Ibrahim and Richard had some money. After Ibrahim spent 2/5 of his money and Richard spent 4/9 of his money, they had the same amount of money left. If Richard had $300 more than Ibrahim, how much did Ibrahim spend?
Let's start by using variables to represent the amount of money Ibrahim and Richard had. Let I be Ibrahim's amount and R be Richard's amount.
After Ibrahim spent 2/5 of his money, he had 3/5 of his money left. Similarly, after Richard spent 4/9 of his money, he had 5/9 of his money left. We know that they had the same amount of money left, so we can set up an equation:
3/5 I = 5/9 R
We also know that Richard had $300 more than Ibrahim, so we can set up another equation:
R = I + 300
Now we can substitute the second equation into the first equation to solve for I:
3/5 I = 5/9 (I + 300)
Multiplying both sides by 45 (the least common multiple of 5 and 9) to get rid of the fractions, we get:
27I = 25(I + 300)
Expanding the brackets, we get:
27I = 25I + 7500
Subtracting 25I from both sides, we get:
2I = 7500
Dividing both sides by 2, we get:
I = 3750
So Ibrahim had $3750 to start with. We can use the second equation to find out how much Richard had:
R = I + 300 = 3750 + 300 = 4050
Now we can check if our solution is correct. After Ibrahim spent 2/5 of his money, he had 3/5 of $3750 = $2250 left. After Richard spent 4/9 of his money, he had 5/9 of $4050 = $2550 left. Indeed, they have the same amount of money left. And we can also see that Richard had $300 more than Ibrahim ($4050 - $3750 = $300). Therefore, Ibrahim spent the difference between his starting amount and what he had left:
3750 - 2250 = 1500
So Ibrahim spent $1500.
Let's assume Ibrahim had x as his initial amount of money.
After spending 2/5 of his money, Ibrahim had x - (2/5)x = 3/5x left.
Similarly, let's assume Richard had y as his initial amount of money.
After spending 4/9 of his money, Richard had y - (4/9)y = 5/9y left.
Given that Ibrahim and Richard had the same amount of money left, we can set up the equation:
3/5x = 5/9y
We also know that Richard had $300 more than Ibrahim, so we can set up another equation:
y = x + 300
We can now solve these two equations simultaneously to find the value of x (the amount Ibrahim had initially) and then calculate how much Ibrahim spent.
From the second equation, we can substitute the value of y into the first equation:
3/5x = 5/9(x + 300)
Simplifying this equation:
27x = 25(x + 300)
Expanding both sides of the equation:
27x = 25x + 7500
Subtracting 25x from both sides:
2x = 7500
Dividing both sides by 2:
x = 3750
Therefore, Ibrahim initially had $3750.
Now, to find out how much Ibrahim spent, we can subtract the amount he had left (3/5x) from the initial amount (x):
Ibrahim spent x - (3/5)x = (2/5)x
Substituting the value of x:
Ibrahim spent (2/5)*3750 = $1500
Therefore, Ibrahim spent $1500.