Login
OR
Don't worry. We'll fix this for you!
In everyday life, we often need to compare quantities to understand their relationship. For example, if a recipe calls for 2 cups of flour and 3 cups of sugar, we want to know how much flour relates to sugar. This comparison is expressed using a ratio. A ratio tells us how many times one quantity contains or is contained within another.
When two such comparisons are equal, we say those ratios are in proportion. Proportions are powerful because they allow us to solve problems where some values are unknown but the relationship remains consistent. For instance, if 4 meters of cloth cost INR 600, how much would 10 meters cost?
This chapter focuses on understanding both ratios and proportions clearly, learning how to calculate and simplify them, and applying these ideas to solve problems involving metric measurements and Indian currency (INR). By mastering these concepts, you'll be well-prepared for competitive entrance exams and practical applications.
A ratio compares two quantities showing their relative sizes. If you have 6 apples and 3 oranges, the ratio of apples to oranges is 6 to 3.
Ratios can be expressed in multiple ways:
All these mean the same thing - how many times one quantity relates to another.
There are two main types of ratios:
Just like fractions, ratios can often be simplified to their lowest terms by dividing both parts by their greatest common divisor (gcd). A simplified ratio is easier to understand and work with.
In the diagram above, the larger blue block represents 6 units, and the smaller green block represents 3 units. Dividing both by 3, we get a simplified ratio of \(2 : 1\), which is easier to interpret: for every 2 units of the first quantity, there is 1 unit of the second.
A proportion states that two ratios are equal. For example, if \(\frac{a}{b} = \frac{c}{d}\), then the ratio \(a:b\) is proportional to \(c:d\).
Proportions help us solve problems when one quantity is unknown but the relationship between quantities is preserved.
graph TD A[Identify given ratios] --> B[Set up proportion \frac{a}{b} = \frac{c}{d}] B --> C[Cross multiply: a \times d = b \times c] C --> D[Solve for unknown quantity]The flowchart above shows the typical steps for solving proportion problems using cross multiplication:
Step 1: Write the ratio in fractional form: \(\frac{150}{600}\).
Step 2: Find the greatest common divisor (gcd) of 150 and 600. Here, \(gcd(150, 600) = 150\).
Step 3: Divide numerator and denominator by 150:
\[ \frac{150 \div 150}{600 \div 150} = \frac{1}{4} \]
Step 4: Express the simplified ratio in colon form:
\(1 : 4\)
Answer: The simplified ratio of 150 m to 600 m is \(1 : 4\).
Step 1: Identify the two ratios: cost per length ratio.
Ratio 1: Cost to length for the first wire = \(\frac{200}{5}\)
Ratio 2: Cost to length for the second (unknown cost \(x\)) = \(\frac{x}{12}\)
Step 2: Set up the proportion:
\[ \frac{200}{5} = \frac{x}{12} \]
Step 3: Use cross multiplication:
\[ 200 \times 12 = 5 \times x \]
Step 4: Calculate:
\[ 2400 = 5x \Rightarrow x = \frac{2400}{5} = 480 \]
Answer: The cost of 12 meters of wire is INR 480.
Step 1: Let the volume of 30% solution be \(x\) litres. Then the volume of 50% solution is \(20 - x\) litres.
Step 2: Set up the proportion of alcohol content:
\[ 0.30x + 0.50(20 - x) = 0.40 \times 20 \]
Step 3: Simplify the equation:
\[ 0.30x + 10 - 0.50x = 8 \]
\[ (0.30 - 0.50)x = 8 - 10 \]
\[ -0.20x = -2 \Rightarrow x = \frac{-2}{-0.20} = 10 \]
Step 4: Find the volume of 50% solution:
\[ 20 - 10 = 10 \text{ litres} \]
Answer: Mix 10 litres each of 30% and 50% solutions to get 20 litres of 40% alcohol solution.
Step 1: Express the ratio of Cost Price (CP) to Selling Price (SP):
\(1500 : 1800\)
Step 2: Simplify the ratio by dividing both terms by 300:
\[ \frac{1500}{300} : \frac{1800}{300} = 5 : 6 \]
Step 3: Calculate the profit:
\[ \text{Profit} = SP - CP = 1800 - 1500 = 300 \]
Step 4: Calculate profit percentage:
\[ \frac{300}{1500} \times 100 = 20\% \]
Answer: The ratio of CP to SP is \(5 : 6\), and the profit percentage is 20%.
Step 1: Convert USD to INR using proportion:
\[ \frac{1 \text{ USD}}{83 \text{ INR}} = \frac{250 \text{ USD}}{x \text{ INR}} \]
Step 2: Cross multiply and solve:
\[ 1 \times x = 83 \times 250 \Rightarrow x = 20750 \text{ INR} \]
Step 3: Find length of wire purchasable with 20750 INR:
\[ \text{Length} = 20750 \times 0.012 = 249 \text{ meters} \]
Answer: 250 USD equals INR 20,750 and can buy 249 meters of copper wire.
When to use: At the start of any problem to reduce complexity and errors.
When to use: Whenever the proportion equation contains an unknown variable.
When to use: When units are different (e.g., meters and centimeters) to avoid incorrect ratios.
When to use: To quickly validate answers or test if two ratios are in proportion.
When to use: When solving profit, loss, or currency conversion problems.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|