Ratio and Proportion

Ratio and Proportion

Ratio

A ratio is a comparison of two quantities that have the same units of measurement. It is usually expressed as a fraction or with a colon “:”.

For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, the ratio of flour to sugar is 2:1 or 2/1.

Ratios can also be used to express the relationship between two or more measurements, such as the ratio of width to length in a rectangle.

Formula

The formula for finding the ratio of two quantities is:
Ratio = A:B
= A/B
Here, the units of the two quantities should be the same. B should not be zero. The ratio A/B has no units.

Proportion

A proportion is a statement that two ratios are equal. It is often written in the form of an equation, such as a/b = c/d, where a, b, c, and d are quantities.

Proportion is the equality of two ratios. It states that two ratios are equivalent.

For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, and you want to make twice as much, the proportion would be:
2/1 = x/2 where x represents the flour needed to make twice as much of the recipe. Solving for x, we get:
x = 4

So, to make twice as much of the recipe, we would need 4 cups of flour.

Proportion can also be used in statistics and physics to compare different measurements or quantities.

Formula

The formula for proportion is a/b = c/d
where a, b, c, and d are four numbers or quantities. This formula states that the ratio of a to b is equal to the ratio of c to d, and it can be read as "a is to b as c is to d".
a/b = c/d
a : b :: c : d
Proportions are denoted by the symbol ‘::’ or ‘=’.

Product of extremes = Product of means

Types of Proportion

1. Direct Proportion

Direct proportion is a mathematical relationship between two variables, in which an increase in one variable results in a proportional increase in the other variable, while a decrease in one variable results in a proportional decrease in the other variable. In other words, when two quantities are in direct proportion, as one quantity increases or decreases, the other quantity changes in the same direction, by the same factor or ratio.

Hence, if “a” and “b” are two quantities, then the direction proportion is written as a ∝ b.

2. Inverse Proportion

Inverse proportion is a mathematical relationship between two variables, in which an increase in one variable results in a proportional decrease in the other variable, while a increase in one variable results in a proportional decrease in the other variable.

Therefore, the inverse proportion of two quantities, say “a” and “b” is represented by a ∝ (1/b).

3. Continued Proportion

Three quantities are said to be in continued proportion; if the ratio between the first and the second is equal to the ratio between the second and the third.

For example: 6, 12, 24 are in continued proportion.
6x24 = 12x12 = 144

3,6,12 are also an example.

Important Properties of Proportion

The following are the important properties of proportion:

1. The cross-product property: In a proportion, the product of the means is equal to the product of the extremes. That is, if a/b = c/d, then ad = bc.

2. The multiplication property: If all four terms of a proportion are multiplied by the same non-zero number, the resulting proportion is still true.

3. The division property: If all four terms of a proportion are divided by the same non-zero number, the resulting proportion is still true.

4. The addition property: If two ratios have the same denominator, the numerator can be added or subtracted, and the resulting fraction can still be in proportion.

5. The subtraction property: If two ratios have the same denominator, the numerator can be added or subtracted, and the resulting fraction can still be in proportion.

6. The inversion property: If a/b = c/d, then b/a = d/c.

7. The identity property: Any number divided by itself is equal to 1. Therefore, a/a = 1.

Difference Between Ratio and Proportion

Unitary Method of Ratio and Proportion

Ratio

The unitary method is a method of solving mathematical problems that involve finding the unit rate or the ratio of two quantities.

To find the ratio using the unitary method, you first divide the larger quantity by the smaller quantity. This will give you the unit rate or the ratio of the two quantities. To apply the unitary method for ratios, we first need to identify the two quantities that are related by the given ratio.

Let's take an example to illustrate this: Suppose we are given that the ratio of the length to the breadth of a rectangular field is 3:2. We need to find the length of the field if the breadth is 20 meters.

Solution: To solve this problem using the unitary method, we need to find the value of one unit of length in terms of the given ratio.
length / breadth = 3 / 2
Substituting the given value of the breadth, we get:
length / 20 = 3 / 2

To solve for the length, we can cross-multiply and simplify:
length x 2 = 3 x 20
length = (3 x 20) / 2
length = 30

Therefore, the length of the rectangular field is 30 meters.

Proportion

To apply the unitary method for proportions, we first need to identify the two quantities that are related to the given proportion.

Let's take an example to illustrate this: Suppose we are given that 20 workers can complete a piece of work in 15 days. We need to find the number of workers required to complete the same work in 10 days.

Solution: To solve this problem using the unitary method, we need to find the value of one unit of work in terms of the given proportion. We can do this by dividing the total work by the product of the number of workers and the number of days it takes to complete the work:

Value of one unit of work = Total work/(Number of workers × Number of days)
= 1/(20 × 15) (assuming the total work is 1 unit)

Now, we can use this value to find the number of workers required to complete the same work in 10 days:

Number of workers = Total work/(Value of one unit of work × Number of days)
= 1/(20 × 15) × 10
= 1/30

So, the number of workers required to complete the same work in 10 days is 30.

Tricks for Ratio and Proportion

Here are a few tricks to help find ratios:

1. Simplify the ratio: You can simplify a ratio by dividing both quantities by a common factor.

For example, 20 : 40 can be simplified to 1 : 2.

2. Convert to fractions: You can convert a ratio to a fraction by dividing the first quantity by the second quantity.

For example, 10 : 5 can be converted to 10/5 or 2/1.

3. Use cross-multiplication: You can use cross-multiplication to compare ratios.

For example, a truck can travel 480 miles on 20 gallons of gas. If the truck has 5 gallons of gas left, how far can it travel?

Solution: By the given information in the question we can write as:
480 miles / 20 gallons of gas = x miles / 5 gallons of gas

To solve for x, we use cross-multiplication,
480/20 = x/5
480 × 5 = 20x
2400 = 20x
x = 120

So, the truck can travel 120 miles with the remaining 5 gallons of gas.

> Use proportions: You can use proportions to compare ratios.
For example, if the ratio of apples to oranges is 2 : 1, you can set up a proportion: 2/1 = x/y.
Where x is the number of apples and y is the number of oranges.

> Compare to a whole: You can also find the ratio by comparing the quantities to a whole.

Example: Suppose that in a class of 30 students, there are 12 boys. What is the ratio of boys to the whole class and ratio of girls to the whole class?

Solution: The ratio of boys to the whole class is: 12 : 30 = 12/30 = 2/5
So, the ratio of boys to the whole class is 2 : 5.
The ratio of girls to the whole class is 18 : 30 = 18/30 = 3/5

Therefore, the ratio of girls to the whole class is 3 : 5.

Ratio to Percentage

Percentage is a way of expressing a number as a fraction of 100. It is often used to express the proportion or ratio of one number to another or to indicate the degree of change or increase or decrease in a given quantity. It is usually denoted by the symbol "%".

Steps involved:

Step 1: Identify the ratio.
Step 2: Express the ratio a : b in the form of a fraction a/b.
Step 3: Multiply the fraction by 100.
Step 4: Simplify and write the answer with the percent symbol (%).

For example, if a person scored 80 out of 100 on a test, their percentage would be 80%, indicating that they got 80 out of every 100 questions correct.

Example: What will be the percentage of strawberry juice packets in a bucket containing juice packets, if its ratio is 3 : 8?

Solution: The given ratio of strawberry juice packets in a bucket containing juice packets of different flavors is 3:8.
So, using the ratio to percent formula: Percentage = [Ratio × 100] %,
Percentage of strawberry juice packets = 3/8 × 100 %.
= 300/8 %
= 37.5%

Therefore, the required percentage is 37.5%.