What is the formula for calculating ratios

Proportions and proportions

What is a relationship?
Let two real, positive numbers a and b be given.
The quotient a: b of the two numbers is also called the ratio of the two numbers a and b in certain contexts.
One reads a: b then as "a to b".
Two examples to illustrate this


A lot of bullets
......Given are 5 * 9 = 45 balls, 20 red and 25 blue.
The ratio of the red balls to the blue ones is 20:25 = 4:5, read 4 to 5.

The drawing contains further ratios in which the whole (45 balls) is taken into account.
The ratio of the red balls to all balls is 20:45 = 4:9.
The ratio of the blue balls to all balls is 25:45 = 5:9
Division of a route
......The route AB is given. The point T divides the distance in proportion 2:1.
This is because a = 2 units of length and b = 1 unit of length.

What is a proportion?
Let two more real, positive numbers c and d be given.
The equation a: b = c: d from two ratios is called proportion or ratio equation.
You read that as a is to b as c is to d.
Two examples to illustrate this


1. Theorem of rays
......If one draws the perpendicular to the segment AB through the points B and T and through point A a half-line as on the left, then the segments c and d arise.

The ratio equation applies a: b = c: d.


Rule of three
5 kg of flour cost € 6.40. What does 300g cost?
solution
The sought costs are x €.
The proportion 5kg applies: 300g = € 6.40: x € or 5000: 300 = 640: x. Then 5000x = 300 * 640 or x = 38.4.
answer
300g flour costs 38.4 ct.

When creating my web pages, I often came across proportions and proportions.
On this page I have explained the topic and put together what I found on my homepage.

Arithmetic with circumstances Top
Transformations
In the first example above, it was found that the ratio of the red balls to the blue ones is 20:25. It is customary to shorten it and indicate the ratio with whole numbers, 20:25 = 4: 5.
You can calculate the ratio, 4: 5 = 0.8. Then 4: 5 = 0.8: 1.
It's also 4: 5 = 1: (5/4) = 1: 1.25. - Furthermore, 4: 5 = 80% = 80: 100.

Generally
For the ratio a: b with the abbreviated fraction a / b, the identities (a / b): 1 = 1: (b / a) = (100a / b)% also apply.


Product equation
If the ratio equation a: b = c: d applies, so does the product equation ad = bc.
Derivation
If the quotients are understood as fractions (a / b = c / d) and both terms are multiplied by the denominators b and d, then ad = bc results.


The reverse is true.
If the product equation ad = bc applies, then also the ratio equation a: b = c: d.
Derivation
Dividing both terms of the product equation ad = bc by bd results in the relationship equation a: b = c: d.

The product equation is described as follows:
In one proportion, the product of the outer links is equal to the product of the inner links.

All ratio equations
If a: b = c: d, there are still seven variants.
 
a: b = c: db: a = d: ca: c = b: dc: a = d: bc: d = a: bd: c = b: ab: d = a: cd: b = c: a

All proportions have the same product equation ad = bc.


Sums and differences
If a: b = c: d, there are five variants with sums and differences.
 
(a + b): b = (c + d): d
a: (a + b) = c: (c + d)
(a-b): b = (c-d): d
a: (a-b) = c: (c-d)
(a + b) :( a-b) = (c + d) :( c-d)

The equations can be simplified so that one always arrives at the product equation ad = bc.


Determination of proportions
One calls the variables in a proportion a: b = c: d the proportional. There are two search equations.
These are the determination of the fourth proportional in a: b = c: x and that of the middle proportional in a: x = x: d.
Interpreted geometrically, it is about the calculation of a rectangle that has the same area as a given rectangle, and the other is the calculation of a square with the same area as the given rectangle.

Proportions chain
If a: b = d: e and b: c = e: f, then a: c = d: f also applies.
proof
From a: b = d: e follows ae = bd, from b: c = e: f follows bf = ce.
From ae = bd and bf = ce or ce = bf, after dividing the right and left terms a: c = d: f, wzbw ..

The proportions a: b = d: e, b: c = e: f and a: c = d: f are combined to form a: b: c = d: e: f.
It is called a chain of proportions or continuous proportion.

Figures and bodies Top
There are simple figures or bodies, the shape of which is determined by a ratio of numbers.
A4 paper size
The A-series paper format is known. The different sheets are obtained by starting from a sheet of 1m² area with the aspect ratio 1: sqrt (2) and by continually halving over A1, A2, ... to arrive at the usual A4 format.

More on my A4 paper page


Rectangles
The following is a compilation of rectangles with different aspect ratios.
1) square, 2) 3-4-5 triangle and its reflection, 3) A-row paper format, 4) rectangle around two interlinked squares, 5) golden rectangle, 6) 30-60-90 triangle and its reflection or the equilateral triangle - reassembled, 7) double square

Taken from my side rectangle

Shapes of a cylinder
......
Two of the three cylinders are slightly alienated.
On the one hand the height h is significantly larger than the circle diameter, on the other it is significantly smaller.
The cylinders are then called rods or disks.
The special thing about the third body is that it is as high as it is wide.
The shape of a cylinder can be determined by the ratio of the circle diameter d = 2r to the height h, namely by d / h describe.
In the drawing, for example, d / h = 0.1 for the rod and d / h = 26 for the disc. A cylinder with d / h = 1 is noticeable. This is the form of the original kilogram in Paris and the copies of the original kilogram in Braunschweig at the Physikalisch-Technische Bundesanstalt (PTB).

Taken from my side cylinder

Ray theorems Top
In geometry, the proportions have the correspondence of the ray laws.
Let two intersecting straight lines and a pair of parallels be given. Sections then arise on the straight lines or on the parallels, which can be related in two ways.


1. Theorem of rays
......

The following applies to the sections on the straight line: a: b = c: d. That is the first sentence of rays.

In words:
If two intersecting straight lines are met by two parallels, the sections on one straight line behave like the sections on the other.


2. Theorem of rays
......
If the parallel sections are included, a: (a + b) = c ': d' applies. That is the second law of rays.

In words:
If two intersecting straight lines are met by two parallels, the sections on the parallels behave like the sections on a straight line. The sections are measured from the intersection. (This addition is important. The relationship a: c '= b: d' is a common mistake.)


......It should be noted:
The parallels can also be on different sides to the intersection of the straight lines.

The ray sentences can also be read there.


The following figure is a variation of the second law of rays and is sometimes also called the third law of rays.
......
The proportion p: q = r: s applies.
In words:
If three straight lines intersecting at one point are met by two parallels, the sections on one parallel behave like the sections on the other.

The formula follows from a: (a + b) = p: r and a: (a + b) = q: s.


Similar triangles
......The figure for the ray theorems can also be seen differently.

It consists of the two triangles ADE and ABC with a common angle and two equal angles. The triangles are therefore similar to each other.
The following applies to similar triangles: Corresponding sides are in the same ratio.
For example, a: c = (a + b) :( c + d). From this it follows a: b = c: d. That is the first sentence of rays.

Observation: The ray sentences can also be replaced by sentences about similar triangles.


One can go further. Similar triangles are generally located not in perspective, so that the figure for the theorem of rays is no longer there.
For example, the diagonal in a regular pentagon can only be determined using similar triangles.

......With the help of the angles you can see that the two yellow triangles are isosceles and similar. Then (d-a): a = a: d or d (d-a) = a². This quadratic equation in d has the (positive) solution d = [1 + sqrt (5)] / 2 * a.

Taken from my page Regular Pentagon

Nevertheless: The ray theorem is of great help in solving geometric problems. I could refer to many pages on my homepage.
I want to limit myself to the side of extreme value problems with constraints. Of the 39 exercises, seven are solved with the help of a ray theorem, namely exercises 02, 03, 19, 24, 26, 31 and 32.

Division of a route Top
Basic task
Divide the distance AB in a ratio of 2: 3.

......> Enter the point C as you like and connect it to point A
> Plot a unit distance from A five times. You get the points P and Q.
> Draw the line BQ.
> Draw the parallel to BQ through point P.
> The point of intersection with the segment AB is the partial point T sought.
The construction is explained by the first theorem of rays.

Harmonic division
 
......The point T divides the segment AB in a certain ratio, namely k = AT: TB. The point T is called the inner subpoint.
......There is a second point U, for which k = AU: UB also applies. It is called the outer subpoint.

You can find it through the following drawing.

......> Choose point C and connect it to point A.
> Draw the line CT.
> Draw the parallel to AC through B. Call the intersection with CT point D.
> Plot the distance BD from B. Call the intersection point D '.
> Draw the line through C and D '. Call the intersection with the straight line AB point U.
The ratio k = AU: UB can be seen in this way.
The following applies: AT: TB = AC: DB (after the 2nd beam set)
AU: UB = AC: BD '(after the 2nd law of rays)
BD = BD '(according to construction)
Conclusion: AT: TB = AU: UB = k
They say: The points T and U share the segment AB harmoniously
or dhe points A, T, B and U are harmonic points.

The following sentence also applies:
The UT segment is the harmonic mean of the AU and BU segments. In the formula language this is called UT = (2AU * BU) / (AU + BU).
A calculation can be found on my page Classic averages.

Continuous division
The continuous division is a special division of a route.
 
......A segment s = AB is continuously divided by the point T if the entire segment is related to the larger section as it is to the smaller section.

In formula language this is called AB:AT = AT:TB or s: x = x: (s-x). This is equivalent to x² + sx-s² = 0.


The name continuous division results from the following property.
If the smaller section s-x is plotted on the larger section x, this section x is also continuously divided by the new subpoint.
Proof:
We have x:(s-x) = (s-x): [x- (s-x)] or x:(s-x) = (s-x):[2x-s].
From this follows the product equation x (2x-s) = (s-x) ² or 2x²-sx = s²-2sx + x² or x² + sx-s² = 0. That's the quadratic equation from above. So it's the same division.
The division can be repeated any number of times, hence the name.

The continuous division is also called the division according to the golden section.
More on my Golden Ratio page.

Proportional sizes Top
This topic is dealt with using the exercise above.
5 kg of flour cost € 6.40. What does 300g cost?
1
The task was solved via a proportion.
The sought costs are x €.
The proportion 5kg applies: 300g = € 6.40: x € or 5000: 300 = 640: x. Then 5000x = 300 * 640 or x = 38.4.
answer
300g flour costs 38.4 ct.


2
Classically, the problem is solved according to the rule of three.
5 kg cost € 6.40
0.3 kg cost x €
------------------------
5 kg cost € 6.40
1 kg costs 6.40 / 5 €
0.3 kg costs (6.40 / 5) * 0.3 €
-----------------------------------
x = (6.40 / 5) * 0.3 = 0.384

Answer: 300g flour costs 38.4 ct.

The first two lines contain the task. The first line is what you know, the next is what you want to know. It must be ensured that the search variable x is on the right.
Then come the three lines that explain the name Dreisatz. The first line is taken from above. In the second line one goes from the majority to the unity, in the third from the unity to the majority.
Finally, the invoice is issued.


There are two quantities in the exercise, namely the set M and the costs K, which are assigned to the set.
The allocation is special: quantity M and costs K are proportional to each other (abbreviated M ~ K), i.e. the costs increase with the quantity. To put it more precisely: The quotient P from the costs K and the quantity M is constant
The equation K = P * M applies, where P is the proportionality factor.
1 kg of flour costs € 1.28, K / M = € 1.28 / 1kg = € 1.28 / kg. So P can be interpreted as a price.

3
Today in schools the task is usually solved with a table. Once the procedure has been practiced, a short table is sufficient.

With the help of a table of values, the graph of the allocation of costs K to quantity M is created.
This is a function with K = P * M.

various Top
Side bisector in a triangle

......The bisectors in any triangle divide in a ratio of 2: 1.

AS: AMa = BS: SMb = CS: SMc = 2:1
 


More on my General Triangle page

Sine law
......
In any triangle, the sides are in the same proportion to the sine values ​​of the opposite angles.
This is summarized in a chain of proportions to a: b: c = sin (alpha): sin (beta): sin (gamma).

A definition of the ellipse
......If P is an arbitrary point on the ellipse, and if one connects it to a focal point and draws the perpendicular to it to the "guideline" 1, then PP applies1:PF2= a:e.
This leads to the definition: The ellipse is the geometric location of all points for which the ratio of their distance d from a guide line to their distance s from the focal point assumes a constant value.

More on my Ellipse page

Potentiation in homeopathy
In homeopathy there are remedies with a "reinforcement" like D8.
D8 corresponds to a dilution of 1:100.000.000.
That means, 1cm³ of active ingredient comes to 100,000,000 cm³ = 100,000 liters = 1,000hl = 100m³ = 4m * 5m * 5m.
The last product can be understood as the volume of a cuboid with a length of 5m, a width of 5m and a height of 4m.
A 1cm³ cube is tiny in this huge space. - No comment.

More on my page Geometric Sequences and Rows

Epicycloids
A circle with radius r rolls on a solid circle with radius R. A point on the rolling circle describes a curve, the epicycloid. Only when the ratio R: r is rational is the curve self-contained.

R: r = 7: 2

R: r = 7: sqrt (2)

More on my Epicycloids page

Trigonometric functions
......Six ratios can be seen from the right triangle.
They lead to the trigonometric functions. The following applies to acute angles
sin (alpha) = a / c, cos (alpha) = b / c, sec (alpha) = c / b, csc (alpha) = c / a, tan (alpha) = a / b and cot (alpha) = b / a.

3-4-5 triangle
......The aspect ratio is a: b: c = 4: 3: 5.
The 3-4-5 triangle is right-angled because the Pythagorean theorem applies: 3² + 4² = 5².

More on my 3-4-5 triangle page

Isosceles trapezoid
...A diagonal divides the trapezoid into two triangles.

The proportion A applies to the areasa: A.c= a:c.


...The diagonals divide the trapezoid into four triangles.

The proportions chain A applies to the areas1:A.2:A.3:A.4=::ac:ac.


More on my trapeze page

Three similar triangles
......The height divides the right-angled triangle into two triangles similar to the starting triangle, since the corresponding angles match.
The proportions chain A applies to the areas1: A2: A3 = c²: a²: b²
More on my right triangle page

Proportions and proportions on the Internet Top

German

Wikipedia
Quotient, theorem of rays, similarity (geometry), rule of three, harmonic division, double ratio, proportionality,
furthermore the potato paradox

English

Wikipedia
Quotient, Ratio, Intercept theorem, Similarity (geometry), Cross-multiplication, Projective harmonic conjugate, Proportionality (mathematics),
also: Potato paradox


credentials    Top
W. Gellert (editor, among others): Small Encyclopedia Mathematics, Leipzig 1986


 Feedback: Email address on my main page

URL of my homepage:
http://www.mathematische-basteleien.de/

© 2015 Jürgen Köller

Top