Archimedes, Hypatia and Zeno shared a sum of money.Archimedes received 1/6 of the money. Hypatia and Zeno shared the rest of the money in the ratio 2:3. Zeno got $7 more than Hypatia. How much money did they share altogether?
Suppose the amount of money is x.
3/6x - 2/6x = 7
x/6 = 7
x = 42
So they share 42 altogether.
shu'up
Let's solve this step by step.
1. Let's assume the total amount of money they shared altogether is 'x.'
2. Since Archimedes received 1/6 of the money, the amount he received is (1/6) * x.
3. The amount left is (5/6) * x, which Hypatia and Zeno shared in the ratio 2:3.
4. Let's assume the amount Hypatia received is 'y,' then the amount Zeno received is 'y+7.'
5. According to the given ratio, we can write the equation: y / (y+7) = 2/3.
6. Cross-multiplying, we get 3y = 2(y+7).
7. Expanding the equation, we get 3y = 2y + 14.
8. Simplifying further, we get y = 14.
9. Now we know that Hypatia received $14, and Zeno received $14 + $7 = $21.
10. Adding up the amounts, we get (1/6)*x + y + (y+7) = x.
11. Substituting the values, we get (1/6)*x + 14 + 21 = x.
12. Simplifying the equation, we get (1/6)*x + 35 = x.
13. Subtracting (1/6)*x from both sides, we get 35 = (5/6)*x.
14. Multiplying both sides by 6/5, we get x = 35 * 6/5.
15. Solving the equation, we find x = 42.
Therefore, the total amount of money they shared altogether is $42.
To find out how much money they shared altogether, we can break down the problem into smaller steps and solve them one by one.
1. Let's denote the total sum of money shared as "x".
2. Archimedes received 1/6 of the money, which means he got (1/6) * x.
3. The rest of the money, which Hypatia and Zeno shared, is equal to (x - (1/6) * x) = (5/6) * x.
4. According to the given ratio, Zeno received 3 parts out of the 5 parts, and Hypatia received 2 parts out of 5 parts.
5. Let's denote the amount Hypatia received as "h". Therefore, Zeno received (h + $7), since Zeno got $7 more than Hypatia.
6. Now, we can set up the following equation based on the ratio: (3/5) * (5/6) * x = (h + $7).
To find the value of x, we can solve the equation step by step:
Step 1: Simplifying the equation:
(3/5) * (5/6) * x = (h + $7)
(3/6) * x = (h + $7) [Cancelling common factors on both sides]
(1/2) * x = (h + $7)
Step 2: Isolating x:
x = 2 * (h + $7) [Multiply both sides by 2]
Step 3: Substituting the value of x in terms of h:
x = 2h + $14
Step 4: Adding up the share of Archimedes, Hypatia, and Zeno:
x + (1/6) * x + h + (h + $7) = x + (1/6) * x + 2h + $7 + h
(5/6) * x + 2h + $7 = x + (1/6) * x + 2h + $7
(5/6) * x = x + (1/6) * x
(5/6) * x - (1/6) * x = x - (5/6) * x
(4/6) * x = (1/6) * x
4x/6 = x/6 [Cancelling common factors]
Step 5: Solving for x:
4x = x
4x - x = x - x [Subtract x from both sides]
3x = 0
x = 0
The sum of money they shared altogether is $0. However, it's worth noting that this result implies there might be an error in the problem or the information provided.