1/3  3/5 = 5/15 3/15 == (5 3)/15 |
a/m b/n = an/mn
bm/mn = = (an bm)/mn |
You can only divide natural integers when the dividend is a multiple of the divisor (24 : 6, 28 : 7). The tasks 13 : 7 or 12 : 5 are insoluble, because there will be a remainder. If 13 Baht are to be distributed between 6 persons, one Baht is left over. These tasks are not included in the calculus of fractions.
If objects can be subdivided, the situation changes. You can distribute one apple equally between 3 children; the apple is cut up and its parts are no longer called apple.
Fractions are memaningless for indivisible objects like persons, machines, hats, clothes, etc., while weights, masses, lengths, money, etc. can be subdivided. A 4 m (metre) long bar can be subdivided into 2 parts, but also into 8 parts. In other words, the 4 m long bar differs from 4 objects, because 4 m is a measured quantity; the number 4 tells the number of measuring units (1 m). When you have 4 objects, the number 4 indicates how many there are. A child counts its coins and is happy about their number; later on, it learns about the unit of the value of money, which can be subdivided.
From now on, it will be assumed that all quantities can be subdivided any number of times; we deal with the regime of divisible quantities and of fractions.
Thus, an object (1) will have 4 equal parts, called a quarter (1/4). You can now compute with parts as before with integers. Thus
1/4 + 1/4 + 1/4 = (1/4)·3 = 3/4.
a : b (say, a over b) means a parts 1/b.
The numerator a denotes the number of parts, the denominator b the magnitude of the part. Instead of a/1 you simply write a. Moreover,
a/a = 1.
Terminology
If a < b, then a/b is a proper fraction, if a > b, it is an improper fraction. 13/5 = 2 + 3/5 is a mixed number.
If you interchange in a fraction the numerator and denominator, you obtain its inverse; b/a is the reciprocal of a/b.
Rules of computation with fractions
You have learned in Arithmetic these rules for integers. Since general numbers represent integers, they also obey these rules.
Expansion and reduction of fractions
| 3/4 = 3·5/4·5 | a/b = a·m/b·m |
In the reverse direction, such formulae yield reduction of fractions.
You expand a fraction by multipying its
numerator and denominator by the same number.
You reduce a fraction by dividing its
numerator and denominator by the same number.
Expansions and reductions do not change the value of a fraction.
| 1. | ax/x = a | 2. | (a² - ab)/(a - b) = a·(a - b)/1·(a - b) = a/1 = a |
During reduction, you must ensure that the factor of the numerator equals the factor of the denominator; you must divide the numerator by a factor if the denominator is divided.You cannot reduce (3a + b)/(3a - b)!
| 3. | Expand 2a/bc to the denominator b²c² | 2a/bc = 2abc/b²c² |
You only can compare fractions, if either the numerators or the denominators are equal. A fraction is larger than another fraction, if the denominators are equal and its numerator is larger, or if the numerators are equal and its denominator is smaller.
| 3/4 > 1/4 | 3/5 > 3/7 |
| 3/8 + 2/8 = 5/8 | a/c + b/c = (a + b)/c | |||
| 3/8 - 2/8 = 1/8 | a/c - b/c = (a - b)/c |
Fractions with equal denominators are added (subtracted) by addition (subtraction) of the numerators and retention of the denominators.
a/n + b/n - c/n - d/n= (a + b - c - d)/n
| 1/3 3/5 = 5/15 3/15 = = (5 3)/15 |
a/m b/n = an/mn
bm/mn = = (an bm)/mn |
Fractions with different denominators are added (subtracted) by giving them the same denominator and then adding (subtracting) their numerators.
a/x - b/y + c/z=ayz/xyz - bxz/xyz + cxy/xyz=(ayz - bxz + cxz)/xyz.
You find the common denominator of fractions in the same way as for integers. For example, you decompose in 3/8+5/20+3/10+2/5 the denominators into prime factors:
| 8 = 2·2·2 = 2³ | The highest power | ||
| 20 = 2·2·5=2²·5 | of 2 is 2³ | ||
| 10=2·5 | 3 and 5 occur | ||
| 5 = 5 | only to power 1 | ||
| N = 2·2·2·5·3 = 2³·3·5 = 120 |
The
common denominator is the product of the highest powers
of the factors of the denominators
Hence
3/8 + 5/20 +
3/10 + 2/3 = 45/120 + 30/120 + 80/120 =
= (45 + 30 + 36 + 80)/120) = 191/120 = 171/120.
For fractions in terms of general numbers, you determine the common denominator in the same manner.
Examples:
a/(a² - ab) - b/(a² - b²) + c/(a² - ab).
| a² - ab = a(a - b) | The first fraction | ||
| a² - b² = (a + b)(a- b) | must be multiplied by | ||
| a² + ab = a(a + b) | a + b, the second by a, | ||
| the third by a - b. | |||
| N = a(a - b)(a + b) = a(a² - b²) |
a(a+b)/N-ba/N+c(a-b)/N = (a²+ab-ab+ac-bc)/N = (a²+ac-bc)/a(a²-b²).
| 1/(x - 2) + 1/(x - 5) + 1/(x - 4) |
These denominators cannot be decomposed. The common denominator is
N = (x - 2)(x - 5)(x - 4).
If the denominators do not have common factors, the common denominator is the product of the individual denominators.
Exercises: Simplify the expressions:
| 1. | 12ap²x(x - 4y)/18a²px²(x - 4y) | |
| 2. | (3x²z - 4x²y)/(2x²y - 2x²z) | |
| 3. | (4p²q²r² - 12pqr + 6p²q²r)/(18pqr - 24pqr² + 6pqr) |
| (3/7)·5 = (3·5)/7 = 15/7 | (a/n)·b = (a·b)/n |
You multiply a fraction by a number by multiplying its numerator by it.
| (3/4)·(5/7) = (3·5)/(4·7)=15/28 | (a/m)·(b/n) = (ab)/(mn) |
The
product of fractions is the fraction of the products
of the numerators and denominatos.
| (3/4)·(4/3)=1 | (a/b)·(b/a)=1 | The product of a fraction by its inverse is 1. |
Examples:
| 1. | [(2x)/(3y)]·y=(2xy)/3y=(2x)/3 | 2. | [(2x)/(3y)]·[(3y)/(4x)]=(2x·3y)/(3y·4x)=1/2 |
| 3. | [(a+b)/4·(a-b)]·{[3(a²-b²)/2a}]={(a+b)·3·(a+b)(a-b)}/[4(a-b)·2a]=3(a+b)²/8a |
| (3/4)/5 = 3/(4·5) | (a/n)/m = a/n·m |
You
divide a fraction by a number by multiplying
its denominator by that number.
| (3/8)/(5/7)=(3/8)·(7/5)=(3·7)/(8·5)=21/40 | (a/b)/(m/n)=(a/b)·(n/m)= (an)/(bm) |
The
fraction of two fractions is the product of one fraction
by the other fraction's inverse.
Examples:
| 1. | (2a/b):4a=2a/b·4a=1/2b | |
| 2. | [(a²-b²)/3]:[(ab)/3]=[(a²-b²)/3]·[3/(ab)]=[(a²-b²)·3]/[3(a+b)]=a-b | |
| 3. | (3/1):5 = 3/(1·5) = 3/5 since 3/1 = 3 |
All rules for fractions are valid, if numerators and denominators contain fractions. They are double fractions. As a rule, you will reduce them to simple fractions.
(7/3)/(4/5) - (1/6)/(1/7) = 35/12 - 7/6 = 35/12 - 14/12 = 21/12 = 7/4 = 13/4
1. (x + 2)/5 - (2x - 3)/2 = 0.2x - 0.6.
You multiply both sides of an equation by the common denominator:
| 2(x + 2) - 5(2x - 3) = 10(0.2x - 0.6) | 2x + 4 - 10x + 15 = 2x - 6 | x = 2½ |
2. 2/(x + 1) - 1/(x - 1) =1 /(2x + 2)
Multiply both sides by the common denominator N = 2(x + 1)(x - 1)
| 4(x - 1) - 2(x + 1) = x - 1 | 4x - 4 - 2x - 2 = x - 1 | x = 5 |
Check: L: 2/(x + 1) - 1/(x - 1) = 2/6 - 1/4 = 1/12; R: 1/(2x + 2) = 1/12.
Both sides yield the same value!
Equations
with fractions often become simpler on
multiplication by the common denominator.
Exercises: Give these fractions common denominators:
| 1. | 5/2ab² | 3/2a²b | ||
| 2. | 2x/3(x - y) | 7x/2y | ||
| 3. | 2p/(p² - q²) | 3q/(p + q) |
Reduce these fractions:
| 4. | (9a² - 25b²)/(9a² - 30ab + 25b²) | |
| 5. | (4a² - 12ay + 9y²)/(4a² - 9y²) |