Fractions Without a Calculator — Lesson & Practice Paper
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Fractions Without a Calculator
A diagnostic ladder from Year 5 to Year 7.
This booklet helps you learn about fractions, step by step. No calculator allowed.
What’s a fraction?
A fraction is a way of describing part of a whole.
The BOTTOM number (called the denominator) tells you how many equal parts the whole is split into. The TOP number (called the numerator) tells you how many of those parts you have.
Example: in 3/4, the whole is split into 4 equal parts, and you have 3 of them.
What you’ll learn
We start with equivalent fractions and simplifying. Then we convert between improper fractions and mixed numbers, then add and subtract fractions, and finally multiply and divide them.
How to use the booklet
There are three levels. Start at Level 1 and work through in order.
If you get stuck, that’s OK — stop there. The level you stop at tells your tutor exactly where you need help.
The ladder:
· Level 1 / Section A — Year 5 (KS2). Equivalent fractions, simplifying, comparing.
· Level 2 / Section B — Year 6 (KS2). Improper/mixed conversion, adding and subtracting.
· Level 3 / Section C — Year 7 (KS3). Multiplying, dividing, and mixed number operations.
TUTOR — MATERIALS TO HAVE READY
Useful objects for this paper:
Paper strips or fraction bars — for Level 1, fold a strip in halves, quarters, eighths to see equivalence.
Pen and paper — for Level 2 and 3, work calculations longhand.
A clock face — clock fractions are intuitive: 15 minutes = 1/4 hour = 15/60, 20 minutes = 1/3 hour.
NO CALCULATOR — this is a non-calculator paper. Times tables and division are tested alongside the fraction work.
LEVEL 1 · YEAR 5 (KS2) · LESSON 1
Equivalent fractions and simplifying
Aim: Recognise equivalent fractions. Simplify a fraction to lowest terms. Compare two fractions.
A fraction has a numerator (top) and a denominator (bottom). 3/4 means 3 parts out of 4 equal parts.
Equivalent fractions have the same value but look different. 1/2 = 2/4 = 3/6 = 5/10 = … Rule: multiply (or divide) BOTH top AND bottom by the same number, and the value stays the same.
Worked example (BUILDING UP)
Find a fraction equivalent to 2/3 with denominator 12.
Denominator goes from 3 to 12 — multiply by 4.
Apply the same to the numerator: 2 × 4 = 8.
So 2/3 = 8/12. ✓
Worked example (SIMPLIFYING)
Simplify 12/18 to lowest terms.
Find the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 12 and 18: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(12, 18) = 6.
Divide top and bottom by 6.
12 ÷ 6 = 2. 18 ÷ 6 = 3.
So 12/18 = 2/3 in lowest terms. ✓
Comparing fractions:
· Same denominator? Compare numerators.
· Same numerator? The fraction with the smaller denominator is bigger (fewer larger pieces).
· Different both? Convert to a common denominator (the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.), then compare numerators.
Worked example (COMPARING)
Which is bigger: 2/3 or 3/5?
Common denominator: LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(3, 5) = 15.
2/3 = 10/15. 3/5 = 9/15.
10/15 > 9/15. So 2/3 is bigger. ✓
Try one
Simplify 15/20 to lowest terms.
Show answer
Find the HCF of 15 and 20: it is 5.
Divide top and bottom by 5: 15÷5 = 3, 20÷5 = 4.
15/20 = 3/4.
Check: 3/4 = 0.75 = 15/20. ✓
LEVEL 2 · YEAR 6 (KS2) · LESSON 2
Improper fractions, mixed numbers, adding and subtracting
Aim: Convert between improper fractions and mixed numbers, then add and subtract them. (We’ll handle different denominators too.)
In Level 1 you learned to recognise equivalent fractions and simplify them. In this lesson we use that skill to add and subtract fractions — including ones bigger than 1 (improper fractions and mixed numbers).
Improper fraction: top is bigger than (or equal to) bottom. Example: 7/3. Mixed number:whole numberA number with no fractional part: 0, 1, 2, 3, 4, … + fraction. Example: 2 1/3.
Both ways of writing the same value. 7/3 = 2 1/3.
Worked example (IMPROPER → MIXED)
Write 11/4 as a mixed number.
Divide top by bottom: 11 ÷ 4 = 2 remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. 3.
Whole numberA number with no fractional part: 0, 1, 2, 3, 4, … part = quotient = 2.
Fractional part = remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1./denominator = 3/4.
So 11/4 = 2 3/4. ✓
Worked example (MIXED → IMPROPER)
Write 3 2/5 as an improper fraction.
Multiply whole numberA number with no fractional part: 0, 1, 2, 3, 4, … by denominator: 3 × 5 = 15.
Add the numerator: 15 + 2 = 17.
Put over the original denominator.
So 3 2/5 = 17/5. ✓
Adding/subtracting fractions:
1. Get a common denominator (the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of the two denominators).
2. Convert each fraction.
3. Add or subtract the numerators only. Keep the denominator.
4. Simplify the result if possible.
Worked example (ADDING)
Calculate 2/3 + 1/6.
Common denominator: LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(3, 6) = 6.
2/3 = 4/6 (multiply top and bottom by 2).
1/6 stays as 1/6.
4/6 + 1/6 = 5/6.
Simplify: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(5, 6) = 1. Already in lowest terms.
Answer: 5/6. ✓
Watch out — you cannot just add tops and bottoms
1/3 + 1/6 is NOT 2/9. Each fraction has a different-sized piece — you cannot add 3 of one with 6 of another without converting first. Always get the common denominator before adding.
Try one
Calculate 3/4 - 1/2.
Show answer
Common denominator: LCM(4, 2) = 4.
Rewrite: 1/2 = 2/4; 3/4 stays the same.
3/4 − 2/4 = 1/4. ✓
LEVEL 3 · YEAR 7 (KS3) · LESSON 3
Multiplying and dividing fractions
Aim: Multiply and divide fractions, including mixed numbers. (We’ll convert mixed numbers to improper fractions first.)
In Level 2 you added and subtracted fractions — the trickiest skill, because the denominators had to match. Multiplying and dividing fractions is actually simpler: the denominators don’t need to match. You can multiply or divide any two fractions straight away.
Multiplying: top × top, bottom × bottom. Then simplify. Dividing: KEEP the first fraction, CHANGE the ÷ to ×, FLIP the second. Then multiply as normal. Mixed numbers: always convert to improper first — never try to multiply or divide mixed numbers directly.
Worked example (MULTIPLYING)
Calculate 2/3 × 4/5.
Multiply tops: 2 × 4 = 8.
Multiply bottoms: 3 × 5 = 15.
8/15. Check lowest terms: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(8, 15) = 1. Done.
Answer: 8/15. ✓
Worked example (DIVIDING)
Calculate 2/3 ÷ 1/6.
Keep: 2/3. Change: ×. Flip: 6/1.
2/3 × 6/1 = 12/3.
12/3 = 4. ✓
WHY KEEP-CHANGE-FLIP WORKS
Dividing is the inverse of multiplying. Multiplying by 1/6 makes things smaller (you take a sixth of them). Dividing by 1/6 must do the opposite — make them 6 times bigger. So ÷ 1/6 is the same as × 6/1. The same logic works for any fraction: ÷ a/b means × b/a (the reciprocal).
Worked example (MIXED NUMBERS)
Calculate 1 1/2 × 2 2/3.
Convert to improper: 1 1/2 = 3/2. 2 2/3 = 8/3.
Multiply: 3/2 × 8/3 = 24/6.
Simplify: 24/6 = 4.
Answer: 4. ✓
Watch out — mixed numbers are NOT a multiplication
2 1/3 means “2 AND a third”, not “2 multiplied by 1/3”. If you tried 2 × 1/3 = 2/3, you would get the wrong answer. Always convert to improper (here: 7/3) before doing arithmetic.
Try one
Calculate 3/4 ÷ 1/2.
Show answer
Keep, change, flip: 3/4 × 2/1.
Multiply: 6/4.
Simplify: 6/4 = 3/2 = 1½.
Check: a half fits into 3/4 one and a half times. ✓
VOCABULARY
Numerator
The TOP number in a fraction. Tells you how many parts you have.
Denominator
The BOTTOM number. Tells you how many equal parts the whole is split into.
Equivalent fraction
A different-looking fraction with the same value. 1/2 = 2/4 = 3/6.
Lowest terms (simplest form)
A fraction where top and bottom share no common factor other than 1.
Improper fraction
A fraction where the top is bigger than (or equal to) the bottom. 7/3.
Mixed number
A whole number combined with a fraction. 2 1/3.
Reciprocal
The flip of a fraction. The reciprocal of 2/3 is 3/2. Used when dividing.
Recap before you start
Equivalent: multiply or divide top and bottom by the same number.
Simplifying: divide top and bottom by their HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..
Comparing: get a common denominator (LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.), then compare numerators.
Improper → mixed: divide, quotient = whole part, remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. = new numerator.
Mixed → improper: (whole × denom + num) over denom.
Add / subtract: common denominator first.
Multiply: top × top, bottom × bottom.
Divide: keep-change-flip.
Mixed numbers in multiply/divide: ALWAYS convert to improper first.
Part Two
Now try the practice questions
45 questions across three sections, 30 minutes when you’re ready. The step-by-step walkthroughs unlock after you mark.
Instructions
Type each numeric answer in the box. For sign questions, use a minus sign (e.g. -7). For radio questions, tap your choice.
You can start the 30-minute timer or work untimed.
When finished, click Mark My Work. Your answers will then be locked, the correct answers will appear under each wrong question with the working, and you will get 5 similar practice questions inline for each one you missed.
Timed practice · 30 minutes
When you’re ready, start the timer
This practice is designed to take 30 minutes. Your tutor will see how long you took. You can work untimed if you prefer.
Section A · Level 1 · Year 5 (KS2)
Equivalent fractions, simplifying to lowest terms, comparing fractions.
Q1.
Complete the equivalent fractions: 1/2 = ?/4. Find the missing numerator.
Your answer:(2 marks)
To get from 2 (denominator) to 4 (new denominator), multiply by 2.
Do the same to the numerator: 1 × 2 = 2.
So 1/2 = 2/4. ✓
Q2.
Complete the equivalent fractions: 2/3 = ?/9. Find the missing numerator.
Your answer:(2 marks)
Denominator goes from 3 to 9, so multiply by 3.
Apply to numerator: 2 × 3 = 6.
So 2/3 = 6/9. ✓
Q3.
Complete the equivalent fractions: 3/4 = 6/?. Find the missing denominator.
Your answer:(2 marks)
Numerator goes from 3 to 6, so multiply by 2.
Apply to denominator: 4 × 2 = 8.
So 3/4 = 6/8. ✓
Q4.
Simplify 6/8 to lowest terms in the form ?/4. Find the missing numerator.
Your answer:(2 marks)
To simplify, divide top and bottom by their HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..
HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(6, 8) = 2.
6 ÷ 2 = 3. 8 ÷ 2 = 4.
So 6/8 = 3/4 in lowest terms. ✓
Q5.
Simplify 9/12 to lowest terms in the form 3/?. Find the missing denominator.
Your answer:(2 marks)
HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(9, 12) = 3.
9 ÷ 3 = 3. 12 ÷ 3 = 4.
So 9/12 = 3/4. ✓
Q6.
Simplify 10/15 to lowest terms in the form ?/3. Find the missing numerator.
Your answer:(2 marks)
HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(10, 15) = 5.
10 ÷ 5 = 2. 15 ÷ 5 = 3.
So 10/15 = 2/3. ✓
Q7.
Which is bigger: 1/2 or 1/3?
Your answer:(2 marks)
Same numerator (1), different denominator.
1/3 means dividing into MORE pieces, so each piece is SMALLER.
So 1/2 > 1/3. Answer: 1/2. ✓
Q8.
Which is bigger: 2/5 or 3/8?
Your answer:(3 marks)
Different denominators — find a common one. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(5, 8) = 40.
2/5 = 16/40 (multiply top and bottom by 8).
3/8 = 15/40 (multiply top and bottom by 5).
16/40 > 15/40, so 2/5 is bigger. ✓
Q9.
Which is the smallest: 1/4, 1/2, or 1/3?
Your answer:(2 marks)
Same numerator (1), so the one with the BIGGEST denominator gives the smallest pieces.
Denominators: 4, 2, 3. Biggest is 4.
So 1/4 is the smallest. ✓
Q10.
Find a fraction equivalent to 1/2 with denominator 10. Type just the numerator.
Your answer:(2 marks)
Denominator goes from 2 to 10, so multiply by 5.
Numerator: 1 × 5 = 5.
So 1/2 = 5/10. ✓
Section B · Level 2 · Year 6 (KS2)
Improper/mixed conversion, adding and subtracting fractions.
Q11.
Write 7/3 as a mixed number in the form A b/c. Find A (the whole numberA number with no fractional part: 0, 1, 2, 3, 4, … part).
Your answer:(2 marks)
Divide: 7 ÷ 3 = 2 remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. 1.
Whole numberA number with no fractional part: 0, 1, 2, 3, 4, … part = quotient = 2.
Fractional part = remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1./denominator = 1/3.
So 7/3 = 2 1/3. A = 2. ✓
Q12.
Write 11/4 as a mixed number. The whole numberA number with no fractional part: 0, 1, 2, 3, 4, … part is A. Find A.
Your answer:(2 marks)
11 ÷ 4 = 2 remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. 3.
Whole part = 2.
Fractional part = 3/4. So 11/4 = 2 3/4. ✓
Q13.
Write 2 1/2 as an improper fraction in the form a/2. Find a.
Your answer:(2 marks)
Multiply whole part by denominator, then add numerator.
2 × 2 + 1 = 4 + 1 = 5.
So 2 1/2 = 5/2. ✓
Q14.
Write 3 1/3 as an improper fraction in the form a/3. Find a.
Your answer:(2 marks)
3 × 3 + 1 = 9 + 1 = 10.
So 3 1/3 = 10/3. ✓
Q15.
Calculate 4/9 + 3/9. Type the numerator of the answer (denominator stays 9).
Your answer:(2 marks)
Same denominator — add numerators, keep denominator.
4 + 3 = 7.
So 4/9 + 3/9 = 7/9. (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(7,9)=1, already in lowest terms.) ✓
Q16.
Calculate 1/4 + 1/6. Express in the form ?/12. Find the numerator.
Your answer:(3 marks)
Common denominator: LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(4, 6) = 12.
1/4 = 3/12 (multiply top and bottom by 3).
1/6 = 2/12 (multiply top and bottom by 2).
3/12 + 2/12 = 5/12. (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(5,12)=1, lowest terms.) ✓
Q17.
Calculate 2/3 + 1/4. Express in the form ?/12. Find the numerator.
Your answer:(3 marks)
LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(3, 4) = 12.
2/3 = 8/12.
1/4 = 3/12.
8/12 + 3/12 = 11/12. ✓
Q18.
Calculate 5/6 - 1/3. Express in lowest terms as 1/?. Find the denominator.
Your answer:(3 marks)
LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(6, 3) = 6.
5/6 stays as 5/6. 1/3 = 2/6.
5/6 − 2/6 = 3/6.
Simplify: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(3, 6) = 3. 3/6 = 1/2. ✓
Q19.
Which is bigger: 3/4 or 5/8?
Your answer:(2 marks)
Convert to common denominator. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(4, 8) = 8.
3/4 = 6/8.
6/8 vs 5/8: 6 > 5, so 3/4 is bigger. ✓
Q20.
Calculate 1 1/2 + 2 1/3. Express as a mixed number A b/c. Find A.
Your answer:(4 marks)
Add whole parts: 1 + 2 = 3.
Add fractional parts: 1/2 + 1/3. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(2,3)=6. So 3/6 + 2/6 = 5/6.
Combine: 3 + 5/6 = 3 5/6.
So A = 3. ✓
Section C · Level 3 · Year 7 (KS3)
Multiplying, dividing, and operations with mixed numbers.
Q21.
Calculate 1/3 × 2/5. Express as ?/15. Find the numerator.
Your answer:(2 marks)
Multiply tops, multiply bottoms.
Top: 1 × 2 = 2.
Bottom: 3 × 5 = 15.
So 1/3 × 2/5 = 2/15. (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(2,15)=1, lowest terms.) ✓
Q22.
Calculate 2/3 × 3/4. Express in lowest terms as 1/?. Find the denominator.
Your answer:(3 marks)
Multiply: (2 × 3) / (3 × 4) = 6/12.
Simplify: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(6, 12) = 6. So 6/12 = 1/2. ✓
Q23.
Calculate 1/2 ÷ 3/4. Express in lowest terms as 2/?. Find the denominator.
Your answer:(3 marks)
Dividing by a fraction = multiplying by its reciprocal (keep-change-flip).
1/2 ÷ 3/4 = 1/2 × 4/3.
= (1 × 4) / (2 × 3) = 4/6.
Simplify: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(4, 6) = 2. 4/6 = 2/3. ✓
Q24.
Calculate 4/5 ÷ 2/3. Express as ?/5 in lowest terms. Find the numerator.
Your answer:(3 marks)
Keep-change-flip: 4/5 ÷ 2/3 = 4/5 × 3/2.
= 12/10.
Simplify: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(12, 10) = 2. So 6/5. ✓
Q25.
Calculate 2 1/2 × 1 1/3. Express as a mixed number A b/c. Find A.
Your answer:(4 marks)
Convert mixed to improper first.
2 1/2 = 5/2. 1 1/3 = 4/3.
5/2 × 4/3 = 20/6 = 10/3.
10/3 = 3 remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. 1, so 10/3 = 3 1/3.
A = 3. ✓
Q26.
Calculate 3 1/3 ÷ 5/9. Give the answer as a whole numberA number with no fractional part: 0, 1, 2, 3, 4, ….
Your answer:(4 marks)
Convert mixed to improper: 3 1/3 = 10/3.
Keep-change-flip: 10/3 ÷ 5/9 = 10/3 × 9/5.
= 90/15.
= 6. ✓
Q27.
Calculate 11/9 × 6/5. Express in lowest terms as ?/15. Find the numerator.
Your answer:(3 marks)
11/9 × 6/5 = 66/45.
Simplify: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(66, 45) = 3. 66 ÷ 3 = 22. 45 ÷ 3 = 15.
So 22/15. ✓
Q28.
Calculate 2/3 + 3/4 - 1/2. Express as ?/12. Find the numerator.
Your answer:(4 marks)
Common denominator: LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(3, 4, 2) = 12.
2/3 = 8/12. 3/4 = 9/12. 1/2 = 6/12.
8/12 + 9/12 − 6/12 = (8 + 9 − 6)/12 = 11/12. ✓
Q29.
Amy ate 3/8 of a pizza. What fraction is left? Express in the form ?/8. Find the numerator.
Your answer:(3 marks)
The whole pizza is 8/8.
8/8 − 3/8 = 5/8.
So 5/8 of the pizza is left. ✓
Q30.
A recipe needs 3/4 cup of sugar for 4 people. How many cups are needed for 12 people? Express as a mixed number A b/c. Find A (the whole numberA number with no fractional part: 0, 1, 2, 3, 4, … part).
Your answer:(4 marks)
12 people is 3 times as many as 4 people (12 ÷ 4 = 3).
So you need 3 times the sugar: 3/4 × 3 = 9/4.
Convert to mixed: 9 ÷ 4 = 2 remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. 1.
So 9/4 = 2 1/4. A = 2. ✓
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