Divide 100 loaves of bread amoung five men so that the amounts are in arithmetic progression ( increase by the same amount each time) and the sum of the three largest is seven times the amount of the two smallest.
Solve the system
a+(a+d)+(a+2d)+(a+3d)+(a+4d)=100
7(2a+d)=3a+9d
5a+10d=100 or
a+2d=20
second equation
14a+7d=3a+9d
11a-2d=0
add equations to get
12a=20
a=20/12=5/3
then solve for d in first equation.
is 10/6 + 65/6 + 120/6 + 175/6 + 230/6 equal to 100
To solve this problem, we need to find an arithmetic progression that satisfies the given conditions.
Let's represent the first term of the arithmetic progression by 'a', and the common difference by 'd'.
Since we are dividing 100 loaves of bread among five men, the sum of the arithmetic progression must be 100. Therefore, we have the equation:
[5/2 * (2a + (n-1)d)] = 100
Next, we know that the sum of the three largest terms is seven times the sum of the two smallest terms:
(a + (a+d) + (a+2d)) = 7 * ((a) + (a+d))
Now, let's solve these two equations simultaneously to find the values of 'a' and 'd':
[5/2 * (2a + (n-1)d)] = 100
(a + (a+d) + (a+2d)) = 7 * ((a) + (a+d))
Simplifying the first equation:
5/2 * (2a + 4d) = 100
5a + 10d = 200
a + 2d = 40
Substituting this value for (a + 2d) in the second equation:
(a + (a+d) + (a+2d)) = 7 * ((a) + (a+d))
3a + 3d = 7a + 7d
4a = 4d
a = d
Since 'd' is the common difference in the arithmetic progression, we can consider it as the smallest term. We are dividing 100 loaves of bread among five men, so each man's share is:
a + (a+d) + (a+2d) + (a+3d) + (a+4d) = 100
5a + 10d = 100
5a + 10a = 100
15a = 100
a = 100/15
Hence, we have found that 'a' is equal to (100/15) and 'd' is also equal to (100/15).
Now, to find the division of bread among the five men, we substitute the values of 'a' and 'd' into the arithmetic progression:
First term (a) = 100/15
Common difference (d) = 100/15
Man 1: a = 100/15
Man 2: a + d = 100/15 + 100/15
Man 3: a + 2d = 100/15 + 200/15
Man 4: a + 3d = 100/15 + 300/15
Man 5: a + 4d = 100/15 + 400/15
To simplify:
Man 1: 100/15
Man 2: 200/15
Man 3: 300/15
Man 4: 400/15
Man 5: 500/15
In decimal form, the division of bread among the five men is approximately:
Man 1: 6.67 loaves
Man 2: 13.33 loaves
Man 3: 20 loaves
Man 4: 26.67 loaves
Man 5: 33.33 loaves
Please note that due to the nature of the problem, there may be slight variations in the exact numbers.