Question

A social agency is charged with providing services to three types of clients: A, B, and C. A total of 600 clients are to be served, with $360,000 available forcounselingand $220,000 available for emergency food and shelter. Type A clients require an average of $400 for counseling and $600 for emergencies. Type Bclients require an average of $1000 for counseling and $400 for emergencies. Type C clients require an average of $600 for counseling and $200 for emergencies.How many of each type of client can be served?

Answer 1

Given the question, the following are the solution steps to answer the question.

STEP 1: Write the given information

`\begin{gathered} Client\text{ type}=A,B,C \\ Total\text{ clients}=600 \\ Amount\text{ available for counselling}=360000 \\ Amount\text{ available for emergency}=22000 \end{gathered}`

STEP 2: Represent the type of clients

Let a be the type of A client

Let b be the type of B client

Let c be the type of C client

STEP 3: Get the needed equations

Based on the information given, we can see that:

For the total clients, we have:

`a+b+c=600----equation\text{ 1}`

For the counselling:

`400a+1000b+600c=360000-----equation\text{ 2}`

For emergency foods:

`600a+400b+200c=220000-----equation\text{ 3}`

STEP 4: Solve the equations simultaneously

`\begin{gathered} From\text{ equation 1;} \\ a=600-b-c \end{gathered}`

Substitute into equation 2 and 3:

`\begin{gathered} 400(600-a-b)+1000b+600c=360000 \\ 600(600-b-c)+400b+200c=220000 \end{gathered}`

By Simplification we have:

`\begin{gathered} 600b+200c+240000=360000-----equation\text{ 4} \\ -200b-400c+360000=220000----equation\text{ 5} \end{gathered}`

Make y the subject of equation 4

`b=(360000-240000-200c)/(600)=(120000-200c)/(600)=(-c+600)/(3)----equation\text{ 6}`

Substitute the value into equation 5

`\begin{gathered} [-200((-c+600)/(3))-400c+360000]=220000 \\ By\text{ simplification,} \\ [(200c-120000)/(3)-400c]=220000-360000 \\ Multiply\text{ through by 3} \\ 200c-120000-1200c=-4200000 \\ -1000c=-420000+120000=-300000 \\ c=(-300000)/(-1000)=300 \end{gathered}`

c = 300

STEP 5: Solve for b

`\begin{gathered} Substitute\text{ 300 for b in equation 6} \\ b=(-300+600)/(3)=(300)/(3)=100 \end{gathered}`

b = 100

STEP 6: Solve for a

Hence,

200 type A clients can be served

100 type B clients can be served

300 type C clients can be served