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Every day, we compare things: the price of two items, the distance traveled by two vehicles, or the amount of ingredients in a recipe. The mathematical concept that helps us compare two quantities quantitatively is called a ratio. When two such ratios have the same value, they are said to be in proportion. Understanding these concepts is essential not only in daily life but also for many competitive exams, where quick and accurate reasoning about quantities can save valuable time.
For example, consider a packet of rice priced at INR 120 per 10 kilograms and another packet priced at INR 200 per 15 kilograms. To find which packet is cheaper per kilogram, you can use the idea of ratios. Similarly, to check if two ratios like 4:6 and 8:12 represent the same relationship, we use proportion.
A ratio is a way to express the relative size of two quantities by showing how many times one quantity contains or is contained within the other. It compares two values of the same kind, such as length to length, weight to weight, or cost to cost.
Ratios are usually written in one of three forms:
3 : 5By convention, the first term is called the antecedent and the second term is called the consequent.
A key point about ratios is that they compare quantities of the same type. Comparing length to weight directly does not produce a meaningful ratio.
For example, if two rods measure 120 cm and 80 cm respectively, the ratio of their lengths is \(120 : 80\). We can simplify this ratio by dividing both terms by their greatest common divisor (GCD), which is 40, resulting in \(3 : 2\). Thus, the rods are in the ratio 3:2, meaning the first rod is one and a half times as long as the second.
Proportion is a statement that two ratios are equal. If the ratios \( A:B \) and \( C:D \) are equal, then they are said to be in proportion. Symbolically, this is written as
\(\frac{A}{B} = \frac{C}{D}\)
A common and powerful tool when working with proportions is cross multiplication. Cross multiplication helps check equality and solve for unknown terms.
graph TD A[Start: Given proportion \frac{A}{B} = \frac{C}{D} with some unknown] B{Is a term unknown?} C[Set up cross-product equation] D[Multiply A x D and B x C] E{Are cross-products equal?} F[No -> Not proportional] G[Yes -> Proportional, solve for unknown if any] A --> B B -->|Yes| C B -->|No| D C --> D D --> E E -->|No| F E -->|Yes| GFor example, to check if ratios \(3 : 4\) and \(6 : 8\) are in proportion, cross multiply:
\(3 \times 8 = 24\) and \(4 \times 6 = 24\)
Since both cross products are equal, the ratios are indeed proportional.
Cross multiplication is also used to find the missing term in proportions such as:
\(\frac{3}{x} = \frac{6}{8}\)
Cross multiply to get \(3 \times 8 = 6 \times x\), so \(24 = 6x\), giving \(x = 4\).
Mixture problems involve combining two or more quantities with different ratios to find the resultant ratio. For example, mixing two solutions of milk and water in different ratios.
The formula to find the ratio of components in the combined mixture is:
Where \(Quantity_1\) and \(Quantity_2\) are the amounts mixed, and \(Ratio_1\) and \(Ratio_2\) are the respective component ratios.
| Component | Quantity | Ratio of Milk : Water | Milk in liters | Water in liters |
|---|---|---|---|---|
| Container A | 10 L | 4 : 1 | 8 L | 2 L |
| Container B | 15 L | 3 : 2 | 9 L | 6 L |
| Total | 25 L | ? | 17 L | 8 L |
The total milk to water ratio after mixing is \(17 : 8\).
In partnership problems, partners invest different amounts for varying durations, and profits are shared according to the ratio of their invested amounts multiplied by the time of investment.
Formula:
This helps determine each partner's share of the total profit based on how much and how long they invested money.
When multiple workers complete a task at different rates, their combined work rate is the sum of individual rates. Total time taken to complete the work together is inversely proportional to this combined work rate.
Work rate is given by:
Understanding ratios between rates helps find combined time or work done.
Ratio and proportion are powerful tools for comparing quantities and solving practical problems encountered in daily life and competitive exams. From simplifying the comparison of lengths to sharing profits and mixing solutions, these concepts train you to think logically and operate efficiently under exam conditions.
Step 1: Write the ratio as \(120 : 80\).
Step 2: Find the greatest common divisor (GCD) of 120 and 80.
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
Common factors: 1, 2, 4, 5, 8, 10, 20, 40
GCD = 40
Step 3: Divide both terms by 40:
\(\frac{120}{40} : \frac{80}{40} = 3 : 2\)
Answer: The simplified ratio is \(3 : 2\).
Step 1: Write as a fraction equation:
\(\frac{3}{x} = \frac{6}{8}\)
Step 2: Cross multiply:
\(3 \times 8 = 6 \times x\)
Which simplifies to \(24 = 6x\).
Step 3: Solve for \(x\):
\(x = \frac{24}{6} = 4\)
Answer: \(x = 4\)
Step 1: Calculate milk and water quantities in each container.
For container A (10 L, ratio 4:1):
Total parts = 4 + 1 = 5 parts
Milk = \(\frac{4}{5} \times 10 = 8\) L
Water = \(\frac{1}{5} \times 10 = 2\) L
For container B (15 L, ratio 3:2):
Total parts = 3 + 2 = 5 parts
Milk = \(\frac{3}{5} \times 15 = 9\) L
Water = \(\frac{2}{5} \times 15 = 6\) L
Step 2: Add quantities:
Total milk = 8 + 9 = 17 L
Total water = 2 + 6 = 8 L
Step 3: Write the resultant ratio:
Milk : Water = \(17 : 8\)
Answer: The combined mixture ratio of milk to water is \(17 : 8\).
Step 1: Calculate the effective investment for each partner:
Partner A: \(50,000 \times 6 = 300,000\)
Partner B: \(70,000 \times 4 = 280,000\)
Step 2: Find the ratio of their investments:
Ratio = \(300,000 : 280,000 = 30 : 28 = 15 : 14\) (simplified by 20,000)
Step 3: Calculate total parts = \(15 + 14 = 29\)
Step 4: Find each partner's share:
Partner A's share = \(\frac{15}{29} \times 36,000 \approx 18,621\) INR
Partner B's share = \(\frac{14}{29} \times 36,000 \approx 17,379\) INR
Answer: Partner A receives approximately INR 18,621, Partner B receives approximately INR 17,379.
Step 1: Let work rate of P = 3 units/day, and Q = 4 units/day.
Combined rate = 3 + 4 = 7 units/day.
Step 2: Total work = Combined rate x time = \(7 \times 12 = 84\) units.
Step 3: Time taken by Q alone = \(\frac{Total\ work}{Q's\ rate} = \frac{84}{4} = 21\) days.
Answer: Q alone takes 21 days to finish the work.
When to use: At the start of any ratio problem to reduce complexity.
When to use: When you see two ratios set equal with missing information.
When to use: When combining items with different composition ratios.
When to use: When partners invest different amounts for different periods.
When to use: When multiple workers contribute to a single task.
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