Ratio
The quantities being compared in a ratio might be physical quantities such as speed or temperature, or may simply refer to amounts of particular objects. A common example of the latter case is the weight ratio of water to cement used in concrete, which is commonly stated as 1:4. This means that the weight of cement used is four times the weight of water used. It does not say anything about the total amounts of cement and water used, nor the amount of concrete being made.
A ratio of 2:3 means that the amount of the first quantity is 2/3 (two thirds) of the amount of the second quantity. If the ratio deals with objects or amounts of objects, this is often expressed as "for every two parts of the first quantity there are three parts of the second quantity".
Proportion
Ratio & Proportion Properties
1. Two quantities are said to be commensurable if their ratio can be expressed as the ratio of two integers.
e.g. ratio of 10 and 12 = 10/12 = 5/6 therefore they are commensurable
2. If a:b and c:d are two ratio then
a:b>c:d if ad>bc
a:b < c:d if ad < bc
a:b=c:d if ad=bc
3. A ratio a:b is of
i) greater inequality if a > b
ii) lesser inequality if a < b
iii) equality if a = b
4. i) Compounded ratios are formed if two or more ratios are multiplied term wise.
e.g. 2:3 x 4:5 becomes 2x4 : 3x5 i.e. 8 : 15
ii) Duplicate ratio of a : b is a^2: b^2
iii) Triplicate ratio of a : b is a^3 : b^3
Ration and Proportion Properties (Contd...)
5. Invertendo
If a:b=c:d then b:a = d:c
a:b=c:d i.e. a/b = c/d
Dividing 1 by each of these ratios,
1/(a/b) = 1/(c/d) i.e. b/a = d/c or b:a = d:c
6. Alternendo
If a/b = c/d, multiplying both sides by b/c we get a/b x b/c = c/d x b/c
a/c = b/d or a:c = b:d
7. Componendo
If a:b = c:d then (a+b):b = (c+d):d
8. Dividendo
If a:b = c:d then (a-b):b = (c-d):d
9. Componendo and dividendo
If a:b = c:d then a+b:a-b::c+d:c-d
Examples
1. A petroleum distributor has two gasohol storage tanks, the first containing 9 percent alcohol and the second containing 12 percent alcohol. They receive an order for 300,000 gallons of 10 percent alcohol. How can they mix alcohol from the two tanks to fill this order?
Let x = volume of type(1) 100-x = volume of type(2) in 100 gallons of mixture. Then 0.09x + 0.12(100-x) = 10 gallons -0.03x + 12 = 10 2 = 0.03x and so x = 66.6667 gallons So we use 66+(2/3) gallons type (1) and 33+(1/3) gallons type (2) per 100 gallons of mixture. For 300,000 gallons, we multiply up by 3000 to get 200,000 and 100,000 of type (1) and type (2) respectively.
2. Suppose 30 liters of a solution with an unknown percentage of alcohol is mixed with 5 liters of a 90% alcohol solution. If the resulting mixture is a 62% alcohol solution, what is the percentage of alcohol in the first solution?
Think of the amount of alcohol in each solution. In the first one, it is 30*(x/100) liters. In the second one, it is 5*(90/100). In the last one, it is 35*(62/100) (because the total amount of the mixture is 35 = 30 + 5 liters). When the first two solutions are mixed, the total amount of alcohol is the sum of the amounts in the two ingredients, so you get an equation: 30*(x/100) + 5*(90/100) = 35*(62/100).
Assignments
1. From a circular sheet of paper with a radius of 20 cm, four circles of radius 5cm each are cut out. What is the ratio of the uncut to the cut portion?
2. Two liquids A and B are in the ratio 5 : 1 in container 1 and in container 2, they are in the ratio 1 : 3. In what ratio should the contents of the two containers be mixed so as to obtain a mixture of A and B in the ratio 1 : 1?
3. The cost of a diamond varies directly as the square of its weight. Once, this diamond broke into four pieces with weights in the ratio 1 : 2 : 3 : 4. When the pieces were sold, the merchant got Rs. 70,000 less. Find the original price of the diamond.
4. In a locality, two-thirds of the people have cable-TV, one-fifth have VCR, and one-tenth have both, what is the fraction of people having either cable TV or VCR?
5. A can is full of paint. Out of this 5 litres are removed and thinning liquid substituted. The process is repeated. Now the ratio of paint to thinner is 49 :15. What is the full capacity of the can?
6. The ratio of sum of squares of first n natural numbers with square of sum of first n natural numbers is 17: 325. the value of n is
7. The value of each of a set of silver coins varies as the square of its diameter, if its thickness remains constant; and it varies as the thickness, if the diameter remains constant. If the diameters of two coins are in the ratio 4 : 3 what should the ratio of their thickness be if the value of the first is 4 times that of the second?
