Theorem of Ratios Between Homogeneous Quantities

The ratio between two homogeneous quantities, A and B, is defined as the measure of A using B as the unit of measure. $$ r = \frac{A}{B} \Longleftrightarrow A = r \cdot B $$ If B is not zero, the ratio A/B is also equal to the ratio of their measures, M(A)/M(B), with respect to any unit of measure. $$ \frac{A}{B} = \frac{M(A)}{M(B)} $$

This relationship is fundamental in mathematics, geometry, and physics.

When dealing with two homogeneous quantities, A and B, the ratio between them, A/B, is essentially the measurement of A using B as the reference unit.

$$ \frac{A}{B} $$

The important thing to note is that if B is not zero, then the ratio A/B is always equal to the ratio of their measures M(A)/M(B), regardless of the unit of measure used.

$$ \frac{A}{B} = \frac{M(A)}{M(B)} $$

This formula demonstrates that the ratio between two homogeneous quantities remains constant and is independent of the specific units used to measure A and B.

This allows for a comparison of quantities in a way that is independent of the particular units of measure.

This concept is also known as the principle of invariance of the ratio of homogeneous quantities.

Independence from Units of Measure. This concept is critical in many scientific and engineering fields, especially in physics and mathematics, where quantities are often compared and measured in different unit systems. The ratio remains consistent, regardless of these conversions, particularly when dealing with proportions and scale ratios.  

A Practical Example

Let’s consider two segments, A and B.

Segment A is 10 centimeters long, while segment B is 5 centimeters long.

To determine the ratio of the lengths of these two segments, we simply divide the length of segment A by the length of segment B. The ratio is:

$$ \frac{10 \ cm}{5 \ cm} = 2 $$

This means that segment A is twice as long as segment B.

Note. It’s crucial to remember that B must not be zero. A ratio involving zero as the divisor is undefined, leading to nonsensical mathematical situations. Division by zero is impossible.

If I change the units of measure, for instance, by measuring the segments in millimeters instead of centimeters, segment A would be 100 mm long (since 1 cm = 10 mm) and segment B would be 50 mm long.

The ratio between the two segments remains the same:

$$ \frac{100 \ mm}{50 \ mm} = 2 $$

This example shows that when comparing two homogeneous quantities, such as the lengths of two segments, their ratio remains constant regardless of the units of measure used.

The Invariance Principle of Ratios in Derived Quantities

The invariance principle of ratios also applies to derived quantities like density (mass/volume) or speed (distance/time).

However, in these cases, one must be cautious because the ratio involves non-homogeneous quantities, so it’s necessary to convert the units of measure appropriately to ensure meaningful comparisons.

Note. In other words, while the ratio between homogeneous quantities is straightforward and doesn’t require additional conversions, the ratio between non-homogeneous quantities demands careful unit conversions to ensure accurate and meaningful comparisons.

Example

Suppose I need to calculate the speed of a car. In this case, the homogeneous quantities are the distance traveled (A) and the time taken (B).

$$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$

I measure the two quantities using kilometers for distance (A) and hours for time (B).

• Distance (A): 120 kilometers
• Time (B): 2 hours

The ratio between these two quantities, which is the speed, is given by:

$$ \text{Speed} = \frac{120 \text{ km}}{2 \text{ hours}} = 60 \text{ km/hour} $$

Now, let’s change the units of measure to meters and seconds:

• Distance: 120,000 meters (since 1 km = 1,000 meters)
• Time: 7,200 seconds (since 1 hour = 3,600 seconds)

The ratio, which still represents speed, becomes:

$$ \text{Speed} = \frac{120,000 \text{ meters}}{7,200 \text{ seconds}} = 16.67 \text{ meters/second} $$

Even though the units of measure have changed, the ratio remains constant.

If I convert 60 km/hour to meters/second (noting that 1 km/hour is approximately 0.27778 meters/second), I get:

$$ 60 \text{ km/hour} \times 0.27778 \text{ m/s per km/hour} = 16.67 \text{ meters/second} $$

This example is a practical demonstration that the ratio between two homogeneous quantities remains constant regardless of the units of measure used.

The Proof

The proof is divided into two parts:

1] The Ratio Between Two Homogeneous Quantities Equals the Ratio Between Their Measures

Let’s start by considering two homogeneous quantities, A and B, with B ≠ 0, and a unit of measure U.

$$ M(A) = a \cdot U $$

$$ M(B) = b \cdot U $$

The measure M(A) is "a" times the unit of measure U, while the measure M(B) is "b" times the unit of measure U.

The ratio between the two quantities is:

$$ r = \frac{A}{B} $$

Therefore, I can express A as "r" times B:

$$ A = r \cdot B $$

When I represent the quantities A and B using the unit of measure U, I have A=a·U and B=b·U:

$$ a \cdot U = r \cdot (b \cdot U) $$

Dividing both sides by the unit of measure, I get:

$$ a = r \cdot b $$

In other words:

$$ r = \frac{a}{b} $$

Thus, the ratio between the quantities r=A/B is equal to the ratio of the coefficients of the unit of measure U:

$$ r = \frac{A}{B} = \frac{a}{b} $$

The unit of measure U disappears after simplification.

The ratio between the measures is:

$$ \frac{M(A)}{M(B)} $$

Knowing that M(A)=a·U and M(B)=b·U:

$$ \frac{M(A)}{M(B)} = \frac{a \cdot U}{b \cdot U} $$

Simplifying, I obtain the desired result:

$$ \frac{M(A)}{M(B)} = \frac{a}{b} $$

In conclusion, the ratio between the quantities A/B is equal to the ratio between the measures M(A)/M(B):

$$ \frac{A}{B} = \frac{M(A)}{M(B)} = \frac{a}{b} = r $$

This proves the thesis based on the initial assumptions.

Note. It’s important to note that the units of measure U do not appear in the ratio between the quantities after algebraic simplification.

2] The Ratio Between Two Homogeneous Quantities is Independent of the Unit of Measure Chosen

Now, consider a different unit of measure, U', to measure the homogeneous quantities A and B:

$$ M(A) = a' \cdot U' $$

$$ M(B ) = b' \cdot U' $$

The ratio between the quantities A and B is:

$$ r = \frac{A}{B} $$

Substituting the quantities with their measures:

$$ r = \frac{M(A)}{M(B)} = \frac{a' \cdot U'}{b' \cdot U'} = \frac{a'}{b'} $$

Again, the ratio is independent of the unit of measure U' that I have chosen.

Additional Observations

Here are a few additional observations about the invariance principle of the ratio between two homogeneous quantities:

• The invariance principle of the ratio of homogeneous quantities is closely related to other fundamental concepts, such as:
  • Direct Proportionality
    When two quantities vary in such a way that their ratio remains constant.
  • Invariance of Ratio
    The idea that the ratio between two homogeneous quantities remains unchanged regardless of the units of measure used.
• Dimensionality and Homogeneity
  A key point is that quantities A and B must be homogeneous, meaning they must have the same dimension or fundamental unit of measure.

  For instance, I can calculate the ratio between two lengths or between two masses, but it doesn’t make physical sense to calculate the ratio between a length and a mass.

And so on.