Tim had 40% more muffins than Rael. Tim gave 20% of his muffins to Rael. Rael in return gave 50% of his muffins to Tim. In the end, Rael had 448 less muffins than Tim. How many muffins did Tim have at first?
Let Rael have x muffins
Tim : 1.4x muffins
lost : Tim 1.4x - 1.4x * 0.2 + 0.5x (x + 1.4x * 0.2)
= 1.12x + 0.64x
= 1.76x
Rael : x + 1.4x * 0.2 - (x + 1.4x * 0.2) * 0.5
= x + 0.28x - 0.64x
= 0.64x
1.76x - 0.64x = 448
1.12x = 448
x = 400
1.4 * 400 = 560
Tim have 560 muffins at first
746
Let's assume that Rael had x muffins.
According to the given information, Tim had 40% more muffins than Rael, which means Tim had x + 40% of x muffins.
To find 40% of x, we can calculate 40/100 * x = 2/5 * x = 0.4x.
So, Tim had x + 0.4x = 1.4x muffins.
Tim gave 20% of his muffins to Rael, which means Tim gave away 20/100 * 1.4x = 0.2 * 1.4x = 0.28x muffins.
After giving away muffins, Tim had 1.4x - 0.28x = 1.12x muffins.
In return, Rael gave 50% of his muffins to Tim, which means Rael gave away 50/100 * x = 0.5x muffins.
So, Rael had x - 0.5x = 0.5x muffins left.
According to the question, Rael had 448 less muffins than Tim at the end.
Therefore, we can write the equation: 1.12x - 0.5x = 448
Simplifying the equation, we get: 0.62x = 448
Dividing both sides of the equation by 0.62, we get: x = 448 / 0.62 = 724
So, Rael had x = 724 muffins.
Tim had 1.4 * 724 = 1013.6 muffins at first.
Since the number of muffins cannot be fractional, Tim had 1013 muffins at first.
Therefore, Tim had 1013 muffins at first.
To solve this problem, we can break it down into steps:
Step 1: Initialize the number of muffins Tim had at first as T.
Step 2: Since Tim had 40% more muffins than Rael, we can express the number of muffins Rael had as R, where R = T - 40% of T.
Step 3: Tim gave 20% of his muffins to Rael, so Tim will have 80% of his original number of muffins left, which is 0.8T.
Step 4: Rael returned the favor by giving 50% of his muffins back to Tim, so Rael will have 50% of his original number of muffins left, which is 0.5R.
Step 5: In the end, Rael had 448 fewer muffins than Tim, so we can write the equation 0.5R = 0.8T - 448.
Now, let's solve the equation to find the value of T:
0.5(T - 40% of T) = 0.8T - 448
Simplify the equation:
0.5T - 0.2T = 0.8T - 448
Combine like terms:
0.3T = 0.8T - 448
Subtract 0.8T from both sides:
-0.5T = -448
Divide both sides by -0.5:
T = (-448) / (-0.5) = 896
Therefore, Tim had 896 muffins at first.