Proportion

**Category : **6th Class

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**

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

*play_arrow*Introduction*play_arrow*Ratio*play_arrow*Proportion*play_arrow*Ratio and Proportion*play_arrow*Ratio, Proportion & Unitary Method*play_arrow*Ratio and Proportion*play_arrow*Ration and Proportion

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