Hello, and welcome to today's mathematics lesson. Today, we are going to explore Chapter Seven: Ratio and Proportion. This is a fascinating topic that helps us compare quantities and understand how things are related to one another. We will learn what ratios are, how to simplify them, how to convert fractional ratios into whole numbers, how to divide quantities in a given ratio, and finally, we will discover the powerful concept of proportion. Let us begin.
Let us start with the fundamental question: what exactly is a ratio? A ratio is a relationship between two quantities of the same kind, having the same unit, and it is obtained by dividing the first quantity by the second. The symbol for ratio is the colon, written as :, and it is placed between the two quantities being compared.
For example, consider fifteen kilograms and twenty kilograms. The ratio between them is written as fifteen kilograms to twenty kilograms, which simplifies to fifteen to twenty, and further reduces to three to four. So we write this as 3 : 4.
Now, there are some important conditions we must remember. First, the two quantities must be of the same kind. You can have a ratio between fifty rupees and eighty rupees, but you cannot have a ratio between fifty rupees and eighty kilograms — that simply does not make sense. Second, before finding a ratio, both quantities must be expressed in the same unit. For instance, to find the ratio between thirty centimetres and two metres, you must first convert two metres to two hundred centimetres. Then the ratio becomes thirty to two hundred, which simplifies to three to twenty.
In any ratio, the first term is called the antecedent, and the second term is called the consequent. In the ratio three to four, three is the antecedent and four is the consequent. A ratio must always be expressed in its simplest form, meaning the antecedent and consequent must be co-prime — their highest common factor must be one. For example, the ratio six to eight is not in simplest form because the HCF of six and eight is two. Dividing both terms by two gives us three to four, which is the simplest form.
Here is a crucial point: a ratio is a pure number. It has no unit. When you divide kilograms by kilograms, or metres by metres, the units cancel out, leaving just a number.
Now, what happens when we encounter fractional ratios? Suppose you need to simplify the ratio one-third to one-fourth. There are two methods to handle this.
The first method is direct division: divide the first quantity by the second. One-third divided by one-fourth equals one-third multiplied by four over one, which gives four over three, or 4 : 3.
The second method uses the LCM. Find the LCM of the denominators three and four, which is twelve. Multiply each term of the ratio by twelve. One-third times twelve is four, and one-fourth times twelve is three. So the ratio becomes 4 : 3.
This works because of an important property: if each term of a ratio is multiplied or divided by the same non-zero number, the ratio remains unchanged.
Let us try a more complex example: four and one-quarter to one and one-twelfth. First, convert to improper fractions: seventeen-fourths to thirteen-twelfths. The LCM of four and twelve is twelve. Multiply each term by twelve: seventeen-fourths times twelve gives fifty-one, and thirteen-twelfths times twelve gives thirteen. The simplified ratio is 51 : 13.
For three fractional quantities, the same principle applies. Take one and one-third, two and one-quarter, and one and five-sixths. Convert to improper fractions: four-thirds, nine-fourths, and eleven-sixths. The LCM of three, four, and six is twelve. Multiplying each term by twelve gives sixteen, twenty-seven, and twenty-two. The whole number ratio is 16 : 27 : 22.
One of the most practical applications of ratio is dividing a quantity among several people or parts. Let us see how this works.
Suppose twenty sweets are to be distributed between A and B in the ratio two to three. This means for every two parts A receives, B receives three parts. Together, they receive two plus three, which equals five parts. So A gets two-fifths of the total sweets, and B gets three-fifths of the total sweets.
Calculating: two-fifths of twenty equals eight sweets for A. Three-fifths of twenty equals twelve sweets for B. Notice that eight plus twelve equals twenty, confirming our answer.
Here is another example: divide two hundred and sixty rupees among A, B, and C in the ratio 1/2 : 1/3 : 1/4. First, convert this fractional ratio to whole numbers. The LCM of two, three, and four is twelve. Multiplying each term by twelve gives 6 : 4 : 3. The sum of these parts is six plus four plus three, which equals thirteen.
Therefore, A's share is six-thirteenths of two hundred and sixty, which equals one hundred and twenty rupees. B's share is four-thirteenths of two hundred and sixty, which equals eighty rupees. C's share is three-thirteenths of two hundred and sixty, which equals sixty rupees.
Sometimes we work with ratios when given differences or sums. Consider two numbers in the ratio 10 : 13, with a difference of forty-eight. The difference between the ratio terms is thirteen minus ten, which equals three. When the difference is three, the first number corresponds to ten. So when the difference is one, the first number is ten-thirds. When the difference is forty-eight, the first number is ten-thirds times forty-eight, which equals one hundred and sixty. Similarly, the second number is thirteen-thirds times forty-eight, which equals two hundred and eight.
Alternatively, using algebra: let the numbers be 10x and 13x. Then thirteen x minus ten x equals forty-eight, so three x equals forty-eight, and x equals sixteen. The numbers are ten times sixteen and thirteen times sixteen, giving one hundred and sixty and two hundred and eight respectively.
When given the LCM or HCF of numbers in a ratio, we use similar reasoning. If two numbers are in the ratio 4 : 5 and their LCM is one hundred and eighty, let the numbers be 4x and 5x. The LCM of four x and five x is twenty x. Setting twenty x equal to one hundred and eighty gives x equals nine. The numbers are thirty-six and forty-five.
If the HCF is given instead, say fifteen for numbers in the ratio 3 : 4, let the numbers be 3x and 4x. The HCF of three x and four x is x itself. So x equals fifteen, and the numbers are forty-five and sixty.
Now we come to one of the most elegant concepts in this chapter: proportion.
When four quantities are related such that the ratio of the first to the second equals the ratio of the third to the fourth, we say these quantities are in proportion. In other words, proportion is the equality of two ratios.
We write this using an equals sign or a double colon. For example, 15 : 20 = 9 : 12, or 15 : 20 :: 9 : 12. Each quantity is called a term or proportional. The first and last terms are the extremes, while the second and third terms are the means.
Here is a fundamental property of proportion: in any proportion, the product of the extremes always equals the product of the means. In our example, fifteen times twelve equals one hundred and eighty, and twenty times nine equals one hundred and eighty as well. This property allows us to find missing terms.
The fourth term of a proportion is called the fourth proportional. To find the fourth proportional of three, four, and eighteen: let it be x. Then 3 : 4 = 18 : x. By the product rule, three times x equals four times eighteen, so x equals seventy-two divided by three, which is twenty-four.
Finally, let us explore continued proportion. Three quantities are in continued proportion when the ratio of the first to the second equals the ratio of the second to the third. That is, a : b = b : c.
In this case, the middle quantity b is called the mean proportional between a and c. From a : b = b : c, we get b² = ac, so b equals the square root of ac. For example, the mean proportional between two and eight is the square root of 2 × 8, which is the square root of sixteen, which is four.
The third quantity c is called the third proportional to a and b. To find the third proportional to twelve and thirty: let it be x. Then 12 : 30 = 30 : x. So twelve times x equals nine hundred, so x equals seventy-five.
When combining ratios, we often need to find a common term. If a : b = 4 : 5, and b : c = 6 : 7, to find a : b : c, we make the b terms equal. Multiply the first ratio by six and the second by five. This gives 24 : 30 and 30 : 35. Therefore, a : b : c = 24 : 30 : 35.
Let us recap the key takeaways from today's lesson. First, a ratio compares two quantities of the same kind and same unit, expressed using a colon between two numbers in their simplest form. Second, fractional ratios can be converted to whole number ratios by multiplying each term by the LCM of the denominators. Third, to divide a quantity in a given ratio, express each share as a fraction of the total parts and multiply by the total quantity. Fourth, proportion is the equality of two ratios, written with an equals sign or double colon, where the product of extremes equals the product of means. Fifth, in continued proportion, the square of the mean proportional equals the product of the first and third terms; therefore, the mean proportional equals the square root of their product. And sixth, when combining ratios, make the common term equal by finding appropriate multipliers.
Ratio and proportion are powerful tools that appear everywhere in mathematics and daily life — from cooking and mixing paints to business partnerships and map scales. Master these concepts, and you will find yourself solving problems with confidence and clarity. Keep practising, stay curious, and I look forward to seeing you in our next lesson. Goodbye for now.