7th Class Mathematics Comparing Quantities Proportion

     Proportion

It is the equality of two ratios i.e. if a: b = c: d, then ad = cd that implies product of extremes = product of means. Four quantities p, q, r, s are in proportion if ps = qr                

Important Points Related to Proportion

If \[\frac{a}{b}=\frac{c}{d}\]then

• \[\frac{a+b}{b}=\frac{c+d}{d}\](componendo)
• \[\frac{a-b}{b}=\frac{c-d}{d}\](dividend)
• \[\frac{a+b}{a-b}=\frac{c+d}{c-d}\] (componendo and dividendo)
• If three numbers a, b, c are in continued proportion and written as a : b :: b: c then \[\frac{a}{b}=\frac{b}{c}\Rightarrow {{b}^{2}}=ac\Rightarrow b=\sqrt{ac}\] hence, b is called mean.  

  The Ratio between two quantities is 7: 9. If the first quantity is 511 then find the other quantity.

(a) 655                                                 

(b) 555

(c) 65                                                    

(d) 656

(e) None of these  

Answer: (c)

Explanation

Let the other quantity be x then                

\[7:9=511:x\Rightarrow x=\frac{511\times 9}{7}=657\]  

  Find two numbers so that their mean proportional is 14 and third proportional is 112.                

(a) 6 and 27                                        

(b) 7 and 28                

(b) 9 and 29                       

(d) 10 and 30

(e) None of these  

Answer: (b)

Explanation

Let the number be \[x\]and \[y,\]then according to question

\[\sqrt{xy}=14\Rightarrow xy=196.........(i)\]

\[\Rightarrow \frac{x}{y}=\frac{y}{112}\Rightarrow {{y}^{2}}=112x.....(ii)\]

From (i) \[y\frac{196}{x}\Rightarrow \frac{{{(196)}^{2}}}{{{x}^{2}}}=112x\Rightarrow {{x}^{3}}=343\Rightarrow x=7\]and \[y=\frac{196}{7}=28\]

Hence, the required numbers are 7 and 28  

  The ration of the volume of three buckets is 3 : 4 : 5. Buckets contains the mixture of water and alcohol. If the mixture contains water and alcohol in the ratio 1 : 4, 1 : 3 and 2 : 5 respectively then find the ratio of water and alcohol when the mixture in all containers are poured in fourth container.

(a) 35:57                                              

(b) 53:157

(c) 157:53                                           

(d) 35:157

(e) None of these  

Answer: (b)  

  Find the ratio among the time taken by three buses to travel the same distances if the ratios of their speed are 5:4:6.

(a) 10 :12 :15                                      

(b) 10 :15 : 12

(b) 15:12:10                                       

(d) 12:15:10

(e) None of these  

Answer: (d)  

• When comparison is made by dividing one quantity by another of the same kind, the result is called ratio. If a and b are two numbers then the ratio of a to a b is denoted by a : b or \[\frac{a}{b}.\]
• Multiply or divide each term of a ratio by the same number, the ratio remains unchanged.
• The equality of two ratios is called proportion. If a : b = c : d then a, b, c and d are called in proportion.
• In a proportion a : b : : c : d then a and d are called extremes and b and c are called mean.
• Product of extremes is equal to product of mean. 

(a)    \[\sqrt{ab}\] is called the mean proportion of a and b.

(b)    If a: b:: b : c then c is called third proportion of a and b.  

• Do you know if p, q and r are in proportion then p: r is the duplicate ratio of p: q.
• If q, r and s are in proportion then p:s is the triplicate ratio of p:q.
• Sine, cosine, tangent, cotangent, secant, cosecant are special ratios which describe trigonometrically function.