Hello, and welcome to today's mathematics lesson. We are going to explore ratio and proportion, including their properties and practical uses. By the end of this session, you will understand how to work with ratios, proportions, and variations, and you will see how these powerful tools help us solve real-world problems.
Let us begin with the fundamental idea of a ratio. The ratio of two quantities of the same kind and in the same units is obtained by dividing the first quantity by the second. If we have two quantities, a and b, where b is not zero, then the quotient a/b is called the ratio between a and b. We write this as a : b.
Here is a crucial point to remember. When a ratio is given as x : y = 2 : 3, you cannot simply assume x = 2 and y = 3. The correct approach is to introduce a constant k, writing x = 2k and y = 3k. This ensures your calculations remain valid for all proportional values.
Let us work through an example. Suppose we need to find what quantity must be added to each term of the ratio 8 : 15 to make it equal to 3 : 5. We let x be the quantity added. Then (8 + x)/(15 + x) = 3/5. Cross-multiplying gives 40 + 5x = 45 + 3x, so 2x = 5, and x = 2½.
Now we turn to the composition of ratios. When we combine two or more ratios by multiplying their corresponding terms, we obtain the compound ratio. For ratios a : b and c : d, the compound ratio is (a × c) : (b × d).
From this idea, several special ratios emerge. The duplicate ratio of a : b is a² : b². The triplicate ratio is a³ : b³. The sub-duplicate ratio of a : b is √a : √b. The sub-triplicate ratio is ∛a : ∛b. Finally, the reciprocal ratio of a : b is b : a.
Consider finding the compound ratio of three ratios: 3a : 2b, 2m : n, and 4x : 3y. We multiply the antecedents: 3a × 2m × 4x, giving 24amx. We multiply the consequents: 2b × n × 3y, giving 6bny. Simplifying, the compound ratio is 4amx : bny.
Let us now define proportion. Four non-zero quantities a, b, c, d are said to be in proportion if a : b = c : d. We write this as a : b :: c : d, read as "a is to b as c is to d."
In this proportion, a and d are called the extremes, while b and c are called the means. A fundamental property follows: the product of the extremes equals the product of the means, so ad = bc. This relationship is crucial for solving proportional problems.
When three quantities are in continued proportion, the ratio of the first to the second equals the ratio of the second to the third. Thus, a, b, c are in continued proportion when a : b = b : c. Here, b is called the mean proportional between a and c, and c is called the third proportional to a and b.
For example, to find the mean proportional between 6.25 and 0.16, we set up 6.25 : x = x : 0.16. This gives x² = 1, so x = ±1.
We now explore powerful properties of proportion that simplify complex problems.
First, invertendo: if a : b = c : d, then b : a = d : c. Second, alternendo: if a : b = c : d, then a : c = b : d. Third, componendo: if a : b = c : d, then (a + b) : b = (c + d) : d. Fourth, dividendo: if a : b = c : d, then (a - b) : b = (c - d) : d.
The most powerful of these is componendo and dividendo combined. If a : b = c : d, then (a + b) : (a - b) = (c + d) : (c - d). This property is especially useful when solving equations with square roots or complex fractions.
Let us apply this to solve (x³ + 3x)/(3x² + 1) = 341/91. Applying componendo and dividendo, we get (x³ + 3x² + 3x + 1)/(x³ - 3x² + 3x - 1) = 432/250. Recognizing these as perfect cubes, (x + 1)³/(x - 1)³ = 6³/5³. Taking cube roots, (x + 1)/(x - 1) = 6/5. Applying componendo and dividendo once more yields 2x/2 = 11/1. Therefore, x = 11.
Finally, we examine variations, which describe how quantities change together.
Two quantities are in direct variation when an increase in one causes a proportional increase in the other. If x and y vary directly, then x/y = constant, or equivalently, x = ky for some constant k. For instance, the cost of pens varies directly with the number purchased.
Two quantities are in inverse variation when an increase in one causes a proportional decrease in the other. If x and y vary inversely, then xy = constant. For example, the time taken to complete a job varies inversely with the number of workers assigned to it.
Consider a hostel with provisions for 50 students lasting 90 days. If 25 new students join, making 75 students total, how long will the provisions last? Since students and days are in inverse variation, 50 × 90 = 75 × a, giving a = 60 days.
Let us recap the key takeaways from this chapter. First, a ratio compares quantities of the same kind, and we use a constant multiplier k for general solutions. Second, compound ratios combine multiple ratios through multiplication, leading to duplicate, triplicate, sub-duplicate, sub-triplicate, and reciprocal forms. Third, proportion states that the product of extremes equals the product of means, with continued proportion involving three or more terms. Fourth, properties like componendo and dividendo provide elegant shortcuts for solving complex proportional equations. Fifth, direct and inverse variation describe how quantities change together, with applications to speed, work, and resource problems.
Sixth, practice with the constant k method ensures accurate solutions to ratio problems.
Mathematics is not merely about memorising formulas; it is about understanding relationships and applying logical reasoning. Ratio and proportion form the foundation for advanced topics you will encounter in higher classes. Practice these concepts thoughtfully, and you will find them powerful allies in your mathematical journey. Thank you for your attention, and keep exploring the beautiful patterns of mathematics.