UKPSC

Let the cost price of both the TV and the mobile phone be $xx$x.

Selling Price of the TV

The TV is sold at a profit of 12%, so its selling price can be expressed as:

$Selling Price (SP) of TV=Cost Price (CP)+12\mathrm{\%} of CP=x+0.12x=1.12x\backslash text\{Selling\; Price\; (SP)\; of\; TV\}\; =\; \backslash text\{Cost\; Price\; (CP)\}\; +\; 12\backslash \%\; \backslash text\{\; of\; CP\}\; =\; x\; +\; 0.12x\; =\; 1.12x$Selling Price (SP) of TV=Cost Price (CP)+12% of CP=x+0.12x=1.12x

Given that the selling price of the TV is ₹20,475, we have:

$1.12x=20,4751.12x\; =\; 20,475$1.12x=20,475

Now, solving for $xx$x:

$x=\frac{20,475}{1.12}=18,275 (Cost Price of TV and Mobile)x\; =\; \backslash frac\{20,475\}\{1.12\}\; =\; 18,275\; \backslash text\{\; (Cost\; Price\; of\; TV\; and\; Mobile)\}$x=1.1220,475​=18,275 (Cost Price of TV and Mobile)

Selling Price of the Mobile Phone

The mobile phone is sold at a loss of ₹1,456, so its selling price is:

$Selling Price (SP) of Mobile=x-1,456\backslash text\{Selling\; Price\; (SP)\; of\; Mobile\}\; =\; x\; –\; 1,456$Selling Price (SP) of Mobile=x−1,456

Substituting the value of $xx$x:

$SP of Mobile=18,275-1,456=16,819\backslash text\{SP\; of\; Mobile\}\; =\; 18,275\; –\; 1,456\; =\; 16,819$SP of Mobile=18,275−1,456=16,819

Total Cost Price and Selling Price

Now, we calculate the total cost price and total selling price:

$Total CP=x+x=2x=2\times 18,275=36,550\backslash text\{Total\; CP\}\; =\; x\; +\; x\; =\; 2x\; =\; 2\; \backslash times\; 18,275\; =\; 36,550$Total CP=x+x=2x=2×18,275=36,550 $Total SP=SP of TV+SP of Mobile=20,475+16,819=37,294\backslash text\{Total\; SP\}\; =\; \backslash text\{SP\; of\; TV\}\; +\; \backslash text\{SP\; of\; Mobile\}\; =\; 20,475\; +\; 16,819\; =\; 37,294$Total SP=SP of TV+SP of Mobile=20,475+16,819=37,294

Total Profit or Loss

Calculating the total profit:

$Total Profit=Total SP-Total CP=37,294-36,550=744\backslash text\{Total\; Profit\}\; =\; \backslash text\{Total\; SP\}\; –\; \backslash text\{Total\; CP\}\; =\; 37,294\; –\; 36,550\; =\; 744$Total Profit=Total SP−Total CP=37,294−36,550=744

Profit Percentage

Finally, calculating the profit percentage:

$Profit Percentage=\left(\frac{744}{36,550}\right)\times 100\approx 2.03\mathrm{\%}\backslash text\{Profit\; Percentage\}\; =\; \backslash left(\; \backslash frac\{744\}\{36,550\}\; \backslash right)\; \backslash times\; 100\; \backslash approx\; 2.03\backslash \%$Profit Percentage=(36,550744​)×100≈2.03%

Thus, the total percentage of profit in this business is approximately 2.03%.

To find out how long it will take A, B, and C to complete the work together, we start by determining their individual work rates:

• A can complete the work in 24 days, so A’s work rate = $\frac{1}{24}\backslash frac\{1\}\{24\}$241​ of the work per day.
• B can complete the work in 36 days, so B’s work rate = $\frac{1}{36}\backslash frac\{1\}\{36\}$361​ of the work per day.
• C can complete the work in 48 days, so C’s work rate = $\frac{1}{48}\backslash frac\{1\}\{48\}$481​ of the work per day.

Now, we find their combined work rate when all three work together:

$Combined work rate=\frac{1}{24}+\frac{1}{36}+\frac{1}{48}\backslash text\{Combined\; work\; rate\}\; =\; \backslash frac\{1\}\{24\}\; +\; \backslash frac\{1\}\{36\}\; +\; \backslash frac\{1\}\{48\}$Combined work rate=241​+361​+481​

Finding a common denominator (LCM of 24, 36, and 48 is 144):

• A’s work in 144 days = $\frac{6}{144}\backslash frac\{6\}\{144\}$1446​
• B’s work in 144 days = $\frac{4}{144}\backslash frac\{4\}\{144\}$1444​
• C’s work in 144 days = $\frac{3}{144}\backslash frac\{3\}\{144\}$1443​

Thus, the combined work rate is:

$\frac{6+4+3}{144}=\frac{13}{144} of the work per day\backslash frac\{6\; +\; 4\; +\; 3\}\{144\}\; =\; \backslash frac\{13\}\{144\}\; \backslash text\{\; of\; the\; work\; per\; day\}$1446+4+3​=14413​ of the work per day

In the first 4 days, all three work together:

$Work done in 4 days=4\times \frac{13}{144}=\frac{52}{144}=\frac{13}{36}\backslash text\{Work\; done\; in\; 4\; days\}\; =\; 4\; \backslash times\; \backslash frac\{13\}\{144\}\; =\; \backslash frac\{52\}\{144\}\; =\; \backslash frac\{13\}\{36\}$Work done in 4 days=4×14413​=14452​=3613​

This leaves:

$Remaining work=1-\frac{13}{36}=\frac{23}{36}\backslash text\{Remaining\; work\}\; =\; 1\; –\; \backslash frac\{13\}\{36\}\; =\; \backslash frac\{23\}\{36\}$Remaining work=1−3613​=3623​

Now, A and B continue working. Let $xx$x be the total days taken to complete the work. A leaves 3 days before the work is done, so they work for $x-3x\; –\; 3$x−3 days. Their combined work rate is:

$\frac{1}{24}+\frac{1}{36}=\frac{3+2}{72}=\frac{5}{72}\backslash frac\{1\}\{24\}\; +\; \backslash frac\{1\}\{36\}\; =\; \backslash frac\{3\; +\; 2\}\{72\}\; =\; \backslash frac\{5\}\{72\}$241​+361​=723+2​=725​

The equation for the remaining work is:

$(x-3)\times \frac{5}{72}=\frac{23}{36}(x\; –\; 3)\; \backslash times\; \backslash frac\{5\}\{72\}\; =\; \backslash frac\{23\}\{36\}$(x−3)×725​=3623​

Multiplying both sides by 72 to clear the fraction:

$5(x-3)=465(x\; –\; 3)\; =\; 46$5(x−3)=46

Solving for $xx$x:

$5x-15=46\hspace{0.25em} ⟹\hspace{0.25em} 5x=61\hspace{0.25em} ⟹\hspace{0.25em} x=\frac{61}{5}=12.2 days5x\; –\; 15\; =\; 46\; \backslash implies\; 5x\; =\; 61\; \backslash implies\; x\; =\; \backslash frac\{61\}\{5\}\; =\; 12.2\; \backslash text\{\; days\}$5x−15=46⟹5x=61⟹x=561​=12.2 days

Thus, the total time taken to complete the work is:

$4+12.2=16.2 days4\; +\; 12.2\; =\; 16.2\; \backslash text\{\; days\}$4+12.2=16.2 days

So, the work will be completed in approximately 16.2 days.