Problems on Ratio

We will discuss here how to solve the problems on Ratio.

1. What should be added to each term of the ratio a : b to make x : y?

Solution:

Let p is to be added to each term of a : b to get the ratio x : y

Therefore,

\(\frac{a + p}{b + p}\) = \(\frac{x}{y}\)

⟹ x(a + p) = y(b + p)

⟹ ax + px = by + py

⟹ py – px = ax – by

⟹ p(y - x) = ax - by

⟹ p = \(\frac{ax - by}{y - x}\)

Therefore, the required number is \(\frac{ax - by}{y - x}\).

2. If a : b = 2 : 3, then find (4a - b) : (2a + 3b)?

Solution:

Given a : b = 2 : 3, then a = 2k, y = 3k (k ≠ 0 is a common multiplier)

Therefore, (4a - b) : (2a + 3b) = \(\frac{4a - b}{2a + 3b}\) = \(\frac{4 ∙ 2k - 3k}{2 ∙ 2k + 3 ∙ 3k}\)

= \(\frac{8k - 3k}{4k + 9k}\)

= \(\frac{5k}{13k}\)

= \(\frac{5}{13}\)

= 5 : 13

3. If x : y = 2 : 5, y : z = 4 : 3 then find x : z.

Solution:

x : y = 2 : 5 ⟹ \(\frac{x}{y}\) = \(\frac{2}{5}\) .......................... (i)

y : z = 4 : 3 ⟹ \(\frac{y}{z}\) = \(\frac{4}{3}\) .......................... (ii)

Multiplying (i) and (ii), we get

\(\frac{x}{y}\) × \(\frac{y}{z}\) = \(\frac{2}{5}\) × \(\frac{4}{3}\)

Therefore, \(\frac{x}{z}\) = \(\frac{8}{15}\)

Therefore, x : z = 8 : 15.

4. If (3x + 5y) : (7x - 4y) = 7 : 4 then find the ratio x : y

Solution:

Given, (3x + 5y) : (7x - 4y) = 7 : 4

⟹ \(\frac{3x + 5y}{7x - 4y}\) = \(\frac{7}{4}\)

⟹ 4(3x + 5y) = 7(7x – 4y)

⟹ 12x + 20y = 49x – 28y

⟹ 12x - 49x = -28y - 20y

⟹ - 37x = - 48y

⟹ 37x = 48y

⟹ \(\frac{x}{y}\) = \(\frac{48}{37}\)

⟹ x : y = 48 : 37

5. If a : b = 5 : 12, b : c = 8 : 3 and c : d = 9 : 16 , what is a : d?

Solution:

a : b = 5 : 12 ⟹ \(\frac{a}{b}\) = \(\frac{5}{12}\) .......................... (i)

b : c = 8 : 3 ⟹ \(\frac{b}{c}\) = \(\frac{8}{3}\) .......................... (ii)

c : d = 9 : 16 ⟹ \(\frac{c}{d}\) = \(\frac{9}{16}\) .......................... (iii)

Multiplying (i), (ii) and (iii), we get

\(\frac{a}{b}\) × \(\frac{b}{c}\) × \(\frac{c}{d}\) = \(\frac{5}{12}\) × \(\frac{8}{3}\) × \(\frac{9}{16}\) = \(\frac{5}{8}\)

Therefore, \(\frac{a}{d}\) = \(\frac{5}{8}\)

Therefore, a : d = 5 : 8

10th Grade Math

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