Ratio and Proportion : Problems, Formulas, and Shortcuts
Ratio means the number of times one quantity contains another quantity. However, the quantities in question should be of same type. You can’t do a ratio between 2 liters of oil and 6gm of gold because they are quantities of different types. A ratio of 2 liters of oil can be done with 6 liters of oil.
Ratio is expressed in the format A:B or A/B.
In A:B, A is called antecedent and B is called consequent.
Types of Ratio:
- Duplicate Ratio: This ratio is the square of two numbers. Example: 25:4 is the duplicate ratio of 5:2.
- Triplicate Ratio: This ratio is the cube of two numbers. Example: 125:8 is the triplicate ratio of 5:2.
- Sub-duplicate Ratio: This ratio is the square root of two numbers. Example: 5:2 is the sub-duplicate of 25:4
- Sub-triplicate Ratio: This ratio is the cube root of two numbers. Example: 5:2 is the sub-duplicate of 125:8
- Inverse Ratio: The ratio in which antecedent and consequent change their places. Example: 25:4 is the inverse of 4:25.
Two ratios are said to be in Proportion when they are equal. For example: 2/5 = 4/10.
This is because 4/10 can be divided by 2 and can be simplified as 2/5. Hence, 2/5 and 4/10 are said to be in proportion. In the equation A:B = C:D, A and D are called extremes, whereas B and C are called means.
In a proportion, product of means is always equal to product of extremes.
In other words, for proportion A:B = C:D;
AD = BC
Types of Proportions:
- Forth Proportion: When a:b = c:d, then d is called the forth proportion of a,b and c.
- Third Proportion: When a:b = b:c, then c is called the third proportion of a and b.
- Second Proportion: When a:b = b:c, then b is called second or mean proportion of a and c.
Enough with the theory. Moving on to….
Shortcut 1: When A : B = m : n, and B : C = p : q. Then A : B : C = mp : pn :nq
Question 1: If A:B = 2:3 and B:C = 4:5, find A:B:C?
Solution 1: A:B:C = 2×4: 3×4: 3×5 = 8:12:15
Question 2: If A:B = 5:3 and B:C = 6:2, find A:B:C?
Solution 2: A:B:C = 30:18:6 (Try to solve this mentally. Its very easy)
Shortcut 2: When A : B = m : n, B : C = p : q, and C : D = r : s. Then A : B : C : D = mpr: npr : nqr : nqs
Question 1: If A:B = 2:3, B:C = 3:4, C:D = 4:5, find A:B:C:D?
Solution 1: 2x3x4: 3x3x4: 3x4x4: 3x4x5 = 24:36:48:60 (Try to solve this mentally)
Shortcut 3: In a container, the ratio between oil and water is P:Q. The total quantity is x litres. To find the amount of water to be mixed so that the ratio becomes A:B, use
x(pb – qa)
a(p + q)
Question 1: A container has oil and water in the ratio 2:3. The total quantity of oil and water is 10 liters. What amount of water should be added, so that the ratio becomes 1:3?
Solution 1: Solving by formula,
Amount of water to be added = 10 (2×3 – 3×1) = 10 (6-3) = 6 liters
1 (2+3) 5
(More shortcuts coming soon)