Question

B receives $\dfrac{1}{8}$ of the profits of a business as wages and the rest is divided between B and A in the proportion of their capitals of \[\$ 6000\] and \[\$ 8000\] respectively. If in a year B receives \[\$ 1000\] totally, what does A receive?
A.\[\$ 800\]
B.\[\$ 1000\]
C.\[\$ 1200\]
D.\[\$ 1500\]

Answer 1

Hint: In the above given question, we are given the capitals of two business partners B and A as \[\$ 6000\] and \[\$ 8000\] respectively. At the end of a year, they made a profit in which B receives an extra $\dfrac{1}{8}$ of the gained profit and the remaining part is shared between A and B, according to the proportion i.e. ratio of their capitals. At the end, B has receives a total of \[\$ 1000\] . We have to find the amount that A has received.

Complete answer:
The shared capital of B and A are \[\$ 6000\] and \[\$ 8000\] respectively.
Now, if we suppose the total gained profit at the end of a year to be \[8x\] ,
Then according to the given problem, B receives the $\dfrac{1}{8}$ part of the total profit i.e. \[8x\] .
That is equal to,
\[ \Rightarrow \dfrac{1}{8} \times 8x\]
i.e. \[ \Rightarrow x\]
Now the remaining part, that is \[8x - x = 7x\] , hence \[7x\] is divided between A and B in the ratio of their capitals.
Now the ratio of their is given by,
\[ \Rightarrow A:B = 8000:6000\]
That is,
\[ \Rightarrow A:B = 8:6\]
Hence,
\[ \Rightarrow A:B = 4:3\]
Therefore, A receives $\dfrac{4}{7}$ part of \[7x\] that is \[4x\] .
Whereas, B receives $\dfrac{3}{7}$ part of \[7x\] that is \[3x\] .
Now, total profit that B has received is given by,
\[ \Rightarrow x + 3x\]
That is,
\[ \Rightarrow 4x\]
Therefore, at the end of a year, A and B both has equally gained an amount of \[4x\] .
Now, it is given that at the end of a year B receives an amount of \[\$ 1000\] .
Hence, at the end of a year, A has also gained an equal amount of \[\$ 1000\] .

Therefore, A receives \[\$ 1000\] so the correct option is B.

Note:
Ratio is the relation between two numbers which shows how much bigger one quantity is than the other. A ratio is defined as the ordered pair of two numbers \[a\] and \[b\] , written as \[a:b\] where \[b\] is not equal to zero. Whereas a proportion is defined as an equation in which two ratios are set to be equal to each other as \[a:b = m:n\] .