Liri had two map sections: one 14 in size, another one 14 in size. “How much is that altogether?”
Erklärus: “With the same denominator it is very simple: add the numerators, the denominator stays the same. 14+14=24.”
Putting parts together
➕ Addition and subtraction
Same denominator: only add or subtract the numerators, the denominator stays unchanged.
➕ Adding fractions with the same denominator
27+37=57
the denominator stays 7
58−18=48simplify →12
When simplifying you divide the numerator AND the denominator by the same number – best of all straight away by the biggest number that goes into both. For 48 that is 4:
48÷4 on top, ÷4 on the bottom →4÷48÷4=12
Different denominators: first bring them to a common denominator (LCM), then add or subtract.
➕ Fractions with different denominators – common denominator
12+13
→ LCM(2,3) = 6
=36+26=56
12 scaled up by 3 → 36 · 13 scaled up by 2 → 26
Work out: 25 + 15 = ?/5
✖️ Multiplication of fractions
Multiplying is easier than adding: numerator times numerator, denominator times denominator – no common denominator needed!
✖️ Multiplying fractions
23×34=2×33×4=612simplify →12
35× 4 =35×41=125
whole number = number/1
Work out: 13 × 12 = 1/?
➗ Division of fractions
Dividing by a fraction means: multiply by the reciprocal (numerator and denominator swapped)!
➗ Dividing fractions
34÷25→ reciprocal →34×52=158
Liri: “So dividing becomes multiplying if I simply turn the fraction upside down?”
Erklärus: “Exactly – ‘dividing by a fraction means multiplying by the reciprocal’.”
🧮 Liri’s fraction calculator – interactive
Enter two fractions and choose an operation – the steps appear automatically:
🪐 Orbital fraction calculator
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🌌 Fractions in the universe
A spaceship that needs 12 of its fuel for the outward flight and 13 for the return flight has 16 left over as a reserve.
Astronomers constantly calculate with fractions of orbital periods to predict when two heavenly bodies will meet again.
❌ Common mistake
In addition the denominators are often added too: 14+14 wrongly becomes 28 instead of the correct 24!
Tip: when the denominators are the same, the denominator stays UNCHANGED – only the numerators are added.
➕ Exercise 1
Work out: 37 + 27 = ?/7
✖️ Exercise 2
Work out: 23 × 14 = ?/12
➗ Exercise 3
Work out: 13 ÷ 16 = ?
All the parts put together
Liri put her map sections together – fraction by fraction, until a bigger picture appeared.
Liri: “Now I can calculate every part of my journey with every other part, no matter how big the pieces are.”
“Addition/subtraction: a common denominator is needed, then work with the numerators. Multiplication: numerator×numerator, denominator×denominator. Division: multiply by the reciprocal.”
What you learned today
12 × 23 = 13 · 12 ÷ 14 = 2
Times: numerator×numerator, denominator×denominator · Dividing: multiply by the reciprocal
🌟
Chapter 32 mastered!
Liri can now add, subtract, multiply and divide fractions.
Erklärus’s notebook – page 32:
“Addition/subtraction: same denominator, then work with the numerators. Multiplication: numerator times numerator, denominator times denominator. Division: multiply by the reciprocal.”
“Addition/subtraction: same denominator, then work with the numerators. Multiplication: numerator times numerator, denominator times denominator. Division: multiply by the reciprocal.”