Question

The table below shows part of the operating budget of a small dairy farm for the last year. The only expense not listed is maintenance. LAST YEARS OPERATING BUDGETExpense Fraction of BudgetFood 1/3Housing 1/3 Medical Care 1/4This year the managers of the farm will change the fraction for the budget for housing to ⅛ but will leave the fraction of the budget for good and medical care the same. Again, the remaining portion of the budget will be before maintenance expenses. What is the difference between the fraction of the budget for maintenance this year and last year?Show your workAnswer

Answer 1

`(5)/(24)`

Step-by-step explanation

Step 1

let the fractions

`\begin{gathered} Food\text{ }\frac{\text{1}}{3} \\ \text{ Housing }\frac{\text{1}}{3} \\ \text{Medical care }(1)/(4) \\ \text{ maintenance= M ( unknown)} \end{gathered}`

so

a)This year the managers of the farm will change the fraction for the budget for housing to 1/8

so

`\begin{gathered} Food\text{ }\frac{\text{1}}{3} \\ \text{ Housing }\frac{\text{1}}{8} \\ \text{Medical care }(1)/(4) \\ \text{ maintenance= M ( unknown)} \end{gathered}`

b) hence,the total budget is

`\begin{gathered} \text{past year= Operatingbudget= food+housidn +Medical care+maintenance} \\ \text{replace} \\ past\text{ year= }(1)/(3)+(1)/(3)+(1)/(4)+m \\ past\text{ year=}(11)/(12)+m \end{gathered}`

this year

`\begin{gathered} \text{past year= Operatingbudget= food+housidn +Medical care+maintenance} \\ \text{replace} \\ thisyear\text{= }(1)/(3)+(1)/(8)+(1)/(4)+m \\ thisyear\text{=}(17)/(24)+m \end{gathered}`

Step 2

find the fractions for maintenance on each year:

as we are using fraction of the total budget, we need to set the sum of the fractions equals 1, hence

a) past year

`\begin{gathered} \text{total = 1, so} \\ past\text{ year=}(11)/(12)+m=1 \\ m=1-(11)/(12) \\ m=\text{ }(1)/(12) \end{gathered}`

b) this year

`\begin{gathered} \text{total = 1, so} \\ this\text{year=}(17)/(24)+m=1 \\ m=1-(17)/(24) \\ m=\text{ }(7)/(24) \end{gathered}`

Step 3

now, find the difference

`\begin{gathered} \text{difference =}(7)/(24)-(1)/(12) \\ \text{difference =}(5)/(24) \end{gathered}`

therefore, the answer is 5/24

I hope this helps you