Alan, Bill and Carl shared $372. After Alan spend 2/ 5 of his share, Bill spent 1 /2 of his share and Carl spent 1/3 of his share, the boys found they had the same amount of money left.


(a) What is the ratio of Alan's share to Bill's share to Carl's share?

(b) How much did they spend altogether?

a + b + c = 372

what is left to each = 1-fraction spent
(3/5) a = (1/2)b = (2/3) c

(a) common denominator is 30
18a = 15 b = 20 c
so a/b = 15/18 = 5/6
a/c = 20/18 = 10/9
b/c = 20/15 = 4/3

(b)
a = 10 c/9
b = 4c/3

a + b + c = 372
10c/9 + 4c/3 + c = 372

10 c/9 + 12 c/9 + 9 c/9 = 372
31 c = 9*372
c = 108

You can take it from there I think

By the way - grade 5 ??

A, B, C are the initial amounts.

then
3/5 A= 1/2 B = 2/3 C then
A=10/9 C
B=4/3 C
but A+B+C=372 so
10/9 C+ 4/3 C+ C=372
10C+12C+9C=9*372
31C=9*372

C= you do it. From this, figure A, and B.

Then, ratios are A/B, A/C, and B/C

how much did they spend? go back to the problem, and do that calculation.

check my logic

To solve this problem, we need to establish equations based on the given information and then solve them.

Let's start by assigning variables:
Let 'x' be Alan's share.
Let 'y' be Bill's share.
Let 'z' be Carl's share.

According to the information given, after spending money:
Alan has 2/5 of his share remaining, which is (3/5)x.
Bill has 1/2 of his share remaining, which is (1/2)y.
Carl has 1/3 of his share remaining, which is (2/3)z.

Now, we can form an equation based on the remaining amounts:
(3/5)x = (1/2)y = (2/3)z ...(1)

We also know that the sum of their remaining amounts is equal to the total money they had initially:
(3/5)x + (1/2)y + (2/3)z = 372 ...(2)

To solve this system of equations, we can use substitution or elimination method. Let's use substitution:

From equation (1), we can express y in terms of x:
(1/2)y = (3/5)x
y = (3/5)(2/1)x
y = (6/5)x

Now, we can substitute the value of y in equation (2):
(3/5)x + (6/5)x + (2/3)z = 372
(9/5)x + (2/3)z = 372

To simplify this equation further, we can multiply both sides by 15 to remove the fractions:
9x + (10/3)z = 558 ...(3)

Now, we need one more equation to solve for x and z. To find this equation, let's consider the total money spent:

Alan spent (2/5)x,
Bill spent (1/2)y = (1/2)(6/5)x = (3/5)x,
Carl spent (1/3)z.

The total money spent is the sum of their individual spendings:

(2/5)x + (3/5)x + (1/3)z = Total money spent

Simplifying this equation, we get:
(5/5)x + (3/5)x + (1/3)z = Total money spent
(8/5)x + (1/3)z = Total money spent ...(4)

Now, we have the two equations:
(9/5)x + (10/3)z = 558 ...(3)
(8/5)x + (1/3)z = Total money spent ...(4)

We have two equations with two variables, and we can solve them using different methods such as substitution or elimination to find the values of x and z.

(a) To find the ratio of Alan's share to Bill's share to Carl's share, substitute the obtained values back into equation (1):

x : y : z = x : (6/5)x : z
= 5 : 6 : z

(b) To find how much they spent altogether, substitute the obtained values back into equation (4) to get the value of "Total money spent."

By solving these equations, we can find the answers to both questions.