worded 'how many … fit into …'.

Convert mixed numbers to improper fractions.
Keep the first fraction exactly as it is.
Flip the second fraction (swap its numerator and denominator) and change $\div$ to $\times$ .
Cancel, then multiply tops and multiply bottoms, as for multiplication.
A whole-number answer is common in 'how many pieces' questions — that is a good sense-check.

How the lesson explains it

Add and subtract

The denominators must match before you add the numerators.

1/4 + 1/2 = 1/4 + 2/4 = 3/4.

Multiply

Multiply the tops and multiply the bottoms, then simplify.

2/3 × 3/5 = 6/15 = 2/5.

Divide

Dividing by a fraction is multiplying by its reciprocal.

1/2 ÷ 1/4 = 1/2 × 4/1 = 2.

Mixed numbers

Turn a mixed number into an improper fraction before you calculate.

1 1/2 is 3/2.

Of an amount

3/4 of 20 means 3/4 × 20. Write 20 as 20/1 if it helps.

3/4 × 20/1 = 60/4 = 15.

Simplify at the end

Cancel a common factor in the top and bottom.

Leave the answer in simplest form unless the question asks for a mixed number.

Add and subtract on one denominator

You can add the numerators only when the denominators already match. 1/5 + 2/5 = 3/5. The bottom stays 5. If the bottoms differ, change both fractions first. For 2/3 + 1/4 the common denominator is 12, because 3 × 4 = 12. Then 2/3 = 8/12 and 1/4 = 3/12.

Add the new numerators and keep 12: 8/12 + 3/12 = 11/12. That fraction is already in simplest form, because 11 and 12 share no common factor greater than 1. Subtraction uses the same preparation. 3/4 - 1/6 needs twelfths: 9/12 - 2/12 = 7/12.

Never add or subtract the denominators. 2/3 + 1/4 is not 3/7. The pieces are different sizes until the bottoms match, so combining the bottoms changes the size of each piece. If a whole number is added, write it as a fraction over 1, or add it to the whole part of a mixed number after the fraction parts are done.

Method: 1. Use a common denominator before you add or subtract. 2. Rewrite each fraction with that denominator. 3. Add or subtract the numerators and keep the denominator.

Multiply tops and bottoms

To multiply fractions, multiply the numerators and multiply the denominators. 2/3 × 3/5 = 6/15. Then simplify. 6 and 15 share a factor of 3, so 6/15 = 2/5. You may cancel before you multiply: the 3 on the top and the 3 on the bottom cancel, leaving 2/1 × 1/5 = 2/5.

A whole number is a fraction over 1. 4 × 2/5 = 4/1 × 2/5 = 8/5. The result can be larger than 1. Leave it improper, or write 13/5, unless the question asks for one form. There is nothing to cancel in that product. A case that does cancel is 4 × 3/8 = 3/2.

Check the size. Multiplying by a proper fraction makes a smaller result. 2/3 of something is less than the something. If your product is bigger than both fractions and both were proper, the multiplication has been replaced by an addition. 2/3 × 3/5 should be 2/5, which is less than 2/3.

Method: 1. Multiply the numerators together and the denominators together. 2. Cancel a shared factor before or after you multiply. 3. Write a whole number as a fraction over 1 if you need to.

Divide by flipping the second fraction

Dividing by a fraction is multiplying by its reciprocal. Keep the first fraction, flip the second, and change the sign to ×. 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. One half contains two quarters, so the result 2 matches the meaning. Flip only the divisor, the fraction you are dividing by.

Dividing by 1/2 doubles. 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2. Halving 3/4 would give 3/8, and that is multiplication by 1/2, not division by it. How many halves fit into a number is twice the number, so the answer is larger than the start.

Simplify after the multiply. 6/4 divides by 2 to give 3/2. You can cancel the 2 before multiplying: 3/4 × 2/1 = 3/2 × 1/1. Either order is fine if the cancel is a common factor of a numerator and a denominator. Do not cancel two numerators with each other.

Method: 1. Keep the first fraction as it is. 2. Flip the second fraction and change the division to a multiplication. 3. Multiply, then simplify the result.

Mixed numbers before you multiply or divide

A mixed number is a whole plus a fraction. 21/2 means 2 + 1/2. For multiplication or division, turn it into one fraction first. Two wholes are 4/2, and 4/2 + 1/2 = 5/2. Then 21/2 × 1/5 = 5/2 × 1/5 = 5/10 = 1/2.

Leaving the whole number alone and only multiplying the fraction gives 21/10, which is much too big. The 2 also has to be multiplied by 1/5. The improper form does that in one step. The same conversion is needed for division. 21/4 ÷ 1/2 = 9/4 × 2/1 = 9/2.

Adding mixed numbers can be done in parts: add the wholes, add the fractions on a common denominator, then carry if the fraction part is 1 or more. That shortcut is for addition and subtraction. It is not safe for a product or a quotient. Finish by simplifying, and write a mixed number only when the question asks for one.

Method: 1. Turn each mixed number into an improper fraction first. 2. Then multiply or divide those improper fractions. 3. Convert back to a mixed number only if the question asks for one.

Mixed numbers: converting and giving the answer in the right form

Many exam questions finish with "Give your answer as a mixed number" or "in its simplest form". To multiply or divide, turn each mixed number into an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the denominator. So 31/2 = 7/2 and 11/5 = 6/5. Work with these, and convert back only at the very end.

To convert back, divide the numerator by the denominator. The whole number of times it goes in is the whole part, and the remainder goes over the same denominator. For 21/5, 5 goes into 21 four times with remainder 1, so the answer is 41/5. Simplify first if you can, and always read the question for the form it wants. An improper fraction left as 21/5 may lose the final mark.

Method: 1. Convert every mixed number to an improper fraction. 2. Carry out the calculation with the improper fractions. 3. Simplify the result by dividing top and bottom by a common factor. 4. Convert back to a mixed number if the question asks for one.

Tip: Check the size of a mixed-number answer: 31/2 × 11/5 is about 3.5 × 1.2, so the answer should be a little over 4.

Adding and subtracting mixed numbers

For addition and subtraction you can either use improper fractions or treat the whole numbers and the fractions separately. With 13/4 + 22/3, add the wholes to get 3, then add 3/4 + 2/3 = 9/12 + 8/12 = 17/12. That is 15/12, so the total is 45/12. The lowest common denominator of 4 and 3 is 12.

Subtraction is where marks are lost. In 41/5 - 13/5 you cannot take 3/5 from 1/5. Do not turn it around. Converting both to improper fractions removes the problem: 21/5 - 8/5 = 13/5 = 23/5. Use improper fractions whenever the fractional part of the first number is smaller than the second.

Method: 1. Find the lowest common denominator of the fractions. 2. Write each fraction as an equivalent fraction with that denominator. 3. Add or subtract the numerators. Use improper fractions if a subtraction needs borrowing. 4. Simplify and convert to a mixed number if asked.

Fractions in worded questions

Exam contexts for fraction calculations include areas of rectangles with mixed-number sides, pieces cut from a length of ribbon, and the fraction of a budget that is left. When two fractions of one whole are spent, add them first. If Sam spends 2/5 on a book and 1/3 on food, the fraction left is 1 - 2/5 - 1/3 = 4/15. Write the whole as 15/15 so that the denominators agree.

Rectangle questions use area = length × width, and the reverse question gives the area and one side and asks for the other, which is a division. A fraction of a fraction, such as three quarters of two fifths, is a multiplication. Say in words what each step is doing and write the units at the end, for example m^2 for an area.

Method: 1. Decide which operation the story needs: add, subtract, multiply or divide. 2. Write the calculation using fractions or improper fractions. 3. Work it out, then simplify. 4. Answer in the form requested and include the units.

How many fit? Dividing in a story

"How many cups of 3/4 litre can be filled from 71/2 litres?" is a division. You are counting how many times 3/4 fits into 71/2. Convert to 15/2 ÷ 3/4, flip the second fraction and multiply: 15/2 × 4/3 = 60/6 = 10.

When the division is not exact, the question wants a whole number of complete pieces, and often asks what is left over. Round the quotient down, not to the nearest whole number, because you cannot count a part piece. For the amount left, take the total length minus the lengths used. Always check the result by multiplying back: 10 × 3/4 = 71/2.

Method: 1. Write the total divided by the size of one piece. 2. Convert mixed numbers to improper fractions, flip the second fraction and multiply. 3. Round down if only complete pieces count. 4. Find any leftover as total minus pieces used, and check by multiplying back.

Tip: Dividing by a fraction less than 1 must give an answer bigger than the number you started with.

Remember

Common denominator to add.
Multiply tops and bottoms.
Divide by multiplying by the reciprocal.

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