An army of ants was migrating. The ants moved 120 grams less than half of their food on their first trip. On their second trip, they managed to move 100 g more than half of the remaining amount of food. They moved 480 g of food on their third trip. 280 g of food was still unmoved. How much food did the army of ants have at first?
Just take it deliberately and slowly.
amount at start ---- x g
amount moved on first day: x/2 -120
amount left at end of first day = x - (x/2-120) = x/2 + 120
second day
amount moved = (1/2)(x/2 + 120) + 100
= x/4 + 160
amount left at end of 2nd day = (x/2 + 120) - (x/4 + 160) = x/4 - 40
amount left after 480 g was moved on 3rd day
= x/4 - 40 - 480 = x/4 - 520
but this equals 280
x/4 - 520 = 280
x/4 = 800
x = 3200 g
Well, it seems like these ants were quite the busy little workers! Let me calculate this for you...
Let's call the initial amount of food x. According to the information given, on their first trip, the ants moved 120 grams less than half of their food, which is (x/2) - 120 grams.
Now, let's move on to the second trip. We know that they managed to move 100 grams more than half of the remaining amount of food. Since (x/2) - 120 grams was left after the first trip, on the second trip they moved ((x/2) - 120) + 100 grams.
After the second trip, there was still 280 grams unmoved, so the equation becomes:
x - [(x/2) - 120] - [((x/2) - 120) + 100] - 480 = 280
Now, let me put my clown calculator to work and solve this equation for you...
After a few silly calculations... *drumroll*... the initial amount of food that the army of ants had is... 2000 grams! So, those ants really knew how to haul a load!
Hope this gives you some food for thought!
Let's break down the given information step-by-step:
1. On their first trip, the ants moved 120 grams less than half of their food.
Let's assume the total amount of food the ants had at first is represented by "F."
The ants moved (1/2)F - 120 grams on their first trip.
2. On their second trip, they managed to move 100 g more than half of the remaining amount of food.
After the first trip, there will be (1/2)F - 120 grams of food remaining.
Therefore, on their second trip, the ants moved [(1/2)F - 120] + 100 grams.
3. They moved 480 g of food on their third trip.
After the second trip, there will be [(1/2)F - 120] - [(1/2)F - 120 + 100] = 480 grams of food remaining.
4. 280 g of food was still unmoved.
After the third trip, there will be 480 grams - 280 grams = 200 grams of food remaining.
5. How much food did the army of ants have at first?
From step 4, we know that 200 grams of food were remaining after the third trip.
Therefore, we can set up an equation: [(1/2)F - 120] - [(1/2)F - 120 + 100] = 200.
Simplifying this equation will help us find the initial amount of food the army of ants had.
Now, I will solve the equation step-by-step:
To solve this problem, let's break it down step by step:
1. Let's assume the initial amount of food that the army of ants had was "x" grams.
2. On their first trip, the ants moved 120 grams less than half of their food. To find out how much they moved, we need to calculate half of the initial amount of food and subtract 120 grams.
Half of the food = x/2
Food moved on the first trip = (x/2) - 120
3. On their second trip, they moved 100 grams more than half of the remaining amount of food. After the first trip, the remaining amount of food is the initial amount of food minus the food moved on the first trip.
Remaining amount of food = x - [(x/2) - 120]
Now, we can calculate the food moved on the second trip:
Food moved on the second trip = (x/2 - 120) + 100 = x/2 - 20
4. On the third trip, the ants moved 480 grams of food. To find the remaining amount of food, we need to subtract the total amount of food moved on the first and second trips from the initial amount of food.
Remaining amount of food = x - [(x/2 - 120) + (x/2 - 20)]
5. We are given that 280 grams of food was still unmoved. So, we can set up the following equation:
Remaining amount of food = 280
Solve this equation to find the initial amount of food (x).
By following this approach, you can calculate the initial amount of food that the army of ants had.