Ratio and Proportion - 03

Explanation:-

Let x = 5k and y = 7k
Then,
= x + y = 36 ⇒ 5k + 7k = 36
⇒ 12k = 36
⇒ k = 3
∴ x = 5k = 5 × 3 = 15

Explanation:-

$$\eqalign{ & {\text{Let C gets }}x{\text{ rupees}} \cr & \therefore {\text{B gets }}x + 8{\text{ rupees}} \cr & \therefore {\text{A gets }}x + 8 + 7{\text{ rupees}} \cr & {\text{Total A}} + {\text{B}} + {\text{C}} \cr & \Rightarrow x + 15 + x + 8 + x = 3x + 23 \cr & \Rightarrow 3x + 23 = {\text{Rs}}{\text{. }}53({\text{given)}} \cr & \Rightarrow 3x = 30 \cr & \Rightarrow x = 10 \cr & \Rightarrow x = {\text{Rs}}{\text{.10}} \cr & \therefore {\text{Ratio of their shares is }} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{A}}\,\,\,\,\,\,\,\,\,:\,\,\,\,\,\,{\text{B}}\,\,\,\,\,\,\,\,:\,\,\,{\text{C}} \cr & {\text{ = }}\left( {x + 15} \right):\left( {x + 8} \right):\left( x \right) \cr & = \,\,\,\,\,\,\,\,\,25\,\,\,\,\,\,\,\,\,:\,\,\,\,\,\,\,18\,\,\,\,\,:\,\,10 \cr}$$

Explanation:-

$$\eqalign{ & \Rightarrow 0.5{\text{A}} = 0.6{\text{B}} = 1.75{\text{C}} \cr & \Rightarrow \frac{5}{{10}} \times {\text{A}} = \frac{6}{{10}} \times {\text{B}} = \frac{{75}}{{100}} \times {\text{C}} \cr & \Rightarrow \frac{1}{2}{\text{A}} = \frac{3}{5}{\text{B}} = \frac{3}{4}{\text{C}} \cr & \Rightarrow 10{\text{A}} = 12{\text{B }} = 15{\text{C}} \cr & \Rightarrow {\text{A}}\,\,\,\,\,\,\,:\,\,\,\,\,\,{\text{B}}\,\,\,\,\,\,\,\,\,:\,\,\,\,\,\,{\text{C}} \cr & 12 \times 15:10 \times 15:10 \times 12 \cr & \Rightarrow 180\,\,\,:\,\,\,\,\,150\,\,\,\,\,\,:\,\,\,\,120 \cr & \Rightarrow 6x\,\,\,\,\,\,:\,\,\,\,\,\,5x\,\,\,\,\,\,\,:\,\,\,\,\,4x \cr & {\text{Total}} = 6x + 5x + 4x = 15x \cr & 15x = 1740 \cr & \Rightarrow x = \frac{{1740}}{{15}} = {\text{Rs}}{\text{. }}116 \cr & \therefore {\text{Share of C is }}4x \cr & = 4 \times 116 \cr & = {\text{Rs}}{\text{. 464}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Remainder}} \cr & = {\text{Rs}}{\text{.}}\left[ {671 + \left( {3 + 7 + 9} \right)} \right] \cr & = {\text{Rs}}{\text{. }}690. \cr & {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left[ {\left( {690 \times \frac{1}{6}} \right) - 3} \right] \cr & = {\text{Rs}}{\text{. }}112. \cr & {\text{B's share}} \cr & = {\text{Rs}}{\text{.}}\left[ {\left( {690 \times \frac{2}{6}} \right) - 7} \right] \cr & = {\text{Rs}}{\text{. }}223. \cr & {\text{C's share}} \cr & = {\text{Rs}}{\text{.}}\left[ {\left( {690 \times \frac{3}{6}} \right) - 9} \right] \cr & = {\text{Rs}}{\text{. }}336. \cr}$$

Explanation:-

$$\eqalign{ & {\text{A}}:{\text{B}} = {\text{3}}:{\text{4}}, \cr & {\text{B}}:{\text{C}} = {\text{5}}:{\text{6}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {5 \times \frac{4}{5}} \right):\left( {6 \times \frac{4}{5}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 4:\frac{{24}}{5}, \cr & {\text{C}}:{\text{D}} = {\text{7}}:{\text{5}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {7 \times \frac{{24}}{{35}}} \right):\left( {5 \times \frac{{24}}{{35}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{24}}{5}:\frac{{24}}{7}. \cr & \therefore {\text{A}}:{\text{B}}:{\text{C}}:{\text{D}} \cr & = 3:4:\frac{{24}}{5}:\frac{{24}}{7} \cr & = 105:140:168:120. \cr}$$ Clearly, C will get the maximum.

Explanation:-

	Rs. 1	:	50 P	:	20 P
Values	13x	:	11x	:	7x
Number of coins	13x×1	:	11x×2	:	7x×5
	13x	:	22x	:	35x

Given total coins
⇒ 13x + 22x + 35x = 420
⇒ 70x = 420
⇒ x = 6
∴ NUmber of 50 paise coins are
= 22x = 22 × 6 = 132

Explanation:-

	Rice	:	Wheat
Weight	4	:	3
Price	5	:	6
Total Price	20	+	18	=	38

∴ 20 + 18 = 38 units
1 unit = Rs. 10
∴ Price of total rice
⇒ 20 × 10 = Rs. 200

Explanation:-

$$\eqalign{ & {\text{A}}:{\text{B}} = 5:12 \cr & {\text{B}}:{\text{C}} = 4:5.50 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 12:16.5 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 12:\frac{{33}}{2}. \cr & {\text{A}}:{\text{B}}:{\text{C}} = 5:12:\frac{{33}}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 10:24:33. \cr & {\text{Sum of ratio terms}} \cr & = 10 + 24 + 33 \cr & = 67. \cr & {\text{C's share}} = {\text{Rs}}{\text{.}}\left( {2010 \times \frac{{33}}{{67}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}990. \cr & B'{\text{s share}} = {\text{Rs}}{\text{.}}\left( {2010 \times \frac{{24}}{{67}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}.720. \cr & {\text{Required difference}} \cr & = {\text{Rs}}{\text{.}}\left( {990 - 720} \right) \cr & = {\text{Rs}}{\text{. }}270. \cr}$$

Explanation:-

$$\eqalign{ & {\text{Second}}:{\text{Third}} \cr & = {\text{9}}:{\text{16,}} \cr & {\text{Third}}:{\text{first}} \cr & = {\text{4}}:{\text{1}} \cr & {\text{ = 16}}:{\text{4}}{\text{.}} \cr & \therefore {\text{Second}}:{\text{Third}}:{\text{first}} \cr & = {\text{9}}:{\text{16}}:{\text{4}}{\text{.}} \cr & {\text{Second number}} \cr & = \left( {{\text{116}} \times \frac{{\text{9}}}{{{\text{29}}}}} \right) \cr & = {\text{36}} \cr}$$

Explanation:-

	Zinc	:	Tin
A	5x	:	2x	=	7x
B	3y	:	4y	=	7y

⇒ A ⇒ 7x = 7 kg
x = 1 kg
∴ Zinc in alloy A ⇒ 5kg
Tin in alloy A ⇒ 2 kg
⇒ B ⇒ 7y = 21 kg
y = 3 kg
Zinc in alloy B ⇒ 3 × 3 = 9 kg
Tin in alloy B ⇒ 3 × 4 = 12 kg
∴ After mix - up the ratio of Zinc and Tin in new alloy $$\eqalign{ & {\text{Zinc}}:Tin \cr & \,\,\,{\text{A}}\,5\,\,:\,\,\,\,\,2 \cr & \,\,\,{\text{B}}\,9\,\,\,:\,\,\,12 \cr & \overline {{\text{A}} + {\text{B}}\,\,\,\,{\text{14}}\,\,\,\,\,{\text{14}}} \cr & \,\,\,\,\,\,\boxed{1\,\,\,\,\,:\,\,\,\,\,1} \cr}$$