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We use fractions to explain the concepts of ratio and proportion. A fraction that can be written in the form of x : y is known as a ratio. On the other hand, when two ratios are set equal to each other, we get a proportion.
In the above ratio, x and y are integers. The concept of ratio and proportion is widely used in mathematics, science, and in our daily lives. In our everyday life, we use ratio and proportion to analyze our financial matters. Similarly, whenever we try a new recipe, we ensure to use the appropriate proportion of the ingredients.
We can simplify the ratio. For example, consider a ratio 35 : 25. Both these integers are multiples of 5, hence we can write the ratio as 7 : 5. A ratio has no unit because both the quantities involved in it have similar units. These same units cancel each other. Remember that a ratio describes the relationship between two quantities, let say x and y, where y is not equal to zero.
Defining Ratio and Proportion
In this section, we will discuss the formal definitions of ratio and proportion. Let us define the ratio first:
A ratio shows the relationship between two values
Or
A ratio describes the relative sizes of two or more quantities
The ratio has many applications in our everyday life. For example, the speed of an airplane is equal to the distance divided by time. It means that the speed is equal to the ratio of distance and time.
A proportion can be defined as:
The equation which says that two ratios are equal to each other is known as proportion
For example, if a car travels 80 kilometers distance in one hour, then it must cover 160 kilometers distance in two hours. Mathematically, we can write it as:
$80 : 160 = 1 : 2$
You can see that we have taken the ratio of similar units. It means that 80 : 160 is one ratio that depicts the distance and 1 : 2 is another ratio that depicts the hours.
- Ratios only exist between similar quantities
- When we use the ratio to compare two quantities, then their units must be the same
- The order of terms in the ratio is also important
- The ratio remains unaffected if both the numbers are multiplied or divided by the same number.
Ratio and Proportion Formulas
Now, let us see what are some of the formulas of ratio and proportion.
Ratio Formula
Suppose we have two quantities (or numbers) x and y and we have to find the ratio between these two quantities. The formula for finding the ratio between these quantities will be:
$x : y = \frac {x}{y}$, where x and y are any two quantities.
Here, x is the first term, also known as an antecedent, and y is the second term also known as a consequent.
For example, what is the ratio between the numbers 6 and 24.
We can write the numbers in ratio form like this:
$6 : 24$
To find the ratio, simply divide 6 by 24:
$ = \frac {6}{24}$
$= \frac {1}{4}$
Hence, the ratio between the numbers 6 and 24 in simplified form is 1 : 4 or $\frac {1}{4}$.
Consider another example below:
Two numbers exist in the ratio of 4:5 and their sum is 90. Find the numbers.
Suppose the numbers are 4x and 5x respectively. Given the information in the example, we can construct the following equation:
$4x + 5x = 90$
$9x = 90$
$x = \frac {90}{9}$
$x = 10$
Now, we will substitute this value of x in 4x and 5x respectively to get the numbers.
First number = 4 (10) = 40
Second number = 5 (10) = 50
Hence, the two numbers are 40 and 50.
Proportion Formula
We know that the proportion is an equality between two ratios a:b and c:d. The terms a and d are known as extreme or extreme terms, whereas, the terms b and c are known as means or mean terms. Mathematically, we can write proportion as:
$a : b = c : d$
In fractional form, we can write it as:
$\frac {a}{b} = \frac {c}{d}$
For instance, consider the following proportion:
$8 : 9 :: 16 : 18$
In the above proportion, 8 and 18 are the extreme values and 9 and 16 are mean values.
Properties of Proportion
Addendo
If a : b = c : d, then a + c : b + d.
Subtrahendo
If a : b = c : d, then a - b : b = c - d : d.
Componendo
If a : b = c : d, then a + b : b = c + d : d.
Alternendo
If a : b = c : d, then a : c = b : d.
Invertendo
If a : b = c : d, then b : a : d : c.
Componendo and dividendo
If a : b = c : d, then a + b : a - b = c + d : c - d.
Third, Fourth and Mean Proportional
- Third proportional : If $a : b = c : d$, then c is known as a third proportional to a, b, and d.
- Fourth proportional: If $a : b = c : d$, then d is known as the fourth proportional to a, b and c.
- Mean proportional: The mean proportional of the ratio a:b is equal to the square root of the product of the terms a and b. It is represented by the following formula:
Mean proportional = $\sqrt {ab}$
Example 1
Three friends invest \$2,000, \$2,500, and \$3,500 in a business. After a year, the business made a profit of \$4000. If the money is withdrawn, how much will each individual receive if their earnings are directly proportional to the money they originally invested?
Solution
The amount that is invested by three friends is directly proportional to the profit earned.
$\frac {x}{2000} = \frac {y} {2500} = \frac{z} {3500} = \frac {x + y + z} {7000} = \frac {4000}{8000}$
Amount that will be received by the first individual = $\frac {x}{2000} = \frac {4}{8}$
$x = \frac {4}{7} \cdot 2000$
$ = \$1000$
Hence, the first individual will receive 1000 dollars.
Amount received the second individual = $\frac {y}{2500} = \frac {4}{8}$
$ y = \frac {4}{8} \cdot 2500$
$ = \$1250$
Hence, the second individual will receive 1250 dollars
Amount received by the third individual = $\frac {z}{3500} = \frac {4}{8}$
$z = \frac {4}{8} \cdot 3500$
$x = \$1750$
Hence, the third individual will receive 1750 dollars.
Example 2
The sum of two numbers is 280. These numbers exist in the ratio of 6: 8. Find the numbers.
Solution
Suppose the numbers are x and y.
The ratio between these numbers is 6:8, so we can say that the numbers are 6x and 8x respectively.
$6x + 8x = 280$
$14x = 280$
Divide both sides by 14 to get the value of x:
$x = 20$
Now, substitute the value of x = 20 in 6x and 8x respectively:
First number = 6x = 6(20) = 120
Second number = 8x = 8 (20) = 160
Hence, the two numbers are 120 and 160.
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