Ratios and Proportions
Understanding Ratios
✔ A ratio is a way to compare two or more quantities. It tells us how much of one quantity
exists compared to another.
✔ A ratio is written as: a : b, which is read as 'a to b' and means a is to b.
✔ Example: If there are 4 apples and 6 oranges, the ratio of apples to oranges is 4:6, which
simplifies to 2:3.
Properties of Ratios
✔ Order Matters: 2:3 is not the same as 3:2.
✔ Simplification: Ratios should be simplified like fractions (e.g., 6:9 simplifies to 2:3 by
dividing both by 3).
✔ Equivalent Ratios: Multiplying both terms by the same number gives an equivalent ratio
(e.g., 2:3 → 4:6).
Understanding Proportion
✔ A proportion is an equation that states that two ratios are equal. It helps us compare
quantities and find missing values.
✔ Basic Form: a:b = c:d or a/b = c/d.
✔ Example: If 2:3 = 4:6, then 2/3 = 4/6, meaning they are in proportion.
Types of Proportions
✔ Direct Proportion: When two quantities increase or decrease together in the same ratio.
✔ Example: If 5 pens cost ₹10, then 10 pens will cost ₹20.
✔ Inverse Proportion: When one quantity increases while the other decreases.
✔ Example: If 4 workers take 10 days to complete a task, then 8 workers will take 5 days.
Continued Proportion
✔ If a, b, and c are in continued proportion, then a:b = b:c.
✔ Third Proportional: If a, b, and c are in continued proportion, c = (b²)/a.
✔ Mean Proportional: If a, b, and c are in continued proportion, then b = √(a × c).
Variation
✔ Direct Variation: When two quantities increase or decrease together (Formula: x ∝ y or x
= k * y).
✔ Example: If 4 books cost ₹80, find the cost of 10 books.
✔ Inverse Variation: When one quantity increases while the other decreases (Formula: x ∝
1/y or x × y = k).
✔ Example: If 6 machines take 15 hours to complete a task, how long will 10 machines take?
