Ratio and Proportion - 04

Explanation:-

Ratio of the expenditures of A, B, C, D are as under: $$\eqalign{ & {\text{A}} \to \frac{{18}}{{40}} = \frac{9}{{20}} = 0.45 \cr & {\text{B}} \to \frac{{21}}{{50}} = 0.42 \cr & {\text{C}} \to \frac{{24}}{{60}} = 0.4 \cr & {\text{D}} \to \frac{{27}}{7} = 0.385. \cr}$$ Clearly, A has done the highest expenditure proportionate to his resources.

Explanation:-

$$\eqalign{ & \,\,\,\,\,\,\,{25^{2.5}}:{5^{3{\text{ }}}} \cr & {(5)^{2.5 \times 2}}:{5^{3{\text{ }}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,{5^5}:{5^3} \cr & \,\,\,\,\,\,{5^3}{.5^2}:{5^{3{\text{ }}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,{5^2}:1 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,25:1 \cr}$$

Explanation:-

$$\eqalign{ & 5.5{\text{a}} = 0.65{\text{b}} \cr & \Rightarrow \frac{{55}}{{10}}{\text{a}} = \frac{{65}}{{100}}{\text{b}} \cr & \Rightarrow 55{\text{a}} = \frac{{65}}{{10}}{\text{b}} \cr & \Rightarrow 550{\text{a}} = 65{\text{b}} \cr & {\text{a}}:{\text{b}} = 65:550 \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 13:110 \cr}$$

Explanation:-

Let the terms of the ratio be x and x + 40. $$\eqalign{ & {\text{Then,}} \cr & = \frac{x}{{x + 40}} = \frac{2}{7} \cr & \Rightarrow 7x = 2x + 80 \cr & \Rightarrow 5x = 80 \cr & \Rightarrow x = 16. \cr & \therefore {\text{Required ratio}} \cr & = 16:56 \cr}$$

Explanation:-

$$\eqalign{ & {\text{Let the third number be }}x. \cr & {\text{Then,}} \cr & {\text{first number}} \cr & = 120\% {\text{ of }}x = \frac{{120}}{{100}}x = \frac{{6x}}{5}. \cr & {\text{And, second number}} \cr & = 150\% {\text{ of }}x = \frac{{150}}{{100}}x = \frac{{3x}}{2}. \cr & \therefore {\text{Required ratio}} \cr & = \frac{{6x}}{5}:\frac{{3x}}{2} \cr & = 12:15 \cr & = 4:5. \cr}$$

Explanation:-

(a + b) : (b + c) : (c + a)
6 : 7 : 8
Le t 6x : 7x : 8x
∴ a + b + b + c + a = 6x + 7x + 8x $$\eqalign{ & \Rightarrow 2a + 2b + 2c = 21x \cr & \Rightarrow 2\left( {a + b + c} \right) = 21x \cr & \Rightarrow a + b + c = \frac{{21}}{2}x \cr & \Rightarrow a + b + c = 14{\text{ given}} \cr & \Rightarrow \frac{{21x}}{2} = 14 \cr & \Rightarrow x = \frac{{28}}{{21}} = \frac{4}{3} \cr & \therefore a + b = 6 \times \frac{4}{3} = 8 \cr & \therefore a + b + c = 14 \cr & \Rightarrow 8 + c = 14 \cr & \Rightarrow c = 14 - 8 = 6 \cr}$$

Explanation:-

$$\eqalign{ & {\text{Let number be }}x\& y \cr & {\text{Given,}} \cr & \,\,x:y \cr & \,\,\,\,3:4 \cr & \,3a:4a \cr & {\text{Now , given that}} \cr & \Rightarrow {\text{8}}{\left( {3a} \right)^2} = {\left( {4a} \right)^2} + 224 \cr & \Rightarrow 72{a^2} = 16{a^2} + 224 \cr & \Rightarrow 56{a^2} = 224 \cr & \Rightarrow {a^2} = 4 \cr & \Rightarrow a = 2 \cr & {\text{Numbers are}} \cr & x = 3 \times 2 = 6 \cr & y = 4 \times 2 = 8 \cr}$$

Explanation:-

	Box	:	Paper bundle
Weight	3	:	22
Let	3x	:	22x

Total weight = 3x + 22x = 25x
25x = 36kg
$$\eqalign{ & x = \frac{{36}}{{25}} \times 1000 \cr & \,\,\,\,\, = 1440{\text{ garms}} \cr}$$ (∴ 1kg = 1000 grams)
∴ Weight of paper bundles = 22x
= 22 × 1440 garms = 31680 garms

Explanation:-

Required ratio
= 120% of Rs. 30 : 110% of Rs. 20
= 36 : 22
= 18 : 11

Explanation:-

	Amit	:	Veer
Income	1	:	3
Expensex	5	:	3
Saving	1000	:	1000

∴ Income ⇒ expenses + Savings $$\eqalign{ & \therefore \frac{{3x - 1000}}{{2x - 1000}} = \frac{5}{3} \cr & \Rightarrow 9x - 3000 = 10x - 5000 \cr & \Rightarrow x = 2000 \cr}$$ ∴ Annual income of Amit is = 3x
= 3 × 2000
= Rs. 6000