James bought a few hamsters. For each day after the first day of the week, the hamsters ate 20 grams of food more than the previous day. The hamsters grew fast, finishing 1260 grams of food in the first week. How much food did the hamster eat on the first day?
Let's assume the hamsters ate x grams of food on the first day.
On the second day, they ate x + 20 grams of food.
On the third day, they ate x + 20 + 20 = x + 40 grams of food.
And so on...
In total, for the first week, they ate x + (x + 20) + (x + 40) + (x + 60) + (x + 80) + (x + 100) + (x + 120) = 7x + 420 grams of food.
It is given that they ate 1260 grams of food in the first week.
Therefore, 7x + 420 = 1260.
Simplifying the equation, 7x = 1260 - 420 = 840.
Dividing both sides by 7, x = 840/7 = 120.
Therefore, the hamsters ate 120 grams of food on the first day.
To find out how much food the hamsters ate on the first day, we can use the concept of arithmetic progression.
Let's say the amount of food the hamsters ate on the first day is represented by 'a' grams. Since each day after the first day, they ate 20 grams more than the previous day, we can express the amount of food eaten on subsequent days as follows:
First day: a grams
Second day: (a + 20) grams
Third day: (a + 20 + 20) grams
And so on...
We know that the hamsters finished a total of 1260 grams of food in the first week. A week consists of seven days, so we can set up the equation:
a + (a + 20) + (a + 20 + 20) + ... + (a + 20(n-1)) = 1260
Here, 'n' represents the number of days in the first week. There are seven days in a week, so n = 7.
By substituting n = 7 and simplifying the equation, we get:
7a + 6(20) + 5(20) + 4(20) + 3(20) + 2(20) + 1(20) = 1260
7a + 120 + 100 + 80 + 60 + 40 + 20 = 1260
7a + 420 = 1260
Subtracting 420 from both sides of the equation gives:
7a = 1260 - 420
7a = 840
Finally, dividing both sides by 7, we find:
a = 840 / 7
a = 120
Therefore, the hamsters ate 120 grams of food on the first day.
the last day is six days from the first day
1st day = f
last day = f + 6(20) = f + 120
arithmetic series average = (1st + last) / 2 = (2 f + 120) / 2 = f + 60
for 7 days ... food = 7(f + 60) = 7 f + 420 = 1260
... 7 f = 840 ... f = ?