Comparing Quantities - Ratios and Percentages

Ratio and Proportion: A ratio is a comparison of two quantities of the same kind by division, expressed as $a:b$ or $\frac{a}{b}$. Imagine a balance scale where one side has 2 weights and the other has 3; the ratio of weights is $2:3$. Proportion occurs when two ratios are equal, such as $a:b = c:d$.

Percentage Basics: Percentage means 'per hundred' and is represented by the symbol $\%$. Visually, a percentage can be understood by looking at a $10 \times 10$ grid of 100 small squares; if 45 squares are colored, it represents $45\%$. To convert a fraction or decimal to a percentage, multiply by 100.

Percentage Change: This measures how much a value has increased or decreased relative to its original value. Visualize a bar graph where the bar for 'this year' is taller than 'last year'; the difference in height relative to the last year's height is the percentage increase.

Profit and Loss: If the Selling Price ($SP$) is greater than the Cost Price ($CP$), a profit is made. If $CP$ is greater than $SP$, a loss occurs. On a horizontal number line, if $SP$ is to the right of $CP$, the distance between them represents profit; if $SP$ is to the left, it represents loss.

Discount and Marked Price: Discount is a reduction given on the Marked Price ($MP$) of an article. Think of a 'Sale' tag on a shirt; the original price on the tag is the $MP$, the amount deducted is the discount, and the final price you pay is the $SP$. Discount is always calculated on the $MP$.

Sales Tax/VAT/GST: These are taxes charged by the government on the sale of an item and added to the bill amount. Imagine a grocery bill where the subtotal is listed first, then a small percentage is added as tax to reach the 'Grand Total'.

Compound Interest ($CI$): Unlike Simple Interest where interest is calculated only on the principal, Compound Interest is 'interest on interest'. Visually, if you plot the growth of money over time, Simple Interest grows as a straight line, while Compound Interest grows as a curve that gets steeper and steeper.