What are some examples of constant proportionality

Proportional allocation

In this chapter we look at what a proportional mapping is.
[Alternative name: direct proportionality]

Understanding this topic requires that you have already read the Assignment lesson. There the necessary basics are explained in detail.

A Assignment uniquely assigns another value to a value.

In mathematics, the following arrow is used to describe assignments: \ ({\ fcolorbox {Red} {} {\ (\ longmapsto \)}} \).

General example

\ (x \ longmapsto y \)
(say: \ (x \) is uniquely assigned to \ (y \))

Here, \ (x \) is referred to as the initial value and \ (y \) as the assigned value.

example 1

1 kg of apples costs 2 euros. 2 kg of apples cost 4 euros ... etc.

The price can be clearly assigned to the number of apples:
\ (\ text {Amount of apples} \ longmapsto \ text {Price of apples} \)

\ (1 \ longmapsto 2 \)
\ (2 \ longmapsto 4 \)
\ (3 \ longmapsto 6 \)
\ (4 \ longmapsto 8 \)
...

Example 2

It takes 1 gardener 6 minutes to mow a specific area of ​​lawn.
If 2 gardeners help together, it only takes them 3 minutes ... etc.

The number of gardeners can be clearly assigned to the working hours:
\ (\ text {Number of gardeners} \ longmapsto \ text {Working hours} \)

\ (1 \ longmapsto 6 \)
\ (2 \ longmapsto 3 \)
\ (3 \ longmapsto 2 \)
\ (4 \ longmapsto 1.5 \)
\ (5 \ longmapsto 1,2 \)
\ (6 \ longmapsto 1 \)
...

Take a closer look at the two examples. Can you tell differences?

Difference 1

  • In example 1: The more Apples, the more You have to pay money.
  • In example 2: The more Gardener, the less Time is needed.

Difference 2

  • Example 1 has one Zero point.
    0 apples cost 0 euros: \ (0 \ longmapsto 0 \).
  • Example 2 owns no zero point.
    It doesn't make sense that it would take 0 gardeners 0 minutes to mow the lawn.

Conclusion

\ (\ Rightarrow \) Example 1 is a proportional assignment.
\ (\ Rightarrow \) Example 2 is an anti-proportional assignment.

Since this chapter is about proportional assignments, let's look at example 1 in more detail. Antiproportional assignments are discussed in the next chapter.

Properties of a proportional mapping

The assignment from example 1 is given

\ (1 \ longmapsto 2 \)
\ (2 \ longmapsto 4 \)
\ (3 \ longmapsto 6 \)
\ (4 \ longmapsto 8 \)

We recognize: As the left value increases, the right value also increases. This enlargement proceeds evenly, i.e. if we double the left value, the right value also doubles; if we triple the left value, the right value triples too ... etc.

Example 1 (continued 1)

1 kg of apples costs 2 euros.
\ (1 \ longmapsto 2 \)

If we double the weight of the apples, the price also doubles.
\ ({\ color {green} {2}} \ cdot 1 \ longmapsto {\ color {green} {2}} \ cdot 2 \)

If we triple the weight of the apples, the price triples too.
\ ({\ color {green} {3}} \ times 1 \ longmapsto {\ color {green} {3}} \ times 2 \)

We can hold on to: Describe proportional assignments even growth.

The following property can be derived from this:

The quotient of the assigned value (\ (y \)) and the initial value (\ (x \)) is always the same.

For a proportional assignment \ (x \ longmapsto y \) the following applies:

\ (y: x = \ text {constant} \)

They say: The number pairs \ (x \) and \ (y \) are quotient equal.

Exception:
The quotient is not defined for the "zero point" \ (0 \ longmapsto 0 \).
Dividing by zero is not allowed!

The quotient of the assigned value (\ (y \)) and the initial value (\ (x \)) is called Proportionality factor.

For a proportional assignment \ (x \ longmapsto y \) the following applies:

\ (y: x = \ text {proportionality factor} \)

Example 1 (continued 2)

If we divide the assigned value by the initial value,

\ (1 \ longmapsto 2 \ qquad \ qquad 2: 1 = {\ color {green} {2}} \)
\ (2 \ longmapsto 4 \ qquad \ qquad 4: 2 = {\ color {green} {2}} \)
\ (3 \ longmapsto 6 \ qquad \ qquad 6: 3 = {\ color {green} {2}} \)
\ (4 \ longmapsto 8 \ qquad \ qquad 8: 4 = {\ color {green} {2}} \)

we find out that the same value always comes out.

This value (here: 2) is called the proportionality factor of the assignment.

If you know the proportionality factor, the assigned value (\ (y \)) can be expressed as a function of the initial value (\ (x \)).

Derivation:

\ (y: x = \ text {proportionality factor} \ qquad | \ cdot x \)

\ (y = \ text {proportionality factor} \ cdot x \)

One can therefore rewrite \ (x \ longmapsto y \) to

\ ({\ fcolorbox {Red} {} {\ (x \ longmapsto k \ cdot x \)}} \)
Here \ (k \) is the proportionality factor.

Example 1 (continued 3)

\ (1 \ longmapsto {\ color {green} {2}} \ cdot 1 \)
\ (2 \ longmapsto {\ color {green} {2}} \ cdot 2 \)
\ (3 \ longmapsto {\ color {green} {2}} \ cdot 3 \)
\ (4 \ longmapsto {\ color {green} {2}} \ cdot 4 \)

Knowing this, we can finally determine when an assignment is proportional.

A mapping is called \ (x \ longmapsto y \) proportional,
if every \ (y \) -value is obtained by multiplying the \ (x \) -value by the same number (proportionality factor).

In mathematical terms:
\ (x \ longmapsto k \ cdot x \)
Here \ (k \) is the proportionality factor.

Representation of proportional assignments

There are essentially four ways to represent a proportional assignment.

  1. Arrow diagram
  2. Assignment table (= table of values)
  3. Coordinate system
  4. Mathematical rule (= assignment rule)

Let's look at an example for each representation.
The following assignment is involved:

- 0 kg of apples cost 0 euros
- 1 kg of apples costs 2 euros
- 2 kg of apples cost 4 euros
- 3 kg of apples cost 6 euros
- 4 kg of apples cost 8 euros

1. Arrow diagram

We already got to know the arrow diagram above.

\ (0 \ longmapsto 0 \)
\ (1 \ longmapsto 2 \)
\ (2 \ longmapsto 4 \)
\ (3 \ longmapsto 6 \)
\ (4 \ longmapsto 8 \)

The number to the left of the arrow is the initial value, the right number is the assigned value.

2. Allocation table (= table of values)

Allocation tables can be displayed both horizontally and vertically. Which representation you choose is up to you. The best way to orientate yourself is to use the representation your teacher uses.

A (horizontal) allocation table has two rows. The output values ​​are in the top row and the assigned values ​​in the bottom row.

\ (\ begin {array} {r | r | r | r | r | r}
\ text {Initial value} & 0 & 1 & 2 & 3 & 4 \ \ hline
\ text {assigned value} & 0 & 2 & 4 & 6 & 8 \
\ end {array} \)

A (vertical) allocation table has two columns. The output values ​​are in the left column and the assigned values ​​in the right column.

\ (\ begin {array} {l | l |}
\ text {initial value} & \ text {assigned value} \ \ hline
0 & 0 \\
1 & 2 \\
2 & 4 \\
3 & 6 \\
4 & 8 \\
\ end {array} \)

Often one also calls an assignment table simply a table of values.

3. Coordinate system

If you look at a piece of squared paper ...

... draw two straight lines that are perpendicular to each other, you get a coordinate system. These straight lines are then called coordinate axes. It is important that you label the coordinate axes correctly (see illustration).

The horizontal coordinate axis stands for the initial values, the vertical coordinate axis for the assigned values ​​of the assignment.

The following assignment is given
\ (1 \ longmapsto 2 \)

How can we graphically represent this assignment?
The assignment corresponds to a point in the coordinate system. This point is obtained by going one unit to the right and two units up from the coordinate origin.

The graphic representation of the assignment
\ (0 \ longmapsto 0 \)
\ (1 \ longmapsto 2 \)
\ (2 \ longmapsto 4 \)
\ (3 \ longmapsto 6 \)
\ (4 \ longmapsto 8 \)
from our example is shown in the adjacent figure.

When we connect the dots, we see:

The Proportional assignment graph is a rising half-line through the zero point.

4. Mathematical rule (= assignment rule)

With the help of a mathematical rule, the second value can be calculated from the first value. In the case of assignments, this mathematical rule is called an assignment rule.

For proportional assignments, the assignment rule is:

\ (y = \ text {proportionality factor} \ cdot x \)

or

\ (y = k \ cdot x \)

if \ (k \) stands for the proportionality factor.

example

Check that the following assignment is proportional.
If necessary, enter an assignment rule!

\ (\ begin {array} {r | r | r | r | r | r}
x & 1 & 2 & 3 & 4 & 5 \ \ hline
y & 3 & 6 & 9 & 12 & 15 \
\ end {array} \)

To check whether an assignment is proportional, divide the values ​​in the bottom line by the values ​​in the top line. If the same number comes out, the allocation is proportional.

\ (\ begin {align *}
3:1 &= 3 \\
6:2 &= 3 \\
9:3 &= 3 \\
12:4 &= 3 \\
15:5 &= 3 \\
\ end {align *} \)

Since dividing the lower row by the upper row results in the same value, the assignment is proportional. The result of the divisions (here: 3) is then the proportionality factor.

The assignment rule is generally:

\ (y = \ text {proportionality factor} \ cdot x \)

or in this case

\ (y = 3 \ cdot x \)

The assignment rule \ (y = 3 \ cdot x \) helps us to calculate the \ (y \) value if an \ (x \) value is given.

For example, if \ (x = 20 \), \ (y \) is calculated as follows

\ (y = 3 \ times 20 = 60 \)

The other way around, of course, works the same way!

For example, if \ (y = 90 \) applies, \ (x \) is calculated as follows

\ (90 = 3 \ cdot x \ qquad |: 3 \)

\ (30 = x \) or \ (x = 30 \)

You can find out more about this topic in the chapter on assignment rules.

Proportional or anti-proportional?

It is often necessary to distinguish a proportional assignment from an anti-proportional assignment. Some differences are listed in the following overview.

 Proportional assignment
(Direct proportionality)
Proportional allocation
(Indirect proportionality)
importanceEven growthOpposite growth
Memorandum"The more the more""The more the less"

example

\ (\ begin {array} {r | r | r | r | r | r | r}
x & {\ color {green} 0} & {\ color {green} 1} & {\ color {green} 2} & {\ color {green} 3} & {\ color {green} 4} & {\ color {green} 5} \ \ hline
y & {\ color {green} 0} & {\ color {green} 2} & {\ color {green} 4} & {\ color {green} 6} & {\ color {green} 8} & {\ color {green} 10} \
\ end {array} \)
\ (\ begin {array} {r | r | r | r | r | r | r}
x & {\ color {green} 1} & {\ color {green} 2} & {\ color {green} 3} & {\ color {green} 4} & {\ color {green} 5} & {\ color {green} 6} \ \ hline
y & {\ color {red} 12} & {\ color {red} 6} & {\ color {red} 4} & {\ color {red} 3} & {\ color {red} 2,4} & { \ color {red} 2} \
\ end {array} \)

graph

Rising straight line
through the zero point

Hyperbola falling from top left to bottom right

Assignment
regulation

\ (x \ longmapsto k \ cdot x \)
\ (k \) is called the proportionality factor.

\ (x \ longmapsto k \ cdot \ frac {1} {x} \)
\ (k \) is called the anti-proportionality factor.

property

The number pairs \ (x \) and \ (y \) are quotient equal, i.e.
\ (y: x = \ text {constant} \)

The number pairs \ (x \) and \ (y \) are product-equal, i.e.
\ (x \ cdot y = \ text {constant} \)

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