Introduction to Ratios

• Ratio Definition: A ratio is a comparison of two or more quantities, showing the relative sizes of 'parts' within a whole. It describes how much of one thing there is compared to another thing.

• Notation: Ratios are typically written using a colon ($a:b$), but can also be expressed with the word 'to' ($a$ to $b$). In a ratio like $3:2$, the numbers $3$ and $2$ are called the terms of the ratio.

• Order Importance: The order of numbers in a ratio is critical and must match the order of the items described. If a mixture has $5$ parts water and $1$ part syrup, the ratio of water to syrup is $5:1$, whereas $1:5$ would represent a completely different mixture.

• The 'Whole': The total number of parts in a situation is found by adding the individual terms of the ratio together. For a ratio $a:b$, the total number of parts is $a + b$.

Simplifying Ratios

• Common Factors: To simplify a ratio, divide all terms by their Greatest Common Divisor (GCD). A ratio is in its simplest form when all terms are integers and share no common factors other than $1$.

• Handling Units: Before writing a ratio, ensure all quantities are in the same units. For example, to compare $50$cm to $2$m, first convert $2$m to $200$cm to get the ratio $50:200$, which simplifies to $1:4$.

Finding Missing Values

• The Multiplier Method: If you know one part of an equivalent ratio, divide it by the corresponding part of the original ratio to find the 'multiplier'. Multiply all other terms by this same value to complete the equivalent ratio.

• Ratio vs. Fraction: While related, they represent different perspectives. A ratio ($a:b$) compares one part to another part, whereas a fraction (rac{a}{a+b}) compares one part to the total whole.

• Part-to-Part vs. Part-to-Whole: A part-to-part ratio compares distinct groups (e.g., boys to girls), while a part-to-whole ratio compares one group to the entire set (e.g., boys to all students).

• Additive Error: A common mistake is trying to find equivalent ratios by adding the same number to both terms. For example, $1:2$ is NOT equivalent to $2:3$ (adding $1$ to both), because the multiplicative relationship has changed.

• Ignoring Units: Students often write ratios using the raw numbers provided without checking if the units match. Always convert to the smallest unit mentioned to avoid working with decimals.

• Fraction Confusion: Assuming that the ratio $1:4$ is the same as the fraction $\frac{1}{4}$ of the whole. In reality, $1:4$ means the first part is $\frac{1}{5}$ of the total.