Cambridge Lower Secondary Mathematics Stage 8 Unit 12: Ratio and Proportion Notes

Understanding the Fundamentals of Simplifying Ratios

A ratio is a mathematical expression used to compare two or more quantities. In mathematical notation, ratios can be represented in three primary ways: the colon form $a:b$, the phrase form "$a$ to $b$", or as a fraction $\frac{a}{b}$. To simplify a ratio, one must divide all parts of the ratio by their greatest common factor (GCF). This process ensures the ratio is in its most reduced form, making comparisons more straightforward. For example, to simplify the ratio $12:18$, one must identify the greatest common factor of $12$ and $18$, which is $6$. Dividing both terms by $6$ results in the simplified ratio of $2:3$.

Equivalent ratios are different numerical ratios that represent the same fundamental comparison. To determine if two ratios are equivalent, such as $4:6$ and $2:3$, one should simplify both ratios to their lowest terms. Since $4:6$ simplifies to $2:3$ by dividing both terms by $2$, the ratios are confirmed as equivalent. It is also possible to have three-part ratios, which compare more than two amounts simultaneously, such as the ratio $2:3:5$.

When dealing with ratios that involve units of measurement, it is critical to convert all quantities to the same units before attempting to simplify. For instance, in a comparison between $2\,m$ and $50\,cm$, one must first convert the meters to centimeters to create a uniform comparison: $200\,cm : 50\,cm$. By dividing both terms by their GCF of $50$, the ratio simplifies to $4:1$. Ratios have extensive application in real-life scenarios, including the creation of recipes, the reading and drawing of maps, producing scale drawings, mixing substances, and calculating probability.

In a specific worked example, to simplify the ratio $24:36$, the greatest common factor is identified as $12$. By dividing both terms by $12$, the ratio yields the result $2:3$. Key vocabulary for this section includes Ratio, Equivalent ratio, Simplify, Comparison, and Greatest common factor (GCF). In summary, ratios are tools to compare quantities that are made more accessible through simplification.

Methods for Sharing in a Ratio

Sharing in a ratio involves dividing a total amount into specific parts according to the proportions defined by a given ratio. The process follows a structured three-step method. First, add the parts of the ratio together to find the total number of parts. Second, divide the total amount being shared by this total number of parts to determine the value of a single part. Third, multiply the value of that single part by each individual part of the ratio to find the specific shares.

For example, to share $\$120$ in the ratio $2:3$, the steps are as follows: add the parts ($2 + 3 = 5$ total parts), divide the total amount by the parts (\120 \div 5 = \$24$ per part), and then calculate the final shares ($2 \times \$24 = \$48$ and $3 \times \$24 = \$72$). This same methodology applies to three-part sharing or sharing among even more categories. If one were to share $\$180$ in the ratio $2:3:4$, the total parts would be $2 + 3 + 4 = 9$. Dividing the total by the parts (\180 \div 9 = \$20$) establishes the value of one part. The final shares are then calculated as $2 \times \$20 = 40$, $3 \times \$20 = 60$, and $4 \times \$20 = 80$.

Sharing in a ratio is used in various real-life contexts, such as splitting money, dividing resources, following specific recipes, and distributing workloads in teamwork. A worked example involves sharing $150$ sweets in the ratio $1:2:3$. The total number of parts is $1 + 2 + 3 = 6$. The value of one part is calculated as $150 \div 6 = 25$. Consequently, the shares are found to be $25$, $50$, and $75$. Key vocabulary for this section includes Share, Ratio parts, Total parts, and Proportion. In summary, dividing amounts requires calculating total parts, finding the unit value, and multiplying for each share.

Exploring Ratio and Direct Proportion

Direct proportion occurs when two quantities increase or decrease at the same rate. A key characteristic of this relationship is that if one value is doubled, the other value will also double. Ratios are used to describe these proportional relationships. For instance, if $3$ books cost $\$12$, then by maintaining the same rate, $6$ books will cost $\$24$. The multiplier that connects two quantities in such a relationship is known as the constant of proportion.

The formal mathematical formula for direct proportion is $y = kx$, where $k$ represents the constant. To solve problems involving proportion, one can use equivalent ratios. For example, if $2$ pens cost $\$4$, one can find the cost of $5$ pens by setting up the ratio $2:4 = 5:x$, which leads to the solution $x = 10$. Alternatively, the unitary method can be employed, which involves finding the value of a single unit first and then scaling up to the desired amount. When represented graphically, direct proportion results in a straight line that passes through the origin $(0,0)$.

Real-life examples of direct proportion include calculating speed, adjusting recipes, determining wages based on hours worked, and performing currency conversions. In a worked example, a recipe requires $4$ cups of flour for every $8$ cookies. To find out how much flour is needed for $12$ cookies, one first identifies the ratio of flour to cookies as $4:8$, which simplifies to $1$ cup per $2$ cookies ($1:2$). For $12$ cookies, the calculation is $12 \div 2 = 6$ cups of flour. Key vocabulary for this section includes Direct proportion, Constant, Equivalent ratios, and Unitary method. Ultimately, direct proportion describes quantities changing at a consistent rate, and ratios serve as the mechanism for solving these problems.

Overall Unit Summary and Synthesis

Throughout this unit on Ratio and Proportion, several core principles have been established. Ratios serve as the fundamental tool for comparing quantities. The process of simplifying these ratios is essential because it makes them easier to understand and compare effectively. Sharing in a ratio is a practical application used to divide amounts fairly according to specific parts. Direct proportion is a specific type of relationship where quantities increase or decrease together at a fixed, consistent rate. Finally, the use of equivalent ratios provides a reliable method for solving a wide variety of proportional problems in both academic and real-world contexts.