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Imagine you and your friend have some money, and you want to share it fairly. How do you decide who gets how much? This is where the concept of ratio comes in. A ratio is a way to compare two quantities, showing how many times one quantity contains another. For example, if you have Rs.300 and your friend has Rs.500, the ratio of your money to your friend's money is 300:500.
Now, suppose you want to mix sugar and water in a recipe so that the sweetness is just right. You might say the sugar to water should be in the ratio 1:4. If you know the amount of sugar, you can find out how much water to add. When two ratios are equal, we say they are in proportion. Proportion helps us solve many real-life problems involving comparisons, mixtures, and scaling quantities.
Understanding ratio and proportion is essential for competitive exams and everyday calculations, such as finance, cooking, and measurement conversions.
A ratio is a comparison of two quantities of the same kind, expressed as "a to b" or written as a:b. It tells us how many times one quantity is compared to another.
For example, if a classroom has 12 boys and 18 girls, the ratio of boys to girls is 12:18. This ratio can be simplified by dividing both numbers by their greatest common divisor (GCD).
In the above diagram, the blue bar represents 3 units and the green bar represents 5 units, illustrating the ratio 3:5 visually.
Ratios can be of two types:
To simplify a ratio, divide both terms by their GCD. For example, 12:18 simplifies to 2:3 because the GCD of 12 and 18 is 6.
A proportion states that two ratios are equal. If we have two ratios \( \frac{a}{b} \) and \( \frac{c}{d} \), then they are in proportion if:
\[ \frac{a}{b} = \frac{c}{d} \]
This means the ratio of \(a\) to \(b\) is the same as the ratio of \(c\) to \(d\).
One of the key properties of proportion is:
\[ a \times d = b \times c \]
This is called the product of means equals product of extremes. It is very useful for solving problems where one term is unknown.
graph TD A[Identify two ratios] --> B[Set up proportion equation] B --> C[Apply cross multiplication] C --> D[Solve for unknown term] D --> E[Check solution for correctness]
This flowchart shows the step-by-step process to solve proportion problems efficiently.
Step 1: Find the greatest common divisor (GCD) 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
GCD is 12.
Step 2: Divide both terms by 12.
24 / 12 = 2, 36 / 12 = 3
Answer: The simplified ratio is 2:3.
Step 1: Write the proportion:
\( \frac{5}{x} = \frac{15}{45} \)
Step 2: Cross multiply:
\( 5 \times 45 = 15 \times x \)
\( 225 = 15x \)
Step 3: Solve for \(x\):
\( x = \frac{225}{15} = 15 \)
Answer: The missing term \(x\) is 15.
Step 1: Assume the original quantities of milk and water are 3x liters and 2x liters respectively.
Step 2: The added mixture has milk and water in ratio 4:1, total 5 parts.
Milk in added mixture = \( \frac{4}{5} \times 10 = 8 \) liters
Water in added mixture = \( \frac{1}{5} \times 10 = 2 \) liters
Step 3: New quantities:
Milk = \(3x + 8\)
Water = \(2x + 2\)
Step 4: The new ratio is:
\( \frac{3x + 8}{2x + 2} \)
Step 5: To find the exact ratio, we need the value of \(x\). Since the original quantities are not given, we can express the new ratio in terms of \(x\) or assume a value for \(x\).
Assuming \(x = 5\) liters (for example):
Milk = \(3 \times 5 + 8 = 15 + 8 = 23\) liters
Water = \(2 \times 5 + 2 = 10 + 2 = 12\) liters
Step 6: Simplify the ratio 23:12.
Since 23 and 12 have no common divisor other than 1, the ratio remains 23:12.
Answer: The new ratio of milk to water is 23:12.
Step 1: Let the cost of 8 kg apples be \(x\) INR.
Step 2: Since cost is directly proportional to weight, set up the proportion:
\( \frac{5}{250} = \frac{8}{x} \)
Step 3: Cross multiply:
\(5 \times x = 8 \times 250\)
\(5x = 2000\)
Step 4: Solve for \(x\):
\(x = \frac{2000}{5} = 400\)
Answer: The cost of 8 kg apples is Rs.400.
Step 1: Let the number of days taken by 15 workers be \(x\).
Step 2: Number of workers and days are inversely proportional because more workers mean fewer days.
So, \(6 \times 10 = 15 \times x\)
Step 3: Calculate \(x\):
\(60 = 15x\)
\(x = \frac{60}{15} = 4\)
Answer: 15 workers will complete the task in 4 days.
When to use: Whenever you have a proportion with one unknown term.
When to use: At the start of any problem involving ratios.
When to use: When dealing with measurements in different units, such as kilograms and grams.
When to use: When mixing two or more substances in given ratios.
When to use: When one quantity increases and the other decreases, such as workers and days to complete a task.
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