6th Class Mathematics Ratio and Proportion Ratio

   Ratio

 

 

Two or more quantities are compared by their difference.

Mariam has 24 note books but Hilary has 78, comparison of both the quantities is their difference, Mariam has, 78 - 24 = 54 note books less than those Hilary has. In another words it is said that, Hilary has 54 more books than those Mariam has.

Another method which is used to compare the two or more quantities is division method. Peter has 20 litres of milk but John has 5 litres, the comparison of the quantities is said to be, Peter has 15 litres more milk than John, but by division of both the quantity it is said that Peter has, \[\frac{20}{5}=4\] times bigger quantity of milk than John. It can be expressed in the ratio form as \[=\frac{1}{4}\]

When Peter has 4 litres milk, John has 1 litre and quantity of milk is increased according to the ratio of y. In the ratio of \[=\frac{4}{1}\] is called antecedent and 1 is consequent.

Two numbers x and y whenever, written in the form of  \[x:y\] is called in ratioand read as \[x\]is to y. In the ratio\[~x\text{ }:\text{ }y,\]\[x\] is the first and \[y\]is second term. The terms of the ratio can not be interchanged or \[x\text{ }:\text{ }y\] cannot be written as \[y:x.\]The ratio form of the numbers can be expressed infraction in this way that the antecedent is numerator and consequent is denominator, therefore, the fractional form of \[x:y=\frac{x}{y}.\]  

 

 

 

The distance between two points on the map is 4 cm, which represents the distance of 1500 km. Find the ratio of actual as well as mapped or imaginary distance?

(a) 2 : 345                                            

(b) 375 : 1

(c) 23 :56                                             

(d) All of these

(e) None of these  

 

Answer: (b)                

Explanation                

Ratio of the actual distance to the given scale \[=\frac{1500}{4}=\frac{375}{1}=375:1.\]  

 

 

Simplest form of Ratio                

If the common factor of antecedent and consequent is not other than 1 then it is called in its simplest form of ratio. Complex form of ratio can be converted into its simplest form by division with same number to both the antecedent and consequent.                

The simplest form of ratio is also called ratio in its lowest term. The ratio 24 : 30 is not in its simplest form because, it has common factor 6, therefore, the simplest or lowest term of ratio,                

\[24:30=\frac{24}{30}=\frac{4}{5}=4:5,\] In this ratio, the factor of antecedent 4 and consequent 5 is 1 only, therefore, called simplest form of ratio.                

 

 

The area of a circular garden inside the rectangular garden is 515 sq. metres. If the area of rectangular garden is 5000 sq. metres. Find the ratio in its simplest form of area of rectangular as well as circular part of the garden?                

(a) 99:1000                                         

(b) 1000:103

(c) 123:4000                                       

(d) All of these

(e) None of these  

 

Answer: (b)                

Explanation

The ratio of area of the rectangular garden to the area of circular garden = \[\frac{5000}{515}=\frac{1000}{103}=1000:103.\]

 

 

Comparison of Ratios

The quantity of two different kinds of object cannot be compared, therefore, three   pencils cannot be compared with four pens but three pencils can be compared  with 10 pencils due to its same kind. The ratio of two same kinds of objects is the   method to express the factor by which one quantity is greater or smaller than the  other quantity. The comparison of two or more fractional form of ratio is compared    by the same method as the comparison of two unlike fractions. Comparison of the following ratios, 5 : 6, 8 : 13, 9 : 16 are compared by first converting them into like fractions, therefore, the fractional form of given ratios are \[\frac{5}{6},\frac{8}{13},\frac{9}{16}\] The LCM of denominators of the fractions

2	6,13,16
	3,13,8

\[=2\times 3\times 13\times 8=624\]

Now, denominator of every fraction should become 624 by multiplying with a same number to both numerator and denominator of each fraction.

Therefore, \[\frac{5}{6}\times \frac{104}{104}=\frac{520}{624},\frac{8}{13}\times \frac{48}{48}=\frac{384}{624}\]and \[\frac{9}{16}\times \frac{39}{39}=\frac{351}{624}.\] Equivalent factions of the given fractions are, \[\frac{520}{624},\frac{384}{624},\frac{351}{624},\]denominators of all the fractions are equal. Therefore, greater fraction has greater numerator and smaller has smaller denominator, therefore, the ascending order of the fractions are,\[\frac{351}{624}<\frac{384}{624}<\frac{520}{624}\]or \[\frac{9}{16}<\frac{8}{13}<\frac{5}{6}\] or 9: 16<8:13<5:6, therefore the smallest ratio among the given ratio is 9 : 16 and greatest ratio is 5 : 6.    

 

 

Compare the following ratios, 56 : 67, 45 : 37, 44 : 79 and write them descending order?                

(a) 56 : 67 > 45 : 37 > 44: 79          

(b) 56 : 67 > 44 : 79 > 45: 37

(c) 45 : 37 > 56 : 67 > 44: 79          

(d) All of these                

(e) None of these

 

Answer: (c)

Explanation

The LCM of denominators of the fractions is 195841. Therefore, the equivalent fractions with common denominator of the given fractions are,\[\frac{238185}{195841},\frac{163688}{195841},\frac{109076}{195841}\]and order of the fractions are, \[\frac{45}{17}>\frac{56}{67}>\frac{44}{79}\]or 45: 37>56:67>44:79.  

 

 

Equivalent Ratio                

The equivalent ratio of a given ratio is obtained by multiplying or dividing the antecedent and consequent of the ratio by the same number. The equivalent ratio of a : b is \[a\times q:b\times q\]whereas, a, b, q are natural numbers and q is greater than 1.

Therefore, the equivalent ratios .of 5 : 8 are, \[\frac{5}{8}\times \frac{2}{2}=\frac{10}{16}\]or 10:16, \[\frac{5}{8}\times \frac{3}{3}=\frac{15}{24}\]or 15:24, \[\frac{5}{8}\times \frac{12}{12}=\frac{60}{96}\] or 60: 96 .  

 

 

Find three equivalent ratios of 7 : 9?                

(a) 14 :18, 21: 27 and 28 : 36        

(b) 12 : 28, 27 : 57 and 46 : 98                

(c) 23 : 34, 45 : 67 and 58 : 78       

(d) All of these                

(e) None of these  

 

Answer: (a)

Explanation         

Three equivalent ratios of 7 : 9 are, 14 :18, 21: 27 and 28 : 36.               

 

Three statements are given below. Which one of the following options is correct about the given statements?                

(i) 5:6

 

In a garden, the ratio of the number of children to the number of women is 9 : 2. If there are 84 more children than woman then find the number of children in the garden?

(a) 108                                                 

(b) 120 

(c) 156                                                  

(d) 182

(e) None of these  

 

Answer: (a)

Explanation                

Let the number of children in the garden is \[x.\]According to the question,                

\[\frac{9}{2}=\frac{x+84}{x}\Rightarrow 9x=2(x+84)\]\[\Rightarrow 9x2x+168\Rightarrow 9x-2x=168\Rightarrow x\frac{168}{7}=24.\]                

Hence the total number of children in the garden = 84 + 24 = 108.  

 

 

Number of non milch animals in a village is 350 and the number of milch animals is 550. Find the ratio of non milch to milch animals in the village?                

(a) 7:1                                                  

(b) 7:11                

(c) 3:21                                                

(d) 5:1                

(e) None of these  

 

Answer: (b)

Explanation

The ratio of non milch to the milch animals in the village \[=\frac{350}{350}=\frac{7}{11}=7:11\]  

 

 

If x = 6y and 8y = 4z, then the ratio of x to z is?

(a) 7:11                                                

(b) 5:7

(c) 3:1                                                   

(d) 9 : 7

(e) None of these  

 

Answer: (c)                

Explanation                

x = 6y and 8y = 4z.

Hence, z = 2y.

The ratio \[x\]to  \[z=\frac{6y}{2y}=3:1\]

Therefore, option (c) is correct and rest of the options is incorrect.  

 

 

The ratio of two numbers is 5 : 9 and their sum is 392. Find the difference of the numbers?

(a) 146                                                 

(b) 112

(c) 166                                                  

(d) 145

(e) None of these  

 

Answer: (b)                

Explanation  

Let us consider the multiplier \[x.\] According to the question \[5x+9x=392\Rightarrow 14x=392\Rightarrow x=14.\]  Hence, both the numbers will be 140 and 252 and their difference = 252 - 140 = 112.