At a farmers market, Maria bought a total of 14 pieces of fruit consisting only of apples and oranges. Apples cost $2 each and oranges cost $3 each. If Maria spent $34 in total, how many oranges did she buy?
A known fruit count and a known cost lead to a linear equation. Name the kind the question asks for, and write the other kind as the total minus that count. Multiply each count by its own price, then use a Desmos regression to find the unknown. The total number of fruits is not the number of either kind.
Hints
- Hint 1
A shared total lets you use one variable for both kinds of fruit. If counts oranges, the apples are . That keeps the two counts adding to .
- Hint 2
Each price belongs to its own fruit: multiply the number of oranges by $3 and the number of apples by $2. Those two costs must add to the amount Maria spent.
- Hint 3
A regression asks Desmos to find a value that makes an equation true. Write your unknown as and replace with . What does the value Desmos finds count?
Step-by-step
Approach 1: Write one cost equation and solve it
Step 1Write the apple count using the orange count
Let the variable stand for the number of oranges. All fruits are either apples or oranges, so the apples are the fruits left after the oranges: . Using for the apples alone would count extra fruit.
- Step 2
Turn the total cost into an equation
The apples cost dollars, and the oranges cost dollars. Maria spent $34 in all, so add the two costs: . Each term in this equation measures dollars, not pieces of fruit.
- Step 3
Find and interpret the orange count
Type in Desmos. The regression symbol tells Desmos to find ; under PARAMETERS, it shows . Since stands for oranges, Maria bought 6 oranges, not 6 apples. Choice C.
Approach 2: Compare with buying only apples
Step 1Find an all-apple starting cost
Use an all-apple baseline, a starting amount for comparison. If all fruits were apples, they would cost dollars. This isn't Maria's purchase; it gives you a cost to compare with $34.
- Step 2
Find the cost of one swap
Replacing one apple with one orange keeps the fruit count at . The orange costs more by dollar, so each swap adds $1 to the cost.
- Step 3
Find the extra amount spent
Compare Maria's $34 with the all-apple cost of $28: . The $6 difference came from swapping apples for oranges.
- Step 4
Count the swaps
Each swap adds $1, and the total extra cost is $6, so the number of swaps is . Each swap gives Maria one orange, so she bought 6 oranges. Choice C.