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In everyday life, we often compare quantities to understand their relationship. For example, when a recipe calls for 2 cups of flour and 3 cups of sugar, it is comparing these two ingredients. This comparison is called a ratio. Ratios help us understand how much of one quantity relates to another.
When two such ratios are equal, they form a proportion. Proportions are useful in solving problems where one part is unknown but the relationship is known. These concepts are fundamental in mathematics and frequently appear in competitive exams, making it essential to master them.
A ratio is a way to compare two quantities of the same kind by division. It tells us how many times one quantity contains another.
For example, if a classroom has 12 boys and 18 girls, the ratio of boys to girls is written as 12:18. This means for every 12 boys, there are 18 girls.
Ratios can be written in three ways:
However, remember that a ratio is not always a fraction of a whole; it is a comparison between two quantities.
Just like fractions, ratios can be simplified by dividing both terms by their highest common factor (HCF).
The above bar diagram visually represents a ratio of 3:5, where the blue bar is shorter (3 units) and the green bar is longer (5 units).
A proportion is an equation stating that two ratios are equal. For example,
\[\frac{a}{b} = \frac{c}{d}\]
This means the ratio \(a:b\) is equal to the ratio \(c:d\).
graph TD A[Start with proportion] --> B[Write as \(\frac{a}{b} = \frac{c}{d}\)] B --> C[Cross multiply: \(a \times d = b \times c\)] C --> D[Solve for unknown variable] D --> E[Check solution by substitution]Step 1: Find the Highest Common Factor (HCF) of 24 and 36.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The HCF is 12.
Step 2: Divide both terms by 12.
\(\frac{24}{12} : \frac{36}{12} = 2 : 3\)
Answer: The simplified ratio is 2:3.
Step 1: Write the proportion as fractions:
\(\frac{3}{4} = \frac{x}{16}\)
Step 2: Use cross multiplication:
\(3 \times 16 = 4 \times x\)
\(48 = 4x\)
Step 3: Solve for \(x\):
\(x = \frac{48}{4} = 12\)
Answer: \(x = 12\)
Step 1: Add the parts of the ratio:
2 + 3 = 5 parts
Step 2: Find the value of one part:
\(\frac{25 \text{ liters}}{5} = 5 \text{ liters per part}\)
Step 3: Calculate quantities:
First liquid = \(2 \times 5 = 10\) liters
Second liquid = \(3 \times 5 = 15\) liters
Answer: 10 liters of first liquid and 15 liters of second liquid.
Step 1: Write the scale as a ratio:
1 cm : 5 km
Step 2: Let actual distance be \(x\) km for 7 cm on the map.
Form proportion:
\(\frac{1}{5} = \frac{7}{x}\)
Step 3: Cross multiply:
\(1 \times x = 5 \times 7\)
\(x = 35\) km
Answer: The actual distance is 35 km.
Step 1: Add the parts of the ratio:
3 + 5 + 7 = 15 parts
Step 2: Find the value of one part:
\(\frac{15000}{15} = 1000\) INR per part
Step 3: Calculate each partner's share:
Partner 1: \(3 \times 1000 = 3000\) INR
Partner 2: \(5 \times 1000 = 5000\) INR
Partner 3: \(7 \times 1000 = 7000\) INR
Answer: The shares are INR 3000, INR 5000, and INR 7000 respectively.
When to use: To make calculations easier and avoid errors.
When to use: When an unknown variable is present in proportion equations.
When to use: When dealing with measurements in different metric units.
When to use: To verify correctness of the solution.
When to use: For complex problems involving multiple ratios or scaling.
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