Ratio and Proportions
Ratio
Ratio is the comparison between two values (they could be or not, different in magnitude). We can represent this with: \(a/b\) or \(a:b\)
Another way to understand this is, imagine that you have two numbers, \(a\) and \(b\), when \(b≠0\). The ratio is: \(r = a/b\). \(r\) is a real number or a fraction expressing how much times \(a\) contains \(b\) and vice versa.
There are few types of ratio, between magnitudes of the same type, different types and unit ratio.
Ratios Between Magnitudes of the Same Type
A ratio between magnitudes of the same type compares quantities with the same unit. For example, lengths of 4 meters and 2 meters, the ratio between them is \(4:2\), which simplifies to\(2:1\).
Ratios Between Magnitudes of Different Types
Magnitudes of different types (like speed to time), ratios can still be formed, but they change based on the context. This way, we will basically play with units. Comparing speed (meters per second) to time (seconds) give us another unit, like acceleration (meters per second squared).
- 10 meters per second / 2 seconds = 5 meters per second² (acceleration)
Unit Ratio (Per Unit Comparison)
A unit ratio involves one of the magnitudes being reduced to a single unit, like unit price (e.g., cost per kilogram). This makes it easier to compare different quantities by scaling one of them down to 1.
- If 10 apples cost $5, the unit ratio for cost per apple is 5/10 = $0.50 per apple.
Proportions
a Proportion is the equality between two or more ratios, if the have \(a/b\) and \(c/d\), we call thar they are in proportion when we express this: \(a/b = c/d\).
this means that the relation between \(a/b\) is pretty the same as in \(c/d\). This is true when the product of the crossing values are the same: \(a.d = b.c\).
This is rule of three! And there are some important aspects to take a look at:
- Cross-Multiplication Property
- if \(a/b = c/d\), then \(a.d = b.c\). This is the fundamentals of proportions, it should be carved on stone.
- If two ratios are equal, they reciprocals are also equal.
- if \(a/b = c/d\), then \(b/a = d/c\).
- Addition of Numerator and Denominator
- You can add the numerator and the denominator of each ratio without changing the proportion: if \(a/b = c/d\), then \(a+b/b = c+d/d\).
Let’s dig into an example. You have a car that consumes 10L of gas each 100Km. How much litres of gas we’ll spend in 300Km?
\[10/100 = x/300 ⇒ 10.300 = 100.x ⇒ 3000 = 100x ⇒ x = 30L\]Thanks, bye!