Ni Hua is an adventurous traveler. He skydives twice in every island he visits and skydives thrice in every peninsula he visits. $\newline$ In the last decade, Ni Hua went skydiving a total of $45$ times in the $19$ islands and peninsulas that he visited. $\newline$ How many islands and peninsulas did Ni Hua visit? $\newline$ Choose $1$ answer: $\newline$ (A) Ni Hua visited $12$ islands and $7$ peninsulas. $\newline$ (B) Ni Hua visited $7$ islands and $12$ peninsulas. $\newline$ (C) There is not enough information to determine the exact number of islands and peninsula Ni Hua visited. $\newline$ (D) The given information describes an impossible situation.

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One-step inequalities: word problems

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Q. Ni Hua is an adventurous traveler. He skydives twice in every island he visits and skydives thrice in every peninsula he visits.

\newline

In the last decade, Ni Hua went skydiving a total of

45

times in the

19

islands and peninsulas that he visited.

\newline

How many islands and peninsulas did Ni Hua visit?

\newline

Choose

1

answer:

\newline

(A) Ni Hua visited

12

islands and

7

peninsulas.

\newline

(B) Ni Hua visited

7

islands and

12

peninsulas.

\newline

\newline

(D) The given information describes an impossible situation.

Denote Islands and Peninsulas: Let's denote the number of islands Ni Hua visited as $I$ and the number of peninsulas as $P$ . We know that Ni Hua skydives twice on each island and thrice on each peninsula. The total number of skydives is $45$ , and the total number of places (islands and peninsulas) visited is $19$ . We can set up two equations based on this information.
Equation for Skydives: The first equation represents the total number of skydives: $2I + 3P = 45$ . This equation comes from the fact that Ni Hua skydives twice on each island ( $2I$ ) and thrice on each peninsula ( $3P$ ), totaling $45$ skydives.
Equation for Places Visited: The second equation represents the total number of places visited: $I + P = 19$ . This equation simply adds the number of islands ( $I$ ) and the number of peninsulas ( $P$ ) to get the total number of places visited, which is $19$ .
Substitute and Solve: To solve this system of equations, we can use substitution or elimination. Let's use substitution. We can solve the second equation for $I$ : $I = 19 - P$ . This equation isolates $I$ in terms of $P$ , making it easier to substitute into the first equation.
Simplify and Solve: Substitute $I = 19 - P$ into the first equation: $2(19 - P) + 3P = 45$ . This substitution will allow us to solve for $P$ .
Calculate Peninsulas: Simplify and solve for $P$ : $38 - 2P + 3P = 45$ . Combine like terms to get $38 + P = 45$ . Then, subtract $38$ from both sides to isolate $P$ : $P = 45 - 38$ .
Substitute for Islands: Calculate $P$ : $P = 7$ . This means Ni Hua visited $7$ peninsulas.
Calculate Islands: Now, substitute $P = 7$ back into the equation $I = 19 - P$ to find $I$ : $I = 19 - 7$ .
Final Answer: Calculate $I$ : $I = 12$ . This means Ni Hua visited $12$ islands.
Final Answer: Calculate $I$ : $I = 12$ . This means Ni Hua visited $12$ islands. We have found that Ni Hua visited $12$ islands and $7$ peninsulas. This matches option (A) in the given choices.