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When comparing two quantities, a ratio is a mathematical phrase that indicates the number of times the first number contains the second. The notation is A: B or A/B. For instance, the ratio of apples to oranges is 12:8 if you have 12 apples and 8 oranges. Dividing two integers by their greatest common divisor (GCD) simplifies the ratio. In this instance, the simplified ratio is 3:2, as the GCD of 12 and 8 is 4.
Ratios are useful in many real-life situations. For instance, if you’re mixing paint and need a ratio of 5 parts blue to 3 parts yellow and want to use 15 parts blue, you’ll need 9 parts yellow to maintain the same ratio (since 15 ÷ 5 = 3 and 3 × 3 = 9).
A ratio is a mathematical expression that compares two quantities, indicating how many times the first number contains the second. It is typically written in the form A: B or as a fraction A/B. Ratios are dimensionless and are used to compare similar units.
In a class of 40 students, 16 are girls and the rest are boys. What is the ratio of girls to boys in the class?
Step 2: Express the ratio.
The ratio of girls to boys is: 16:24
Step 3: Simplify the ratio.
To simplify, divide both numbers by their greatest common divisor (GCD), which is 8:
16 ÷ 8 = 2
24 ÷ 8 = 3
So, the simplified ratio of girls to boys is: 2:3
This means that for every 2 girls in the class, there are 3 boys.
Step | Description | Example | Result |
Identify the Conversion Factor | A conversion factor shows the relationship between two units. | 1 foot = 12 inches. | Conversion factor: 12 inches / 1 foot |
Mistake | What to Watch For |
Incorrect unit placement | Ensure the units you want to cancel are positioned diagonally across multiplication (e.g., m over m, ft over ft). |
Using the wrong conversion factor | Double-check your conversions (e.g., 1 kg = 1,000 g, not 100 g). |
Failing to match units |
Section | Details |
Definition | A proportion is an equation showing that two ratios are equal. Example: 2:3 = 4:6 |
Formula | If a/b = c/d, then a × d = b × c (cross-multiplication) |
Example Problem |
Common Mistakes
Wrong cross-multiplication
No matching units
Forgetting to simplify ratios
When comparing three or more quantities, the ratio is expressed as:
a: b: c: ...
Also Read - How to Calculate Profit Percentage
This indicates the relative sizes of each quantity to the others. For instance, if a recipe requires 2 cups of flour, 3 cups of sugar, and 4 cups of milk, the ratio of flour to sugar to milk is 2:3:4.
To calculate and simplify ratios with more than two terms:
Set Up the Conversion Ratio
Multiply the quantity by the conversion factor so units cancel appropriately. |
5 feet × (12 inches / 1 foot) |
60 inches |
Apply the Conversion to Ratios | Convert both quantities in the ratio to the same unit before simplifying. | 3 meters : 150 cm → 3 m × 100 = 300 cm → 300:150 | Simplified ratio: 2:1 |
Always convert both quantities to the same unit before forming a ratio.
If 5 pens cost ₹20, how much do 8 pens cost?
Step 1: Set Ratio | 5/20 = 8/x |
Step 2: Cross Multiply | 5x = 8 × 20 = 160 |
Step 3: Solve for x | X = 160/ 5 = ₹32 ⇒ 8 pens cost ₹32 |
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Example:
Given the quantities 10, 15, and 20:
Thus, the simplified ratio is 2:3:4.
Two ratios are equivalent if they represent the same relationship. To check if two ratios are equivalent:
Example:
Are 2:3 and 4:6 equivalent?
Since both fractions are equal, the ratios are equivalent.
The inverse ratio of a:b is b: a.
Example:
A ratio expresses the relationship between two or more quantities. When comparing two quantities, the ratio can be written as a fraction.
Example 1: Consider the ratio of 3 to 4, written as 3:4. This can be expressed as the fraction 3/4.
Example 2: If there are 10 apples and 5 oranges, the ratio of apples to oranges is 10:5, which simplifies to 2:1 or 2/1.
Expressing ratios as fractions allows for easier mathematical operations such as addition, subtraction, multiplication, and division.
A percentage represents a ratio out of 100. To convert a ratio to a percentage, follow these steps:
Example 1: For the ratio 3:4, first express it as the fraction 3/4. Then, multiply by 100:
(3/4) × 100 = 75%
Example 2: If there are 5 red balls and 15 blue balls, the ratio of red to blue balls is 5:15, which simplifies to 1:3. Expressed as a fraction, this is 1/3. Multiplying by 100 gives:
(1/3) × 100 = 33.33%
This means that approximately 33.33% of the balls are red.
Comparing quantities to determine their relationship, for example, "for every two apples, there are three oranges", is the essence of calculating ratios. Ratios assist us in making just and clear decisions, whether sharing a bill, doubling a recipe, or assessing data. This little mathematical skill has a lot of practical uses.
1. What is a ratio in simple terms?
A ratio compares two or more quantities to show how much of one thing there is compared to another, for example, 2:3 means for every 2 of one item, there are 3 of another.
2. How do I simplify a ratio?
To simplify a ratio, divide all parts of the ratio by their greatest common divisor (GCD). For example, 10:20 simplifies to 1:2.
3. Can a ratio be written as a fraction or a percentage?
Yes! A ratio like 3:4 can be written as the fraction 3/4, and as a percentage by multiplying the fraction by 100—so, 3/4 = 75%.
4. Why are ratios useful in real life?
Ratios help in everyday situations like cooking, budgeting, comparing prices, or analysing data, anytime you're comparing two or more values.
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