james and john invested ¥700 and ¥300 respectively, at the end the of one year, they divided their profits such that a third of the profit is divided equally for the efforts they have put into the business and the remaining amount of profit is divided in the ratio of the investments they made in the business. if james received ¥800 more than john did, what was the profit made by their business in that year?

To find the profit made by their business in that year, we need to follow a step-by-step approach.

Step 1: Determine the total investment made by James and John.
James invested ¥700, and John invested ¥300. So the total investment is ¥700 + ¥300 = ¥1000.

Step 2: Calculate the ratio of their investments.
James invested ¥700, and John invested ¥300. So the ratio of their investments is 700:300, which can be simplified as 7:3.

Step 3: Divide the profit equally for the efforts they put into the business.
One-third of the profit is divided equally, so let's assign this amount as "x".

Step 4: Divide the remaining profit in the ratio of their investments.
The remaining amount of profit after dividing one-third equally is two-thirds of the total profit. Let's assign this amount as "2x".

According to the given information, James received ¥800 more than John. So we can set up the equation: 7x - 3x = ¥800.

Simplifying the equation, we have:
4x = ¥800.

Step 5: Solve the equation to find the value of "x".
Dividing both sides of the equation by 4, we get:
x = ¥200.

Step 6: Calculate the total profit.
Since one-third of the profit is ¥200, the total profit is 3 times that amount:
Total profit = 3 * ¥200 = ¥600.

Therefore, the profit made by their business in that year is ¥600.