Quick Revision Notes & Formula Sheet

Class 6 Mathematics

Definition: A ratio is a way of comparing two quantities of the same kind by division.
Notation: The ratio of two quantities $a$ and $b$ is written as a : b (read as "$a$ is to $b$").
Terms of a Ratio: In the ratio a : b:
- '$a$' is called the First term or Antecedent.
- '$b$' is called the Second term or Consequent.

Same Units: Both quantities must be in the same units before finding their ratio. (e.g., convert rupees to paise, or meters to centimeters).
- Example: Ratio of $5\text{ kg}$ to $500\text{ g}$ $\rightarrow$ Convert $5\text{ kg}$ to $5000\text{ g}$, then find ratio ($5000 : 500$).
No Unit: A ratio has no units of its own because the units cancel out during division.
Order Matters: The order of terms is very important.
- Ratio a : b is not the same as b : a (unless $a = b$).
Simplest Form: A ratio is always expressed in its lowest terms (simplest form). A ratio a : b is in simplest form if the Highest Common Factor (HCF) of $a$ and $b$ is $1$.

Definition: Ratios that look different but represent the same value when reduced to their simplest form are called equivalent ratios.
How to find: By multiplying or dividing both the antecedent and consequent by the same non-zero number.
- If a : b is given, equivalent ratios can be written as: a × k : b × k OR (a ÷ k) : (b ÷ k) (where $k$ is any non-zero number).

Definition: An equality of two ratios is called a proportion.
Notation: If two ratios a : b and c : d are equal, we write them as: a : b :: c : d (read as "$a$ is to $b$ as $c$ is to $d$").
Meaning: It means $\frac{a}{b} = \frac{c}{d}$.
Terms of Proportion:
- In a : b :: c : d, $a, b, c,$ and $d$ are called the four terms in order.
- Extreme terms (Extremes): $a$ and $d$ (First and Last terms).
- Middle terms (Means): $b$ and $c$ (Second and Third terms).

For four numbers to be in proportion, the Product of Extremes must be equal to the Product of Means.
- If a : b :: c : d, then: Product of Extremes (a × d) = Product of Means (b × c)

Definition: The method in which we first find the value of one unit and then find the value of the required number of units is called the unitary method.
Steps to solve:
1. Step 1: Find the value of many items by division to get the value of one unit.
2. Step 2: Multiply the value of one unit by the required number of items to get the final answer.

Always write the formula or statement clearly before solving numerical problems.
Do not forget to write units in the final step of word problems (like Unitary Method), even though ratios do not have units.