Question

A,b,c,d each have some money. the amt A has is 1/3 total of b,c and d. the amt b has is 1/4 total of a,c,d. the amt c has is 1/5 total of a,b,d. if d has $1725, how much do they have altogether?

Answer 1

$4500

Explanation:

The amount a has is 1/3 total of b, c, and d

`a=1/3(b+c+d)` eq. 1

The amount b has is 1/4 total of a, c, and d

`b=1/4(a+c+d)` eq. 2

The amount c has is 1/5 total of a, b, and d

`c=1/5(a+b+d)` eq. 3

`d=1725`

substitute the value of
`d=1725` in above equations

`3a=b+c+1725`

`4b=a+c+1725`

`5c=a+b+1725`

Rearrange the equations to form matrices

`3a-b-c=1725`

`-a+4b-c=1725`

`-a-b+5c=1725`

`X=\left[\begin{array}{c}a&b&c\\\end{array}\right]`

`A=\left[\begin{array}{ccc}3&-1&-1\\-1&4&-1\\-1&-1&5\end{array}\right]`

`B=\left[\begin{array}{c}1725&1725&1725\\\end{array}\right]`

As we know

`AX=B`

To find the values of matrix X, we take inverse of matrix A and multiply it with matrix B

`X=A^(-1) B`

`\left[\begin{array}{c}a&b&c\\\end{array}\right]=\left[\begin{array}{ccc}3&-1&-1\\-1&4&-1\\-1&-1&5\end{array}\right]^(-1) \left[\begin{array}{c}1725&1725&1725\\\end{array}\right]`

Solving using calculator yield the following results

`X=\left[\begin{array}{c}1125&900&750\\\end{array}\right]`

so,

`a=1125\\b=900\\c=750\\d=1725\\`

Finally, altogether they have

`1125+900+750+1725=4500`

Verification:

Lets verify if have got the right answer!

Substitute the amount of b, c, and d in eq. 1

`a=1/3(b+c+d)`

`a=1/3(900+750+1725)`

`a=1/3(3375)`

`a=1125` (proved)

Substitute the amount of a, c, and d in eq. 2

`b=1/4(a+c+d)`

`b=1/4(1125+750+1725)`

`b=1/4(3600)`

`b=900` (proved)

Substitute the amount of a, b, and d in eq. 3

`c=1/5(a+b+d)`

`c=1/5(1125+900+1725)`

`c=1/5(3750)`

`c=750` (proved)