Today our topic of discussion is Ratio and Proportion.
Ratio and Proportion
Algebraic Ratio and Proportion
It is important for us to have a clear conception of ratio and proportion in mathematics. Arithmetical ratio and proportion have been elaborately discussed in class VII. In this chapter, algebraic ratio and proportion will be discussed.
We regularly use the concept of ratio and proportion in many applications: the production of construction materials, food staff and goods, applying fertilizer in land, making beautiful shapes and design to make things attractive and good looking, and so on. Many problems of daily lives can also be solved by using the concept of ratio and proportion.
After studying this chapter, the students will be able to –
‣ exp ain algebraic ratio and proportion.
‣ use different types of rules of transformation related to proportion.
‣ learn successive proportion.
‣ use ratio, proportion, successive proportion in solving real life problems.
Ratio and Proportion
Ratio
If we have two quantities (or numbers) of the same unit, we can express one quantity as a multiple (or a fraction) of the other quantity. This fraction is called the ratio of two quantities. Thus, a ratio says how much of one thing there is compared to another thing.
The ratio of two quantities p and q is written as p: q=p/q. These two quantities, p and q are of same kind and same unit. Here, p is called antecedent and q is called subsequent of the ratio.
Sometimes we use the ratio as an approximate measure. For example, the number of cars on the road at 10 A.M. is two times higher than the number of cars at 8 A.M. In this case, it is not necessary to know the exact number of cars to determine the ratio.
Again, in many occasions, we say that the area of your house is three times to the area of mine. Also in this case, it is not necessary to know the exact area of the house. We use the concept of ratio in many other cases of practical life.
Proportion
If four quantities (or numbers), are such that the ratio of the first and the second quantities is equal to the ratio of the third and the fourth quantities. those four quantities form a proportion. If a, b, c, d are four such quantities, we write a:b=c :d. The four quantities need not to be of same kind. The two quantities of the ratio need to be of the same kind.
In the above figure, let the bases of two triangles be a and b, respectively and each of their height is h unit. If the areas of these triangle are A and B square units, respectively, we can write.
A/B={(1/2)ah}/{(1/2)bh}=a/b
or, A: B =a: b
bh i.e. the ratio of two areas is equal to the ratio of two bases.
Continued proportion
By continued proportion of a,b,c, it is meant that a: b=b: c. a, b, c will be continued proportion, If and only if b2-ac. In case of continued proportional, all the quantities are to be of same kind. In this case, c is called the third proportional of a and b and b is called the mid-proportional of a and c.
Example 1. A and B traverse the fixed distance in t1 and t2 minutes. Find the ratio of the average velocity of A and B
Solution: Let the average velocities of A and B be v1 metre and v2 metre per minute respectively. So, in time t1 minutes A traverses v1.t1 metre and in t2 minutes B traverses v2.t2 metra. According to the problem, v1.t1 = v2.t2
∴ v1/v2 = t2/t1 time.Here, the ratio of the velocities is inversely proportional to the ratio of time.
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