Ratio and Proportion - Unitary Method

A Ratio is a comparison of two quantities of the same kind by division, written as $a:b$ or $\frac{a}{b}$. For example, if you have $3$ red pens and $9$ blue pens, the ratio of red to blue pens is $3:9$. Visually, this simplifies to $1:3$, meaning for every $1$ red pen, there are $3$ blue pens.

Quantities in a ratio must be expressed in the same units before comparison. If you compare $50$ cm to $2$ m, you must convert $2$ m to $200$ cm first, making the ratio $50:200$ or $1:4$. Imagine a balance scale where both sides must use the same weight standard to be accurate.

A ratio is in its simplest form if the HCF (Highest Common Factor) of its terms is $1$. For example, the ratio $10:15$ is simplified to $2:3$ by dividing both parts by $5$. This is like looking at a large grid of $10 \times 15$ squares and seeing it as a simpler $2 \times 3$ pattern.

Proportion exists when two ratios are equal, denoted by $a:b = c:d$ or $a:b :: c:d$. This can be visualized using two similar shapes of different sizes; if the ratio of length to width is the same for both, their dimensions are in proportion.

In a proportion $a:b :: c:d$, the terms $a$ and $d$ are called 'Extremes' (the outer values) and $b$ and $c$ are 'Means' (the inner values). A fundamental property is that the product of extremes always equals the product of means: $a \times d = b \times c$. Imagine a 'cross-multiplication' $X$ connecting the terms.

The Unitary Method is a technique to solve problems by first finding the value of a single unit. Think of it as a two-step process: 'Divide' to find the unit value, then 'Multiply' to find the total value for the required number of units.

To find the value of one unit (the unit rate), divide the total value by the total number of units. For instance, if $5$ boxes weigh $25$ kg, then $1$ box weighs $25 \div 5 = 5$ kg. Visualize breaking a large block into smaller, equal-sized pieces to see the value of just one piece.

To find the total value of multiple units, multiply the unit value by the number of units required. If you know $1$ apple costs $₹ 10$, then $7$ apples will cost $10 \times 7 = ₹ 70$. This is like building a tower of identical blocks, where the total height is the height of one block multiplied by the number of blocks used.