Mathematics RU

Practice · Chapter 5

Operations with fractions

Adding, subtracting, multiplying and dividing fractions, mixed numbers and minus signs. The problems never run out: your answer is checked at once, and a hint and a full solution are there when you need them.

How to solve it

A fraction $\frac{a}{b}$ is $a$ pieces of a whole cut into $b$ equal parts. Every rule follows from that: only equal pieces can be added, so a common denominator comes first, while multiplying and dividing work straight away.

Step by step

  1. Turn a mixed number into an improper fraction: $2\tfrac{1}{3} = \frac{2 \cdot 3 + 1}{3} = \frac{7}{3}$.
  2. Adding and subtracting: find a common denominator, the least common multiple of the denominators. Multiply the top and the bottom of each fraction by the missing factor.
  3. Add or subtract the numerators and keep the common denominator.
  4. Multiplying: top times top, bottom times bottom. Dividing is multiplying by the flipped fraction.
  5. Reduce the answer: divide the numerator and the denominator by their greatest common divisor.
The denominator of the first fraction. The denominator of the second fraction. The extra factors: how many times the common denominator is larger than this fraction's denominator. The common denominator $m$, the least common multiple of $b$ and $d$. Example: $\frac{3}{4} + \frac{1}{6}$. The common denominator is $12$, the factors are $3$ and $2$: $\frac{9}{12} + \frac{2}{12} = \frac{11}{12}$.

Common mistakes

  • Adding the denominators: $\frac{1}{2} + \frac{1}{3} \ne \frac{2}{5}$. A half and a third together are more than a half, and $\frac{2}{5}$ is less.
  • Multiplying only the top or only the bottom changes the fraction. The factor always goes both up and down.
  • Flipping the wrong fraction when dividing: $\frac{1}{4} : \frac{7}{8} = \frac{1}{4} \cdot \frac{8}{7}$ — the divisor, the second fraction, is flipped.
  • Losing a minus: $-\frac{1}{2} + \left(-\frac{5}{9}\right)$ is a sum of two negatives, so the answer is negative too.
  • Not reducing the answer: $\frac{6}{8}$ is $\frac{3}{4}$.

Example

Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier