Ratio

Ratio

Hello ScienceBee, in this lesson, we look into the world of ratios. 
 

 

What is a ratio?

A ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

Video: Introduction to ratios 
 
The numbers in a ratio may be quantities of any kind, such as counts of persons or objects, or such as measurements of lengths, weights, time, etc.
A ratio may be either specified by giving both constituting numbers, written as "a to b" or "a:b", or by giving just the value of their quotient " ", since the product of the quotient and the second number yields the first, as required by the above definition.
Consequently, a ratio may be considered as an ordered pair of numbers, as a fraction with the first number in the numerator and the second as denominator, or as the value denoted by this fraction. Ratios of counts, given by natural numbers, may be either natural or rational numbers, ratios of measurements generally result in real numbers.

 

What is a rate?

When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.

Video: Introduction to rates

 

Notation and terminology for ratio

The ratio of numbers A and B can be expressed as:
  • the ratio of A to B
  • A is to B (often followed by "as C is to D")
  • A∶B
  • a fraction with A as numerator and B as denominator, that represents the quotient: A divided by B: . This can be expressed as a simple or a decimal fraction, or as a percentage, etc.
The numbers A and B are sometimes called terms of the ratio with A being the antecedent and B being the consequent.
A statement expressing the equality of two ratios A∶B and C∶D is called a proportion, written as A∶B = C∶D or A∶B::C∶D. This latter form, when spoken or written in the English language, is often expressed as (A is to B) as (C is to D).
A, B, C and D are called the terms of the proportion. A and D are called its extremes, and B and C are called its means. The equality of three or more ratios, like A∶B = C∶D = E∶F, is called a continued proportion.
Ratios are sometimes used with three or even more terms, e.g., the proportion for the edge lengths of a "two by four" that is ten inches long is therefore 
a good concrete mix (in volume units) is sometimes quoted as 
For a (rather dry) mixture of 4/1 parts in volume of cement to water, it could be said that the ratio of cement to water is 4∶1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement.

 

Proportions and percentage ratios

Video: Proportions and Percents
 
If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3:2 is the same as 12:8. It is usual either to reduce terms to the lowest common denominator, or to express them in parts per hundred (percent).
If a mixture contains substances A, B, C and D in the ratio 5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total and multiply by 100, we have converted to percentages: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25:45:20:10).
If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, , or 40% of the whole is apples and , or 60% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion.
If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction. For example, older televisions have a 4:3 aspect ratio, which means that the width is 4/3 of the height (this can also be expressed as 1.33:1 or just 1.33 rounded to two decimal places). Modern widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places. One of the popular widescreen movie formats is 2.35:1 or simply 2.35. Representing ratios as decimal fractions simplifies their comparison. When comparing 1.33, 1.78 and 2.35, it is obvious which format offers wider image. Such a comparison works only when values being compared are consistent, like always expressing width in relation to height.

 

Irrational ratios

Ratios may also be established between incommensurable quantities (quantities whose ratio, as value of a fraction, amounts to an irrational number). The earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal d to the length of a side s of a square, which is the square root of 2, formally Another example is the ratio of a circle's circumference to its diameter, which is called π, and is not just an algebraically irrational number, but a transcendental irrational.
Also well-known is the golden ratio of two (mostly) lengths a and b, which is defined by the proportion 
or, equivalently
Taking the ratios as fractions and as having the value x, yields the equation
or  which has the positive, irrational solution   Thus at least one of a and b has to be irrational for them to be in the golden ratio. An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive Fibonacci numbers: even though all these ratios are ratios of two integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio.
Similarly, the silver ratio of a and b is defined by the proportion 
corresponding to
This equation has the positive, irrational solution   so again at least one of the two quantities a and b in the silver ratio must be irrational.

Video: Trigonometric equations example 7 irrational ratio

 

Basic ratios

This video discusses basic ratios. 

 

Equivalent ratios

This video discusses equivalent ratios. 

 

Solving ratio problems with graph

This video discusses solving ratio problems with graph. 

 

Let's Review

  1. A ratio is a ____ between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is ____ to ____ (that is, 8:6, which is equivalent to the ratio 4:3). 

  2. When two quantities are measured with the same unit, as is often the case, their ratio is a ____ number. A quotient of two quantities that are measured with different units is called a ____ .

  3. A statement expressing the equality of two ratios A∶B and C∶D is called a ____ , written as A∶B = C∶D or A∶B::C∶D. This latter form, when spoken or written in the English language, is often expressed as (A is to B) as (C is to D). A, B, C and D are called the ____ of the proportion. A and D are called its ____, and B and C are called its ____ . The equality of three or more ratios, like A∶B = C∶D = E∶F, is called a ____  ____ .

     

    Answer

    1.  A ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). 

    2. When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.

    3. A statement expressing the equality of two ratios A∶B and C∶D is called a proportion, written as A∶B = C∶D or A∶B::C∶D. This latter form, when spoken or written in the English language, is often expressed as (A is to B) as (C is to D). A, B, C and D are called the terms of the proportion. A and D are called its extremes, and B and C are called its means. The equality of three or more ratios, like A∶B = C∶D = E∶F, is called a continued proportion.

     

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