At the beginning of January 2002, the price of ground beef was $1.70 per pound and the price of tuna fish was $2.20 per pound. For the following 15 months, the price of ground beef increased at the rate of $0.03 per month and the price of tuna fish decreased at $0.02 per month. In approximately how many months after the beginning of January 2002 was the price of ground beef and tuna fish the same, and what was the price? Choose 1 answer: (A) 7.5 months and $1.90 (B) 7.5 months and $2.00 (C) 10 months and $1.90 (D) 10 months and $2.00

Question

At the beginning of January 2002, the price of ground beef was  $1.70 per pound and the price of tuna fish was  $2.20 per pound. For the following 15 months, the price of ground beef increased at the rate of  $0.03 per month and the price of tuna fish decreased at  $0.02 per month. In approximately how many months after the beginning of January 2002 was the price of ground beef and tuna fish the same, and what was the price? Choose 1 answer: (A) 7.5 months and  $1.90 (B) 7.5 months and  $2.00 (C) 10 months and  $1.90 (D) 10 months and  $2.00

Bytelearn · Accepted Answer

Initial Prices and Rates: Let's denote the initial price of ground beef as $1.70 and the initial price of tuna fish as $2.20. The rate of increase for ground beef is $0.03 per month, and the rate of decrease for tuna fish is $0.02 per month. We need to find the number of months after which the prices of ground beef and tuna fish will be the same. Calculate Price Formulas: To calculate the price of ground beef after $n$ months, we use the formula: Price of ground beef = Initial price of ground beef + (Rate of increase per month * $n$). This gives us: $1.70 + 0.03n$. Set Equations Equal: Similarly, to calculate the price of tuna fish after $n$ months, we use the formula: Price of tuna fish = Initial price of tuna fish - (Rate of decrease per month * $n$). This gives us: $2.20 - 0.02n$. Solve for Number of Months: To find when the prices are the same, we set the two expressions equal to each other: $1.70 + 0.03n = 2.20 - 0.02n$. Calculate Price at Equilibrium: Combining like terms, we get $0.03n + 0.02n = 2.20 - 1.70$, which simplifies to $0.05n = 0.50$. Dividing both sides by 0.05 to solve for $n$, we find $n = 0.50 / 0.05 = 10$. This means that it takes 10 months for the prices of ground beef and tuna fish to be the same. To find the price at which they are the same, we substitute $n = 10$ into one of our original equations. Using the ground beef equation: $1.70 + 0.03*10 = 1.70 + 0.30 = $2.00$.

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