Q $1$ $\newline$ Medical residents at a hospital are choosing their individual specialties. Among them, $30 \%$ choose cardiology, $18 \%$ choose oncology, $32 \%$ choose endocrinology, and the remaining $x \%$ choose hematology. Once the doctors pick their first specialty, they are then each asked to choose a second specialty from the previous four options in case their original specialty is already filled. They may not pick their original specialty again. $20 \%$ of those who originally picked cardiology choose oncology as their second choice. If no other field chooses oncology as their second choice, and the hospital boasts $150$ medical residents, then what is the total number of residents who named oncology as either their first or second choice, in terms of $x$ ? $\newline$ (A) $x^{2}-124$ $\newline$ (B) $12 x-124$ $\newline$ C) $x^{2}+12 x-124$ $\newline$ (D) $x^{2}-12 x-124$

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Grade 7

Identify independent and dependent variables

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Q. Q

1

\newline

Medical residents at a hospital are choosing their individual specialties. Among them,

30 \%

choose cardiology,

18 \%

choose oncology,

32 \%

choose endocrinology, and the remaining

x \%

choose hematology. Once the doctors pick their first specialty, they are then each asked to choose a second specialty from the previous four options in case their original specialty is already filled. They may not pick their original specialty again.

20 \%

of those who originally picked cardiology choose oncology as their second choice. If no other field chooses oncology as their second choice, and the hospital boasts

150

medical residents, then what is the total number of residents who named oncology as either their first or second choice, in terms of

x

\newline

(A)

x^{2}-124

\newline

(B)

12 x-124

\newline

x^{2}+12 x-124

\newline

(D)

x^{2}-12 x-124

Calculate Cardiology Residents: Calculate the number of residents who chose cardiology as their first specialty. $\newline$ $30\%$ of $150$ residents chose cardiology. $\newline$ Number of residents choosing cardiology = $30\%$ of $150$ $\newline$ = $(30/100) \times 150$ $\newline$ = $45$ residents
Calculate Oncology Residents: Calculate the number of residents who chose oncology as their first specialty. $\newline$ $18\%$ of $150$ residents chose oncology. $\newline$ Number of residents choosing oncology = $18\%$ of $150$ $\newline$ = $(18/100) \times 150$ $\newline$ = $27$ residents
Calculate Cardiology to Oncology: Calculate the number of residents who chose cardiology and then oncology as their second choice. $\newline$ $20$ \% of those who chose cardiology pick oncology as their second choice. $\newline$ Number of residents choosing oncology as second choice = $20$ \% of $45$ $\newline$ = $(20/100) \times 45$ $\newline$ = $9$ residents
Total Oncology Residents: Calculate the total number of residents who chose oncology as either their first or second choice. $\newline$ Total number of residents choosing oncology $=$ First choice oncology $+$ Second choice oncology $\newline$ $= 27$ residents (first choice) $+ 9$ residents (second choice) $\newline$ $= 36$ residents
Express Oncology in Terms of $x$ : Express the total number of residents who chose oncology in terms of $x$ . Let $x\%$ be the percentage of residents who chose hematology as their first specialty. Since the percentages must add up to $100\%$ , we have: $30\%$ (cardiology) + $18\%$ (oncology) + $32\%$ (endocrinology) + $x\%$ (hematology) = $100\%$ $x\%$ = $100\%$ - ( $30\%$ + $18\%$ + $32\%$ ) $x\%$ = $100\%$ - $x$ $6$ $x\%$ = $x$ $8$
Calculate Hematology Residents: Calculate the number of residents who chose hematology as their first specialty. $\newline$ Number of residents choosing hematology = $x\%$ of $150$ $\newline$ = $(20/100) \times 150$ $\newline$ = $30$ residents
Total Oncology Residents: Since no other field chooses oncology as their second choice, the total number of residents who named oncology as either their first or second choice remains the same. $\newline$ Total number of residents choosing oncology = $36$ residents
Express Oncology in Terms of $x$ : Express the total number of residents who chose oncology as either their first or second choice in terms of $x$ . $\newline$ Since $x$ represents the percentage of residents who chose hematology, and we have found that $x\% = 20\%$ , we can substitute $x$ with $20$ in our total from Step $4$ . $\newline$ Total number of residents choosing oncology = $36$ residents $\newline$ This total is independent of $x$ , as no other specialty chose oncology as a second choice.