Chapter 8: RATIO AND PROPORTION


Introduction


Ratio means two quantities in the same units like Hari's height is 100 cm and Ram's height is 200 cm so the ratio of their heights is 100/200 or 1:2. the ratio has no units.

Proportion is equality of two ratio's a:b = c:d we write as a:b::c:d and a,d are extremes and b,c are means. The product of means = product of extremes.
                                                a * d = b * c

When we have to divide a quantity 'x' into a:b we need to break it into two parts one is a/a+b and second is b/b+a.



Componendo - Dividendo rule:

if a:b = c:d then (a+b)/(a-b) = (c+d)/(c-d)



Proportionality:

If x ∝ y i.e. x is directly proportional to y which means when x increases y too increases and when x decreases y to decreases.

If
x ∝ 1/y i.e. x is inversely proportional to y which means if x increases y decreases and when x decreases y increases.

x ∝ y then x = k*y where k = constant
x ∝ 1/ y then x * y= k where k = constant



Questions & Answers:


Q. If a:b = 3:4 and b:c is 5:4 then find a:b:c

A. The value common to both is b so b has to made same in both ratios. The LCM of 4,5 is 20 so we convert the ratios by multiplication


a:b = (3*5):(4*5) and b:c = (5*4)/(4*4)

a:b = 15:20 and b:c = 20:16 now b value is same in both sides and so we can merge on basis of 'b'.

a:b:c = 15:20:16


Q.Find fourth proportional between 4,9,12

A. Assume fourth proportional as 'x'. so we get 4:9::12:x .

As we know from above formula: 4*x = 9*12 so we can get value of x.




Q. Third proportional to 16, 36.

A. Assume third proportional is 'x'. so we get 16:36::36:x

As we know from formula that 16*x = 36*36 we can get 'x'.


Q. Mean proportional between 0.08 and 0.18

A. mean proportional = square-root(0.08*0.18)

= square-root(8/100 * 18/100)

= square-root(144/10000)

=12/100

=0.12




Q. If x : y = 3 : 4, find (4x + 5y) : (5x - 2y)

A. 4x+5y : 5x-2y = 4(x/y)+5 / 5(x/y) - 2  we get this by dividing by y to both numerator and denominator.

substituting x/y in equation we can solve it.


Q.   Divide Rs. 600 in the ratio 4:2

A. the ratio 4:2 means first value shall be 4/4+2 and second value shall be 2/4+2 i.e. 600 has to be divided into 4/6 and 2/6 parts. so we get 400,200.


Q. A bag contains 50 p, 25 P and 10 p coins in the ratio 5: 9: 4, amounting to Rs. 206. Find the number of coins of each type.

A. Since ratio is 5:9:4 this means actual number of coins can be 5x, 9x and 4x.

We know that, 5x*(1/2) + 9x*(1/4) + 4x*(1/10) = 206 as we converted the paisa values to rupee.




Q. A mixture contains alcohol and water  in  the  ratio  4  : 3.  If 5  litres of water  is added  to  the mixture, the ratio becomes 4: 5. Find the quantity of alcohol in the given mixture.

A. Alcohol and water were 4x and 3x and now 4x / 3x+5 = 4 / 5.

Quiz

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