Partnership - 01

Explanation:-

$$\eqalign{ & {\text{3A}} = {\text{4B}} \cr & {\text{B}} = {\text{2C}} \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{4}{3} \cr & \frac{{\text{B}}}{{\text{C}}} = \frac{2}{1} \cr & {\text{A}}\,\,\,\,\,\,\,\,{\text{B}}\,\,\,\,\,\,\,\,{\text{C}} \cr & {\text{4}}\,\,\,\,\,\,\,\,\,\,{\text{3}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,{\text{2}}\,\,\,\,\,\,\,\,\,\,{\text{1}} \cr & \boxed{{\text{8}}\,\,:\,\,6\,\,\,:\,\,\,3\,} \cr}$$

Explanation:-

∵ Loss will be divided according to their investment ratio =

A	:	B
3000	:	2400
5	:	4

$$\eqalign{ & {\text{Loss of B}} = \frac{4}{{\left( {5 + 4} \right)}} \times 720 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 320}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Given share of }}{{\text{2}}^{{\text{nd}}}} \cr & = \frac{5}{{13}}{\text{of}}\left( {{{\text{1}}^{{\text{st}}}} + {{\text{3}}^{{\text{rd}}}}} \right) \cr & {\text{or, }}\frac{{{{\text{2}}^{{\text{nd}}}}}}{{{{\text{1}}^{{\text{st}}}}{\text{ + }}{{\text{3}}^{{\text{rd}}}}}} = \frac{5}{{13}} \cr & \therefore {1^{{\text{st}}}} + {{\text{2}}^{{\text{nd}}}} + {{\text{3}}^{{\text{rd}}}} = 13 + 5 = 18 \cr & \because 18{\text{units}} = 1620 \cr & \therefore {\text{1 unit}} = \frac{{1620}}{{18}} \cr & \therefore 5{\text{ units}} = \frac{{1620}}{{18}} \times {\text{5}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{\text{50}} \cr & {\text{Hence share of }}{{\text{2}}^{{\text{nd}}}} = {\text{Rs}}{\text{.}}\,{\text{450}} \cr}$$

Explanation:-

Note : In such type of question we can assume ratio as per our need to avoid fraction

Capital →	A 7 × 3	:	B 9 × 3
New Ratio, →	A 21x	:	B 27x

Total capital invested by A in 9 months
$$\eqalign{ & = 21x \times 3 + 7x \times 6 \cr & = 105x \cr}$$
Total capital of B invested in 9 months
$$\eqalign{ & = 27x \times 4 + 18x \times 5 \cr & = 198x \cr}$$
$$\eqalign{ & {\text{According to the question,}} \cr & \left( {105x + 198x} \right) = {\text{Rs}}{\text{. 10201}} \cr & 303x = {\text{Rs}}{\text{. 10201}} \cr & x = \frac{{10201}}{{303}} \cr & {\text{Hence,}} \cr & {\text{Share of A}} = 105 \times \frac{{10201}}{{303}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,3535 \cr & {\text{Share of B}} = 198 \times \frac{{10201}}{{303}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{6666}} \cr}$$

Explanation:-

Total profit = Rs. 4000
According to the question,
$$\eqalign{ & {\text{20% of B's capital}} = {\text{Rs}}{\text{.}}\,{\text{4000}} \cr & {\text{1% of B's capital}} = {\text{Rs}}{\text{.}}\,\frac{{{\text{4000}}}}{{20}} \cr & {\text{B's total capital}} \cr & = {\text{Rs}}{\text{. }}\frac{{{\text{4000}}}}{{20}} \times 100 \cr & = {\text{Rs}}{\text{. }}20000 \cr}$$
Let total capital required for business = 100 units.

	A	:	B	:	C
Capital	30	:	40	:	30
	× 500	:	× 500	:	× 500
	15000	:	20000	:	$$\boxed{15000}$$

Hence, Required capital for C = Rs. 15000

Explanation:-

Table correction

Total capital invested by A in 1 year
$$\eqalign{ & = 12 \times 4000 \cr & = {\text{Rs}}{\text{. 48000}} \cr}$$
Total capital invested by B in 1 year
$$\eqalign{ & = 4 \times 6000 + 8 \times 8000 \cr & = 24000 + 64000 \cr & = {\text{Rs}}{\text{.}}\,{\text{88000}} \cr}$$
Total capital invested by C in 1 year
$$\eqalign{ & = 9 \times 8000 + 3 \times 6000 \cr & = 72000 + 18000 \cr & = {\text{Rs}}{\text{. 90000}} \cr}$$

	A	:	B	:	C
Capital	48000	:	88000	:	90000
	24	:	44	:	45

According to the question,
$$\left( {24 + 44 + 45} \right){\text{units}}$$     = $${\text{Rs}}{\text{.}}\,{\text{16950}}$$
$$\eqalign{ & {\text{113 units}} = {\text{Rs}}{\text{. 16950}} \cr & 1{\text{ unit}} = \frac{{16950}}{{113}}{\text{ = Rs}}{\text{. 150}} \cr & {\text{Hence,}} \cr & {\text{Profit of A}} = 150 \times 24 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{3600}} \cr & {\text{Profit of B}} = 150 \times 44 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{6600}} \cr & {\text{Profit of C}} = 150 \times 45 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{6750}} \cr}$$

Explanation:-

A invests Rs. 2400 for 12 months
B invests Rs. 3600 for 8 months
And Rs. 3000 for 4 months
C invests Rs. X for 8 months
Ratio of profit of A, B and C
$$\Rightarrow {\text{Profit of A}}$$   : $${\text{Profit of B}}$$   : $${\text{Profit of C}}$$
$$\Rightarrow {\text{2400}} \times {\text{12}}$$   : $$\left( {3600 \times 8} \right)$$   + $$\left( {3000 \times 4} \right)$$   : $${\text{X}} \times 8$$
$$\eqalign{ & \Rightarrow 28800:40800:8{\text{X}} \cr & \Rightarrow 3600:5100:{\text{X}} \cr}$$
Given profit of C = Rs. 8000
And total profit of A, B and C = Rs. 22500
$$\eqalign{ & \therefore \frac{{{\text{X}} \times 22500}}{{3600 + 5100 + {\text{X}}}} = 8000 \cr & \Rightarrow \frac{{{\text{X}} \times 22500}}{{8700 + {\text{X}}}} = 8000 \cr}$$
$$\Rightarrow 22500{\text{X}}$$   = $$69600000$$   + $$8000{\text{X}}$$
$$\eqalign{ & \Rightarrow 14500{\text{X}} = 69600000 \cr & \Rightarrow {\text{X}} = {\text{Rs}}.\,4800 \cr}$$

Explanation:-

	A	:	B	:	C
Capital →	24000	:	32000	:	18000
	24	:	32	:	18
	12	:	16	:	9

$$\eqalign{ & {\text{Let the total profit}} = 100x \cr & {\text{Extra share of A}} \cr & = 100x \times \frac{{10}}{{100}} \cr & = 15x \cr & {\text{Extra share of B}} \cr & = 100x \times \frac{{12}}{{100}} \cr & = 12x \cr & {\text{Remaining profit}} \cr & = \left[ {100x - \left( {15x + 12x} \right)} \right] \cr & = 73x \cr}$$
According to the question,
Note: Remaining profit will be distributed in the ratio of their capitals.
∴ Share of C
$$\eqalign{ & \Leftrightarrow \frac{{73x}}{{\left( {12 + 16 + 9} \right)}} \times 9 = {\text{Rs}}{\text{. }}65700 \cr & \Leftrightarrow \frac{{657x}}{{37}} = {\text{Rs}}{\text{. }}65700 \cr & \Leftrightarrow x = {\text{Rs}}{\text{. }}\frac{{65700 \times 37}}{{657}} \cr & \Leftrightarrow x = {\text{Rs}}{\text{. 3}}700 \cr & {\text{Hence, required profit}} \cr & = 100x \cr & = 100 \times 3700 \cr & = {\text{Rs}}{\text{. 3}}70000 \cr}$$

Explanation:-

Total capital invested by A in 1 year
$$\eqalign{ & = 36000 \times 12 \cr & = {\text{Rs}}{\text{. 432000}} \cr}$$
Total capital invested by B in 1 year
$$= 45000 \times 4$$   + $$\left( {45000 - 20000} \right) \times 5$$     + $$\left( {55000 + 25000} \right) \times 3$$
$$= 180000 + 125000 + 240000$$
$$= {\text{Rs}}{\text{.}}\,{\text{545000}}$$

	A	:	B
Ratio of Capital →	432000	:	545000
Ratio of Profit →	432	:	545

$$\eqalign{ & {\text{According to the question,}} \cr & \left( {432 + 545} \right){\text{units}} = {\text{Rs}}{\text{. 117240}} \cr & {\text{977 units}} = {\text{Rs}}{\text{. 117240}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. }}\frac{{117240}}{{977}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 120}} \cr & {\text{Difference in profit}} \cr & = \left( {545 - 432} \right) \times 120 \cr & = {\text{ 13560}} \cr}$$
It means B will get Rs. 13560 more than A

Explanation:-

Ratio of capital invested by
$${\text{A, B and C}} = 15:10:6$$
Total capital invested by A in 1 year
$$\eqalign{ & = 15x \times 4 + 30x \times 8 \cr & = 300x \cr}$$
Total capital invested by B in 1 year
$$\eqalign{ & = 10x \times 6 + 5x \times 6 \cr & = 90x \cr}$$
Total capital invested by C in 1 year
$$\eqalign{ & = 6x \times 12 \cr & = 72x \cr}$$
Ratio of profits:

A	:	B	:	C
300x	:	90x	:	72x
50x	:	15x	:	12x

According to the question,
$$\Leftrightarrow \left( {50x + 15x + 12x} \right)$$     = $${\text{Rs}}{\text{. 34650}}$$
$$\eqalign{ & \Leftrightarrow 77x = {\text{Rs}}.{\text{ }}34650 \cr & \Leftrightarrow x = {\text{Rs}}{\text{. }}\frac{{34650}}{{77}} \cr & \Leftrightarrow x = {\text{Rs}}{\text{. }}450 \cr & {\text{Profit of A}} = {\text{Rs}}{\text{. }}450 \times 50 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 22500}} \cr & {\text{Profit of B}} = {\text{Rs}}{\text{. }}450 \times 15 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 6750}} \cr & {\text{Profit of C}} = {\text{Rs}}{\text{. }}450 \times 12 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 5400}} \cr}$$