The concepts of Ratio and Proportion are fundamental to the study of various topics in the quantitative ability section including Numbers, Geometry, Speed, Distance, and Time concepts. Moreover, if you know ratios and proportions formulas, tips and tricks, then it becomes easy to understand and solve these problems. So go ahead and read on.
Ratios and Proportion Concepts
a) Ratios Concept
The ratio is defined as the comparison of two or more similar variables. A ratio is dimensionless and has no units. In the ratio 1 : 3, the two numbers are called the elements. The first element is called the antecedent and the second element is called the consequent. A ratio can also have more than two elements, as 1:3:7.
Some important rules to remember when dealing with ratios are:
a) Adding/Subtracting a quantity to each element of a ratio: Consider the ratio a: b. Let x be a positive number.
- If a>b, then the ratio (a+x) : (b+x) is less than a : b
- If a<b, then the ratio (a+x) : (b+x) is greater than a : b
- If a=b, then the ratio (a+x) : (b+x) = a : b = 1
If x is a negative number, then the above inequalities get reversed.
b) Duplicate and Triplicate ratios: The ratio a2 : b2 is called the duplicate ratio of a:b, and the ratio a3 : b3 is called triplicate ratio of a:b. Also, 2√a : 2√b and 3√a : 3√b are called the subduplicate and sub-triplicate ratios respectively of a : b.
c) Comparisons of ratios with unity
- If a > b, a : b is more than 1.
- If a < b, a : b is less than 1.
- If a = b, a : b is equal to 1.
d) If the ratio of the two numbers is a : b, the numbers can be assumed to be ak and bk, where k is a constant. When all the elements of a ratio are multiplied or divided by the same number, the resulting ratio is equal to the original ratio.
Commonly made mistakes when calculating Ratios
We must always ensure that the quantities are in the same units before we find the ratio between them. Else, you might end up making mistakes of this kind.
When two ratios are equal, a proportion is formed. If a : b = c : d, then a, b, c and d are in proportion. This is also written as a : b :: c : d . This is equivalent to saying a/b = c/d or a.d = c.b (the product of extremes is equal to the product of means).
Consider a : b = c : d. Then, the following Ratio and Proportion Formulas apply.
- Componendo - Add 1 to both the sides
- Dividendo - Subtract 1 from both the sides
- Componendo and Dividendo
Ratios and Proportions Formula
Here are some of the important Ratio and Proportion Formulas
- If a : b = c : d, we write, a:b ::c:d and we say that a, b, c, d are in proportion.
- Here a and d are called extremes, while b and c are called mean terms.
- Product of means = Product of extremes that is a :b :: c: d = (b*c)=(a*d)
- We say that (a:b)>(c:d) = a/b > c/d.
- The compounded ratio of the ratios (a:b) ,(c:d) , (e:f) is (ace :bdf).
- Duplicate ratio of (a:b) is ( a to the power of 2 : b to the power of 2)
- Sub duplicate ratio (a:b) is (square root of a: square root of b).
- Triplicate ratio of (a:b) is ( a to the power of 3 : b to the power of 3)
- Sub triplicate ratio of (a:b) is (cube th root of a: cube th root of b).
- If a/b =c/d , then a+b / a-b = c+d / c-d .
- We say that x is directly proportional to y, if x=ky for some constant k and we write, x is directly proportional to y.
- We say that x is inversely proportional to y, if xy=k for some constant k and we write, x is indirectly proportional to y.
- Suppose a container contains x units of liquid from which y units are taken out and replaced by water. After n operations, the quantity of pure liquid in the final mixture =[ x*(1-y/x) to the power of n] units.
Ratios and Proportions Problems
Question 1: If in a hostel 45 days, food is available food is available for 50 students, for how many days will this food be sufficient for 75 students?
For 50 students, food is sufficient for 45 days
For 1 student, food is sufficient for 45 x 50 days
and for 75 students, food is sufficient for (45 x 50)/75 days. i,e., for 30 days
Question 2: A and B are two alloys of gold and copper prepared by mixing metals in proportions 7:2 and 7:11 respectively. If equal quantities of the alloys are melted to from a third alloy C, the proportion of gold and copper in C will be?
Gold in C =(7/9 + 7/18) = 21/18 = 7/6
Copper in C = (2/9 + 11/18) = 15/18 = 5/6
∴ Gold : Copper = 7/6 : 5/6 = 7 : 5