6th Class Mathematics Ratio and Proportion Proportion

  Proportion

 

 

The equality of two ratios is called proportion. If a tray of cake is distributed among eight boys and each boy gets equal part of the cake then cake is distributed in proportion. The smallest form of ratio, 12 : 96 is 1: 8 and 19 :152 is 1: 8 therefore,12 : 96 = 19 : 152 is in proportion. if\[x:y=m:n\]then \[x,y,m\]and n are said in proportion and written as, \[x:y::m:n.\]In the proportion, \[x:y::m:n,\]\[x\] and n are first and last term and therefore called extreme terms and middle term y and m are called means. Product of extreme terms of a proportion, \[x\times n\]is always equal to the product of middle terms, \[y\times m,\]therefore,\[x\times n=y\times m.\]lf\[x\times n\ne y\times m\]then, they are not in proportion.    

 

 

The age ratio of Peter and his father is 2 : 5. Find the age of Peter if his fathers age is 40 years?

(a) 17                                                    

(b) 16           

(c) 18                                                    

(d) 19

(e) None of these             

 

Answer: (b)                

Explanation

2 : 5 Peter age :\[\text{40}\Rightarrow \frac{\text{2}}{\text{5}}\text{=}\frac{\text{peter}\,\text{age}}{\text{40}}\] Therefore, age of Peter \[\text{=}\frac{\text{40 }\!\!\times\!\!\text{ 2}}{\text{5}}\text{=}\frac{\text{80}}{\text{5}}\text{=16}\,\text{yearas}\]  

 

 

Continued Proportion

Three numbers a, b and c are said to be in continued proportion even if a, b, b, c are in proportion. The continued proportion a, b, b, c is written as, a : b : : b : c. In the continued proportion, a : b : : b : c, a and c is called extreme terms and twice b is called middle or means term. The product of the extreme and middle terms is always equal. Therefore, \[a\times b=b\times b\]or \[a\times c=b\times c\]or\[a\times c={{b}^{2}}\]or\[{{b}^{2}}=ac.\]    

 

 

The terms a, 5 and 10 are in continued proportion then find the value of a from the options given below?

(a) \[\frac{7}{2}\]                                            

(b) \[\frac{3}{4}\]

(c)  \[\frac{5}{2}\]                                            

(d) All of these

(e) None of these  

 

Answer: (c)

Explanation  

\[a:5::5:10\Rightarrow \] Product of extreme terms = Product of middle terms\[\Rightarrow a\times 10=5\times 5\Rightarrow 10a={{5}^{2}}\Rightarrow a=\frac{{{5}^{2}}}{10}=\frac{25}{10}=\frac{5}{2}\]

 

 

Mean

Proportion The middle term of a continued proportion is called its mean. If a, b and c are in continued proportion then, b is called its mean proportional between a and c, and mean proportion is calculated by \[{{b}^{2}}=ac\]or \[b=\sqrt{ac}.\]    

 

 

Find the mean proportion between 5 and 125?

(a) 35                                                    

(b) 25

(c) 45                                                    

(d) All of these 

(e) None of these                          

 

Answer: (b)                                             

Explanation

Let us consider the mean proportion between  5 and 125 is x, therefore,  

\[x=\sqrt{5\times 125}=\sqrt{625}=25.\]                

 

 

The ratio of man and woman in a joint family is 9 : 8, if number of men in the family is 18. Find the number of women in the family?

(a) 16                                                    

(b) 18

(c) 19                                                    

(d) All of these

(e) None of these  

 

Answer: (a)

Explanation

The ratio of 18 : Number of women = 9 : 8 .

Hence,                

\[\frac{\text{18}}{\text{women}}\text{=}\frac{\text{9}}{\text{8}}\Rightarrow

\text{number}\,\text{of}\,\text{woman}\,\text{=}\frac{\text{18 }\!\!\times\!\!\text{ 8}}{\text{9}}\text{=16}\]                  

 

 

The ratio of mixture of water and lime is 5 : 9, if mixture contains 20 litre of water then find the amount of lime (In kg) in the mixture?                      

(a) 35kg                                                

(b) 36kg                                       

(c) 56 kg                                               

(d) 18 kg               

(e) None of these

 

Answer: (b)  

Explanation                

\[\frac{\text{20}}{\text{Quantity}\,\text{of}\,\text{Lime}}\text{=}\frac{\text{5}}{\text{9}}\]                

\[\text{Quantity}\,\text{of}\,\text{Lime}=\frac{20\times 9}{5}=36\]  

 

 

A shopkeeper gets Rs 150 as a gross income on selling 500 kg of wheat. What is the ratio of the quantity of wheat to income of the shopkeeper?

(a) 8:7                                                  

(b) 7:9

(c) 10:3                                                

(d) All of these

(e) None of these  

 

Answer: (c)

Explanation

The ratio of quantity of wheat to the ratio of gross income \[=\frac{500}{150}=10:3\]      

 

 

  The length of the playground is 500 metres and the ratio of length to width of the playground is 9 : 8. Find the width of the playground?                

(a) 444.444 metres         

(b) 555.555 metres                

(c) 224.345 metres                          

(d) All of these                

(e) None of these  

 

Answer: (a)

Explanation                  

\[\begin{align}   & \frac{\text{Length of the playground}}{\text{Width of the playground}}\text{=}\frac{\text{9}}{\text{8}} \\  &  \\ \end{align}\]\[\Rightarrow \frac{\text{500}}{\text{Width of the playground}}\text{=}\frac{\text{9}}{\text{8}}\]                

\[\text{Width of the playground=}\frac{\text{500 }\!\!\times\!\!\text{ 8}}{\text{9}}\text{=444}\text{.44metres}\text{.}\]  

 

 

Find the value of \[x,\] if \[100:x=550:340?\]

(a) 62.81                                              

(b) 61.81

(c) 66.78                                              

(d) All of these

(e) None of these  

 

Answer: (b)                

Explanation  

\[x=\frac{340\times 100}{550}=61.81\]