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

● Ratio and proportion

• Basic Concept of Ratios
• Important Properties of Ratios
• Ratio in Lowest Term
  
• Types of Ratios
• Comparing Ratios
• Arranging Ratios
  
• Dividing into a Given Ratio
  
• Divide a Number into Three Parts in a Given Ratio
• Dividing a Quantity into Three Parts in a Given Ratio
  
• Problems on Ratio
  
• Worksheet on Ratio in Lowest Term
  
• Worksheet on Types of Ratios
  
• Worksheet on Comparison on Ratios
• Worksheet on Ratio of Two or More Quantities
  
• Worksheet on Dividing a Quantity in a Given Ratio
• Word Problems on Ratio
  
• Proportion
  
• Definition of Continued Proportion
  
• Mean and Third Proportional
  
• Word Problems on Proportion
  
• Worksheet on Proportion and Continued Proportion
  
• Worksheet on Mean Proportional
  
• Properties of Ratio and Proportion

10th Grade Math

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