CBSE Class 10 Mathematics Ratio and Proportion Notes Set 22

Ratio and Proportion

A ratio a : b compares two quantities of the same kind (a/b). The duplicate ratio of a : b is a² : b², the triplicate ratio is a³ : b³ and the sub-duplicate ratio is √a : √b. If a : b = c : d the four numbers are in proportion, so ad = bc.
Useful properties: componendo gives (a + b)/b = (c + d)/d, dividendo gives (a − b)/b = (c − d)/d and componendo-dividendo gives (a + b)/(a − b) = (c + d)/(c − d). If a/b = c/d = e/f then each ratio equals (a + c + e)/(b + d + f).

Variation

y varies directly with x if y/x is a constant k (y = kx); y varies inversely with x if xy is a constant. If y varies directly with x² then y/x² is constant, and so on.
When a quantity depends on two things (price depends on pages and on years), combine the variations: price = k × pages/years, and find k from the first case.

Ratio in Ages and Mixtures

In age problems the difference of two ages stays the same with time while the ratio changes. Write ages as multiples of one variable (3k and 5k) and use the past or future condition to find k.
In mixtures, write each ingredient as a fraction of the vessel, add the amounts of the same ingredient from different vessels and then take the final ratio. Adding the same number to both terms of a ratio moves it towards 1, subtracting moves it away from 1.

Worked Examples

Illustration 1. The ratio of ages of A and B is 5 : 3. Four years later it will be 3 : 2. Find their present ages.
Answer:
Let the ages be 5k and 3k: (5k + 4)/(3k + 4) = 3/2, so 10k + 8 = 9k + 12 and k = 4.
A = 20 years and B = 12 years.

Illustration 2. If x : y = 3 : 4, find (2x + y) : (x + 3y).
Answer:
Take x = 3 and y = 4: (6 + 4) : (3 + 12) = 10 : 15 = 2 : 3.

Illustration 3. y varies inversely with x and y = 6 when x = 4. Find y when x = 8.
Answer:
xy = 24, so y = 3.