Ratio and Proportion - Important Points
Proportion
The equality of two ratios is called proportion.
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
Thus, a : b :: c : d <=> (b x c) = (a x d).
Fourth Proportional
If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional
a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional
Mean proportional between a and b is √ab
Componendo and Dividendo
If a/b = c/d, then (a + b) / (a – b) = (c + d) / (c – d)
This rule is known as componando and dividendo rule.
Variation
We say that x is directly proportional to y, if x = ky for some constant k.
We say that x is inversely proportional to y, if xy = k for some constant k.
Example 1: a : b = 3 : 4; b : c = 6 : 7. Find a : b : c.
Solution: a b c
3 4
6 7
a : b : c = 3 × 6 : 6 × 4 : 4 × 7 = 9 : 12 : 14
Example 2: A sum of Rs. 19840 has been divided among A, B and C in the ratio of 5 : 4 : 7. Find the share of B.
Solution: B's share = [4/(5 + 4 +7)] x 19840
= Rs.4960
Example 3: 36% of first number is equal to 28% of the second number. What is the ratio of the first number to the second number?
Solution: Let the numbers be x and y.
36% of x = 28% of y
36/100 of x = 28/100 of y
x : y = 7 : 9
Example 4: In a bowl there is 30 litre mixture of milk and water. The ratio of milk and water is 7 : 3. How much water must be added to it so that the ratio of milk to the water be 3 : 7?
Solution: Milk quantity in the mixture = 7/10 × 30 = 21 litres
So, Water = 30 - 21 = 9 litres
New ratio = 3 : 7
3 parts of milk is 21 litres
Water quantity in the mixture = 7/3 × 21 = 49 litres
49 - 9 = 40 litres water is to be added in the new mixture
Example 5: A bag contains one rupee, 50 paise and 25 paise coins. If these coins are in the ratio of 5 : 6 : 8, and the total amount of coins is Rs. 420, find the number of 50 paise coins in the bag.
Solution: Let the number of one rupee, 50 paise, 25 paise coins be 5x, 6x and 8x respectively
The value of one rupee coins= Rs. 1 × 5x = Rs. 5x
The value of 50 paise coins = Rs. 0.50 × 6x = Rs. 3x
The value of 25 paise coins = Rs. 0.25 × 8x = Rs. 2x
Total value = 5x + 3x + 2x = Rs. 420
=> x = 42
Number of 50 paise coins = 42 x 6 = 252